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Research ArticleA CAS Approach to Handle the AnisotropicHookersquos Law for Cancellous Bone and Wood
Sandeep Kumar
Department of Mathematics Government PG Degree College New Tehri Tehri Garhwal Uttarakhand 249 001 India
Correspondence should be addressed to Sandeep Kumar drsandeepkumarmathhotmailcom
Received 29 August 2013 Accepted 1 October 2013 Published 23 March 2014
Academic Editors J Deng and D Sun
Copyright copy 2014 Sandeep Kumar This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The present research entirely relies on the Computer Algebric Systems (CAS) to develop techniques for the data analysis of thesets of elastic constant data measurements In particular this study deals with the development of some appropriate programmingcodes that favor the data analysis of known values of elastic constants for cancellous bone hardwoods and softwood species Moreprecisely a ldquoMathematicardquo code which has an ability to unfold a fourth-order elasticity tensor is discussed Also an effort towardsthe fabrication of an appropriate ldquoMAPLErdquo code has been exposed that can calculate not only the eigenvalues and eigenvectors forcancellous bone hardwoods and softwood species but also computes the nominal average of eigenvectors average eigenvectorsaverage eigenvalues and the average elasticity matrices for these materials Further using such a MAPLE code the histogramscorresponding to average elasticity matrices of 15 hardwood species have been plotted and the graphs for I II III IV V and VIeigenvalues of each hardwood species against their apparent densities are also drawn
1 Introduction
The study of material symmetry of 3-dimensional spaceis of great interest due to having crucial theoretical aswell as practical significance This is because a symmetricalspace includes crystals and all homogeneous fields withoutexceptions electric magnetic gravitational and so forth
The variation of material properties with respect todirection at a stagnant point in a material is called materialsymmetry for instance if the material properties are samein all directions at some fixed point they are called isotropicwhereas if the material properties show variation at the samepoint they are called anisotropic [1]Of course the familiaritywith material symmetries is the best way to categorize thematerials However according to [2] many materials areanisotropic and inhomogeneous due to the varying composi-tion of their constituents In such materials it becomes tick-lish to identify the symmetries ormore particularly the elasticsymmetries The variable composition method to identifymaterialrsquos elastic symmetry becomes complicated and henceto overcome this difficulty an approach is developed by[3] called ldquoaveraging anisotropic elastic constant datardquo Inthis approach the identification of elastic symmetries and
method of variable composition are analyzed separatelyWith the aid of this fabulous approach a sheer volume ofresearch towards material symmetry has been put forward byvarious researchers for example estimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constants[4] anisotropic Hookersquos law for cancellous bone and wood[2] the validation of generalized Hookersquos law for coronaryarteries [5] constitutive relationships of fabric density andelastic properties in cancellous bone architecture [6] analysisof elastic anisotropy of wood material for engineering appli-cations [7] a multidimensional anisotropic strength criterionbased on Kelvinrsquos modes [8] spectral decomposition of a 4th-order covariance tensor application to diffusion tensor MRI[9] a normal distribution of tensor valued random variablesapplications to diffusion tensor MRI [10] and so forth
Concerning computer assisted analysis of bonematerial afew of the articles like directmechanical assessment of elasticsymmetries and properties of trabecular bone architecture[11] relationships between morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructions [12] and intrinsic mechanicalproperties of trabecular calcaneus determined by finite
Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2014 Article ID 487314 28 pageshttpdxdoiorg1011552014487314
2 Chinese Journal of Engineering
Table 1 Voigtrsquos mappings for stress (left) and strain tensors (right) respectively
1205901112059022
12059033
12059023
1205901312059012
12059023
1205901212059013
120590111205902212059033120590231205901312059012
1205981112059822
120598331205982312059813
12059812
12059812
12059823
12059813
12059811
1205982212059833212059823212059813212059812
Table 2 Index conversion rule of Voigt
119894 119895 11 22 33 23 or 32 13 or 31 12 or 21119901 1 2 3 4 5 6
Table 3 Voigt to Kelvin and vice versa
120585 = 1 1 1 radic2 radic2 radic2
element models using 3D synchrotron microtomography[13] have explored the subject material up to some greatextent
Now as far as the present proposed research concernedwe turn our concentration towards cancellous bone andwood (soft and hard) According to [4] the type of materialsymmetry exposed by bone tissue is often ambiguous Soin order to attain accuracy we have to measure the elasticconstants of bone specimen which may be represented asorthotropic However the orthotropy of the bone specimenmay or may not be of higher degree in all the directionswith respect to a fixed point in the specimen Thereby for allpractical purposes the bone specimen can be represented astransversely isotropic or even isotropic
Now the question arises iswhat is the need of studying themechanical properties of cancellous bone and woodThe rig-orous answer is that the knowledge about mechanical prop-erties of cancellous bone is essential for the determinationof bone fracture risk in osteoporosis and other pathologicalconditions involving impaired bone strength [6] Also themechanical properties of cancellous bone are determined bythe properties of its bone tissue and its architecture [4 14ndash21] Further the wood is a cellulosic semicrystalline cellu-lar material and the mechanical properties (eg elasticitystrength and rheology etc) are generally higher along thebole of a tree than across the bole [7]
Mechanically clear wood obeys the law of elasticorthotropic material like the bone does In general the woodysubstance also exposes the high toughness and stiffnessproperties and these properties vary according to the typeof wood and the direction in which the woody substance isexamined as the woody substance shows a higher degree ofanisotropy
In addition to deal with anisotropic Hookersquos law oneshould be familiar with the algebra of the 4th-order tensorWe would like to emphasize that the 4th-order tensor algebrais not only involved in the study of material symmetrybut its crucial appearance has been set up in the field ofdiffusion tensor MRI [9 10] visualization and processing of
tensor fields [22] biomechanics [13] tissue mechanics [1]geophysics [23] and so forth
Though an adequate amount of research work has beendedicated towards the algebra of 4th-order tensors thenalso in the following section we shall present a brief digestregarding this issue as this issue hits the present study up tosome great level
2 The Algebra of Fourth-Order Tensor
The splendid word ldquoTensorrdquo has been derived from the Latinword ldquoTensusrdquo which is the past participle of ldquotendererdquo andstands for ldquoStretchrdquo This word was used in anatomy inthe early 1704 to represent muscle and stretches Howeverin Mathematics it existed in 1846 when William RowanHamilton has explored his quaternion Algebra Even thoughthe Hamilton sense regarding tensor did not survive Therecent meaning of tensor is due to Voigt who used the termldquoTensortriplerdquo in crystal elasticity around 1899
Likewise the second-order tensor a fourth-order tensorA (say) is a linear map from LinV to LinV That is it assignsto each second-order tensor A the second-order tensor AUthat is
A Lin (V) 997888rarr Lin (V) U 997891997888rarr AU forallU isin Lin (V)
(1)
where V is a vector space and Lin(V) is a linear space ofdimension 119899
2 and is equivalent to
Lin (V) ≑ 119860 V 997888rarr V 119906 997891997888rarr 119860119906 forall119906 isin V (2)
Also any 4th-order tensor A has the representation
A = 119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 (3)
where
119860119894119895119896119897
= ⟨119890119894otimes 119890
119895A (119890
119896otimes 119890
119896)⟩ (4)
and the set of all four vectors that is the set of all linear mapsfrom Lin(V) to Lin(V) is a linear space of dimension 119899
4 andis denoted by Lin(V)
The outer (or open) product of the two 4th-order tensorsis given by the composition
(AB)U = A (BU) forallU isin Lin (V) (5)
In abstract index notations we have
(AB)119894119895119896119897
= 119860119894119895119901119902
119861119901119901119902119896119897
(6)
Chinese Journal of Engineering 3
In[2]= SymIndex[2 dim ] =
Apply[ (Sort[List[]] amp) Array[ List Array[dim amp 2]] 2 ]
SymIndex[2 3]
SymIndex[2 6]
In[3]=SymIndex[4] =
Apply[ Flatten[ Sort[ Map[ Sort Partition[List[] 2]]]] amp
Array[ List Array[3 amp 4]] 4]
Algorithm 1
In[4]=MakeName[myexp ] =
ToExpression ( StringJoin Map[ ToString[ ] amp List[ myexp]])
MakeTensor[ mystring 2 dim ] =
Apply[(MakeName[mystring ] amp) SymIndex[2 dim] 2]
MakeTensor[ mystring 4 ] =
Apply[(MakeName[mystring ] amp) SymIndex[4] 4]
In[5]=( stress = MakeTensor[sigma 2 3]) MatrixForm
In[6]=( strain = MakeTensor[epsilon 2 3]) MatrixForm
In[7]=( C4 = MakeTensor[C 4] ) MatrixForm
Algorithm 2
Moreover the transpose of A isin Lin(V) is a 4th-order tensorA119879 and is defined by
⟨UA119879V⟩ = ⟨VAU⟩ forallUV isin Lin (V) (7)
On Lin(V) we can define the inner product as
⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897
119861119894119895119896119897
(8)
The induced norm is
A = [tr (A119879A)]12
(9)
and the metric is
119889119864(AB) = A minus B (10)
Hence with respect to the inner product given by (8) theset 119890
119894otimes 119890
119895otimes 119890
119896otimes 119890
1198971le119894119895119896119897le119899
forms an orthonormal basis ofLin(V)
21 Symmetries of a 4th-Order Tensor In case of a second-order tensor the only known symmetry represents an invari-ance under the mutual rotation of two indices while for the4th-order tensor there are several notions of symmetry thatrepresent invariance under exchanging pair of indices Thusfor the 4th-order tensor generally we have three types ofsymmetries namely major minor and total symmetries andthese notions of symmetries widely play an important role inthe theory of elasticity [24]
211 Major Symmetry of a Fourth-Rank Tensor For A isin
Lin(V) we say that A is symmetric if A = A119879 or incomponent form 119860
119894119895119896119897= 119860
119896119897119894119895and we say that A is skew-
symmetric if A = minusA119879 or 119860119894119895119896119897
= minus119860119896119897119894119895
Moreover we have the decomposition
A =1
2(A + A
119879) +
1
2(A minus A
119879) (11)
Then the set of all symmetric fourth-order tensors is
Sym (V) = A isin Lin (V) | A = A119879 (12)
is a subspace of Lin(V) of dimension 1198992(1198992+ 1)2
In the theory of elasticity this symmetry is often evokedas ldquomajor symmetryrdquo Since the major symmetry representsinvariance under exchanging the pair (119894 119895) and (119896 119897) then ifA isin Sym(V) then exist120582
119894119894=12119899
2 isin R and 119880119894119894=12119899
2 is anorthonormal basis of Lin(V) such that
A =
1198992
sum
119894=1
120582119894119880119894otimes 119880
119894 (13)
The real number 120582119894is called the eigenvalues of A associated
with the eigentensor 119880119894
Such a representation is called the spectral decompositionofAThe trace and the determinant of the fourth-order tensorare defined respectively as
tr (A) =
1198992
sum
119894=1
120582119894 det (A) =
1198992
prod
119894=1
120582119894 (14)
4 Chinese Journal of Engineering
In[8] =indexrule2to1 = 1 1 -gt 1 2 2 -gt 2 3 3 -gt 3 2 3 -gt
4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt
5 2 1 -gt 6
indexrule1to2 = Map[Rule[[[2]] [[1]] ] amp indexrule2to1]
In[9] =Index6[1] = Range[ 6] indexrule1to2
Algorithm 3
In[10]=HookeVto4[ myC ] =
Array[ myC[[ 1 2 indexrule2to1 3 4 indexrule2to1]] amp Array[3 amp 4]]
In[11]=(C2 = MakeTensor[C 2 6]) MatrixForm
In[12]=C2
In[13]=(C2to4 = HookeVto4[C2] ) MatrixForm
In[14]=C2to4
In[15]=Hooke4toV[ myC ] =
Apply[ Part[ myC ] amp
Array[ Join[ 1 indexrule1to2 2 indexrule1to2] amp
Array[6 amp 2]] 2]
In[16]=C2back = Hooke4toV[C2to4] ) MatrixForm
In[17]=(C4to2 = Hooke4toV[C4] ) MatrixForm
In[18]=(C4back = HookeVto4[C4to2]) MatrixForm
In[19]=Hooke2toV[myc2 ] = Table[
Which[
i lt= 3 ampamp j lt= 3 myc2[[i j]]
4 lt= i ampamp j lt= 3 myc2[[i j]] 2 and(12)
i lt= 3 ampamp 4 lt= j myc2[[i j]] 2 and(12)
4 lt= i ampamp 4 lt= j myc2[[i j]] 2
] i 6 j 6]
HookeVto2[mycV ] = Table[
Which[
i lt= 3 ampamp j lt= 3 mycV[[i j]]
4 lt= i ampamp j lt= 3 mycV[[i j]] lowast 2 and(12)
i lt= 3 ampamp 4 lt= j mycV[[i j]] lowast 2 and(12)
4 lt= i ampamp 4 lt= j mycV[[i j]] lowast 2
] i 6 j 6]
Hooke4to2[myC ] = HookeVto2[Hooke4toV[ myC ]]
Hooke2to4[myC ] = HookeVto4[Hooke2toV[ myC ]]
In[20]=(CV = Hooke2toV[ C2 ] ) MatrixForm
In[21]=(C2back = HookeVto2[ CV ] ) MatrixForm
In[22]=(CV = Hooke4to2[C4]) MatrixForm
In[23]=C4back = Hooke2to4[CV]) MatrixForm
Algorithm 4
212 Minor Symmetry The second type of symmetry iscalled ldquominor symmetryrdquo and is defined by
⟨UAV⟩ = ⟨U119879AV119879⟩ forallUV isin Lin (V) (15)
In component form 119860119894119895119896119897
= 119860119895119894119896119897
= 119860119894119895119897119896
1 le 119894 119895 119896 119897 le 119899Now the invariance under the exchange of first pair of
the indices is called the ldquofirst minor symmetryrdquo and theinvariance under the exchange of second pair of indices iscalled ldquosecond minor symmetryrdquo
The set of all 4th-order tensors that satisfy the minorsymmetry is denoted by
Sym (V) = A isin Lin (V) | A satisfies minor symmetry (16)
213 Total Symmetry A fourth-order tensorA is said to havetotal symmetry if in addition to satisfying the major andminor symmetries it also satisfies
A119906 otimes U otimes V = A119906 otimes U119879 otimes V forallU isin Lin (V) 119906 V isin V
(17)
Chinese Journal of Engineering 5
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 13
12 22 23
13 23 33
120590ij
Cijkl
120598ij
lowast=
Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862
2313= 119862
1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the
components having gray background expose symmetry
1
2
3
12
34
56
Column Row
12
34
56
Figure 2 Histogram showing average elasticity matrix of Quipo
2468101214
12
34
56
Column Row
12
34
56
Figure 3 Histogram showing average elasticity matrix of white
246810121416
12
34
56
Column Row
12
34
56
Figure 4 Histogram showing average elasticity matrix of Khaya
or in abstract index notations119860119894119895119896119897
= 119860120590(119894)120590(119895)120590(119895)120590(119897)
for somepermutation 120590 of [1 119899]
The set of the fourth-rank tensors bearing total symmetryis denoted by
Symtotal(V) = A isin Lin (V) | A satisfies total symmetry
(18)
Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886
The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]
119860119904
119894119895119896119897=
1
3(119860
119894119895119896119897+ 119860
119894119896119895119897+ 119860
119894119897119896119895)
119860119886
119894119895119896119897=
1
3(119860
119894119895119896119897minus 119860
119894119896119895119897minus 119860
119894119897119895119896)
(19)
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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2 Chinese Journal of Engineering
Table 1 Voigtrsquos mappings for stress (left) and strain tensors (right) respectively
1205901112059022
12059033
12059023
1205901312059012
12059023
1205901212059013
120590111205902212059033120590231205901312059012
1205981112059822
120598331205982312059813
12059812
12059812
12059823
12059813
12059811
1205982212059833212059823212059813212059812
Table 2 Index conversion rule of Voigt
119894 119895 11 22 33 23 or 32 13 or 31 12 or 21119901 1 2 3 4 5 6
Table 3 Voigt to Kelvin and vice versa
120585 = 1 1 1 radic2 radic2 radic2
element models using 3D synchrotron microtomography[13] have explored the subject material up to some greatextent
Now as far as the present proposed research concernedwe turn our concentration towards cancellous bone andwood (soft and hard) According to [4] the type of materialsymmetry exposed by bone tissue is often ambiguous Soin order to attain accuracy we have to measure the elasticconstants of bone specimen which may be represented asorthotropic However the orthotropy of the bone specimenmay or may not be of higher degree in all the directionswith respect to a fixed point in the specimen Thereby for allpractical purposes the bone specimen can be represented astransversely isotropic or even isotropic
Now the question arises iswhat is the need of studying themechanical properties of cancellous bone and woodThe rig-orous answer is that the knowledge about mechanical prop-erties of cancellous bone is essential for the determinationof bone fracture risk in osteoporosis and other pathologicalconditions involving impaired bone strength [6] Also themechanical properties of cancellous bone are determined bythe properties of its bone tissue and its architecture [4 14ndash21] Further the wood is a cellulosic semicrystalline cellu-lar material and the mechanical properties (eg elasticitystrength and rheology etc) are generally higher along thebole of a tree than across the bole [7]
Mechanically clear wood obeys the law of elasticorthotropic material like the bone does In general the woodysubstance also exposes the high toughness and stiffnessproperties and these properties vary according to the typeof wood and the direction in which the woody substance isexamined as the woody substance shows a higher degree ofanisotropy
In addition to deal with anisotropic Hookersquos law oneshould be familiar with the algebra of the 4th-order tensorWe would like to emphasize that the 4th-order tensor algebrais not only involved in the study of material symmetrybut its crucial appearance has been set up in the field ofdiffusion tensor MRI [9 10] visualization and processing of
tensor fields [22] biomechanics [13] tissue mechanics [1]geophysics [23] and so forth
Though an adequate amount of research work has beendedicated towards the algebra of 4th-order tensors thenalso in the following section we shall present a brief digestregarding this issue as this issue hits the present study up tosome great level
2 The Algebra of Fourth-Order Tensor
The splendid word ldquoTensorrdquo has been derived from the Latinword ldquoTensusrdquo which is the past participle of ldquotendererdquo andstands for ldquoStretchrdquo This word was used in anatomy inthe early 1704 to represent muscle and stretches Howeverin Mathematics it existed in 1846 when William RowanHamilton has explored his quaternion Algebra Even thoughthe Hamilton sense regarding tensor did not survive Therecent meaning of tensor is due to Voigt who used the termldquoTensortriplerdquo in crystal elasticity around 1899
Likewise the second-order tensor a fourth-order tensorA (say) is a linear map from LinV to LinV That is it assignsto each second-order tensor A the second-order tensor AUthat is
A Lin (V) 997888rarr Lin (V) U 997891997888rarr AU forallU isin Lin (V)
(1)
where V is a vector space and Lin(V) is a linear space ofdimension 119899
2 and is equivalent to
Lin (V) ≑ 119860 V 997888rarr V 119906 997891997888rarr 119860119906 forall119906 isin V (2)
Also any 4th-order tensor A has the representation
A = 119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 (3)
where
119860119894119895119896119897
= ⟨119890119894otimes 119890
119895A (119890
119896otimes 119890
119896)⟩ (4)
and the set of all four vectors that is the set of all linear mapsfrom Lin(V) to Lin(V) is a linear space of dimension 119899
4 andis denoted by Lin(V)
The outer (or open) product of the two 4th-order tensorsis given by the composition
(AB)U = A (BU) forallU isin Lin (V) (5)
In abstract index notations we have
(AB)119894119895119896119897
= 119860119894119895119901119902
119861119901119901119902119896119897
(6)
Chinese Journal of Engineering 3
In[2]= SymIndex[2 dim ] =
Apply[ (Sort[List[]] amp) Array[ List Array[dim amp 2]] 2 ]
SymIndex[2 3]
SymIndex[2 6]
In[3]=SymIndex[4] =
Apply[ Flatten[ Sort[ Map[ Sort Partition[List[] 2]]]] amp
Array[ List Array[3 amp 4]] 4]
Algorithm 1
In[4]=MakeName[myexp ] =
ToExpression ( StringJoin Map[ ToString[ ] amp List[ myexp]])
MakeTensor[ mystring 2 dim ] =
Apply[(MakeName[mystring ] amp) SymIndex[2 dim] 2]
MakeTensor[ mystring 4 ] =
Apply[(MakeName[mystring ] amp) SymIndex[4] 4]
In[5]=( stress = MakeTensor[sigma 2 3]) MatrixForm
In[6]=( strain = MakeTensor[epsilon 2 3]) MatrixForm
In[7]=( C4 = MakeTensor[C 4] ) MatrixForm
Algorithm 2
Moreover the transpose of A isin Lin(V) is a 4th-order tensorA119879 and is defined by
⟨UA119879V⟩ = ⟨VAU⟩ forallUV isin Lin (V) (7)
On Lin(V) we can define the inner product as
⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897
119861119894119895119896119897
(8)
The induced norm is
A = [tr (A119879A)]12
(9)
and the metric is
119889119864(AB) = A minus B (10)
Hence with respect to the inner product given by (8) theset 119890
119894otimes 119890
119895otimes 119890
119896otimes 119890
1198971le119894119895119896119897le119899
forms an orthonormal basis ofLin(V)
21 Symmetries of a 4th-Order Tensor In case of a second-order tensor the only known symmetry represents an invari-ance under the mutual rotation of two indices while for the4th-order tensor there are several notions of symmetry thatrepresent invariance under exchanging pair of indices Thusfor the 4th-order tensor generally we have three types ofsymmetries namely major minor and total symmetries andthese notions of symmetries widely play an important role inthe theory of elasticity [24]
211 Major Symmetry of a Fourth-Rank Tensor For A isin
Lin(V) we say that A is symmetric if A = A119879 or incomponent form 119860
119894119895119896119897= 119860
119896119897119894119895and we say that A is skew-
symmetric if A = minusA119879 or 119860119894119895119896119897
= minus119860119896119897119894119895
Moreover we have the decomposition
A =1
2(A + A
119879) +
1
2(A minus A
119879) (11)
Then the set of all symmetric fourth-order tensors is
Sym (V) = A isin Lin (V) | A = A119879 (12)
is a subspace of Lin(V) of dimension 1198992(1198992+ 1)2
In the theory of elasticity this symmetry is often evokedas ldquomajor symmetryrdquo Since the major symmetry representsinvariance under exchanging the pair (119894 119895) and (119896 119897) then ifA isin Sym(V) then exist120582
119894119894=12119899
2 isin R and 119880119894119894=12119899
2 is anorthonormal basis of Lin(V) such that
A =
1198992
sum
119894=1
120582119894119880119894otimes 119880
119894 (13)
The real number 120582119894is called the eigenvalues of A associated
with the eigentensor 119880119894
Such a representation is called the spectral decompositionofAThe trace and the determinant of the fourth-order tensorare defined respectively as
tr (A) =
1198992
sum
119894=1
120582119894 det (A) =
1198992
prod
119894=1
120582119894 (14)
4 Chinese Journal of Engineering
In[8] =indexrule2to1 = 1 1 -gt 1 2 2 -gt 2 3 3 -gt 3 2 3 -gt
4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt
5 2 1 -gt 6
indexrule1to2 = Map[Rule[[[2]] [[1]] ] amp indexrule2to1]
In[9] =Index6[1] = Range[ 6] indexrule1to2
Algorithm 3
In[10]=HookeVto4[ myC ] =
Array[ myC[[ 1 2 indexrule2to1 3 4 indexrule2to1]] amp Array[3 amp 4]]
In[11]=(C2 = MakeTensor[C 2 6]) MatrixForm
In[12]=C2
In[13]=(C2to4 = HookeVto4[C2] ) MatrixForm
In[14]=C2to4
In[15]=Hooke4toV[ myC ] =
Apply[ Part[ myC ] amp
Array[ Join[ 1 indexrule1to2 2 indexrule1to2] amp
Array[6 amp 2]] 2]
In[16]=C2back = Hooke4toV[C2to4] ) MatrixForm
In[17]=(C4to2 = Hooke4toV[C4] ) MatrixForm
In[18]=(C4back = HookeVto4[C4to2]) MatrixForm
In[19]=Hooke2toV[myc2 ] = Table[
Which[
i lt= 3 ampamp j lt= 3 myc2[[i j]]
4 lt= i ampamp j lt= 3 myc2[[i j]] 2 and(12)
i lt= 3 ampamp 4 lt= j myc2[[i j]] 2 and(12)
4 lt= i ampamp 4 lt= j myc2[[i j]] 2
] i 6 j 6]
HookeVto2[mycV ] = Table[
Which[
i lt= 3 ampamp j lt= 3 mycV[[i j]]
4 lt= i ampamp j lt= 3 mycV[[i j]] lowast 2 and(12)
i lt= 3 ampamp 4 lt= j mycV[[i j]] lowast 2 and(12)
4 lt= i ampamp 4 lt= j mycV[[i j]] lowast 2
] i 6 j 6]
Hooke4to2[myC ] = HookeVto2[Hooke4toV[ myC ]]
Hooke2to4[myC ] = HookeVto4[Hooke2toV[ myC ]]
In[20]=(CV = Hooke2toV[ C2 ] ) MatrixForm
In[21]=(C2back = HookeVto2[ CV ] ) MatrixForm
In[22]=(CV = Hooke4to2[C4]) MatrixForm
In[23]=C4back = Hooke2to4[CV]) MatrixForm
Algorithm 4
212 Minor Symmetry The second type of symmetry iscalled ldquominor symmetryrdquo and is defined by
⟨UAV⟩ = ⟨U119879AV119879⟩ forallUV isin Lin (V) (15)
In component form 119860119894119895119896119897
= 119860119895119894119896119897
= 119860119894119895119897119896
1 le 119894 119895 119896 119897 le 119899Now the invariance under the exchange of first pair of
the indices is called the ldquofirst minor symmetryrdquo and theinvariance under the exchange of second pair of indices iscalled ldquosecond minor symmetryrdquo
The set of all 4th-order tensors that satisfy the minorsymmetry is denoted by
Sym (V) = A isin Lin (V) | A satisfies minor symmetry (16)
213 Total Symmetry A fourth-order tensorA is said to havetotal symmetry if in addition to satisfying the major andminor symmetries it also satisfies
A119906 otimes U otimes V = A119906 otimes U119879 otimes V forallU isin Lin (V) 119906 V isin V
(17)
Chinese Journal of Engineering 5
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 13
12 22 23
13 23 33
120590ij
Cijkl
120598ij
lowast=
Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862
2313= 119862
1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the
components having gray background expose symmetry
1
2
3
12
34
56
Column Row
12
34
56
Figure 2 Histogram showing average elasticity matrix of Quipo
2468101214
12
34
56
Column Row
12
34
56
Figure 3 Histogram showing average elasticity matrix of white
246810121416
12
34
56
Column Row
12
34
56
Figure 4 Histogram showing average elasticity matrix of Khaya
or in abstract index notations119860119894119895119896119897
= 119860120590(119894)120590(119895)120590(119895)120590(119897)
for somepermutation 120590 of [1 119899]
The set of the fourth-rank tensors bearing total symmetryis denoted by
Symtotal(V) = A isin Lin (V) | A satisfies total symmetry
(18)
Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886
The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]
119860119904
119894119895119896119897=
1
3(119860
119894119895119896119897+ 119860
119894119896119895119897+ 119860
119894119897119896119895)
119860119886
119894119895119896119897=
1
3(119860
119894119895119896119897minus 119860
119894119896119895119897minus 119860
119894119897119895119896)
(19)
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 3
In[2]= SymIndex[2 dim ] =
Apply[ (Sort[List[]] amp) Array[ List Array[dim amp 2]] 2 ]
SymIndex[2 3]
SymIndex[2 6]
In[3]=SymIndex[4] =
Apply[ Flatten[ Sort[ Map[ Sort Partition[List[] 2]]]] amp
Array[ List Array[3 amp 4]] 4]
Algorithm 1
In[4]=MakeName[myexp ] =
ToExpression ( StringJoin Map[ ToString[ ] amp List[ myexp]])
MakeTensor[ mystring 2 dim ] =
Apply[(MakeName[mystring ] amp) SymIndex[2 dim] 2]
MakeTensor[ mystring 4 ] =
Apply[(MakeName[mystring ] amp) SymIndex[4] 4]
In[5]=( stress = MakeTensor[sigma 2 3]) MatrixForm
In[6]=( strain = MakeTensor[epsilon 2 3]) MatrixForm
In[7]=( C4 = MakeTensor[C 4] ) MatrixForm
Algorithm 2
Moreover the transpose of A isin Lin(V) is a 4th-order tensorA119879 and is defined by
⟨UA119879V⟩ = ⟨VAU⟩ forallUV isin Lin (V) (7)
On Lin(V) we can define the inner product as
⟨(119860) B⟩ ≑ tr (A119879B) = 119860119894119895119896119897
119861119894119895119896119897
(8)
The induced norm is
A = [tr (A119879A)]12
(9)
and the metric is
119889119864(AB) = A minus B (10)
Hence with respect to the inner product given by (8) theset 119890
119894otimes 119890
119895otimes 119890
119896otimes 119890
1198971le119894119895119896119897le119899
forms an orthonormal basis ofLin(V)
21 Symmetries of a 4th-Order Tensor In case of a second-order tensor the only known symmetry represents an invari-ance under the mutual rotation of two indices while for the4th-order tensor there are several notions of symmetry thatrepresent invariance under exchanging pair of indices Thusfor the 4th-order tensor generally we have three types ofsymmetries namely major minor and total symmetries andthese notions of symmetries widely play an important role inthe theory of elasticity [24]
211 Major Symmetry of a Fourth-Rank Tensor For A isin
Lin(V) we say that A is symmetric if A = A119879 or incomponent form 119860
119894119895119896119897= 119860
119896119897119894119895and we say that A is skew-
symmetric if A = minusA119879 or 119860119894119895119896119897
= minus119860119896119897119894119895
Moreover we have the decomposition
A =1
2(A + A
119879) +
1
2(A minus A
119879) (11)
Then the set of all symmetric fourth-order tensors is
Sym (V) = A isin Lin (V) | A = A119879 (12)
is a subspace of Lin(V) of dimension 1198992(1198992+ 1)2
In the theory of elasticity this symmetry is often evokedas ldquomajor symmetryrdquo Since the major symmetry representsinvariance under exchanging the pair (119894 119895) and (119896 119897) then ifA isin Sym(V) then exist120582
119894119894=12119899
2 isin R and 119880119894119894=12119899
2 is anorthonormal basis of Lin(V) such that
A =
1198992
sum
119894=1
120582119894119880119894otimes 119880
119894 (13)
The real number 120582119894is called the eigenvalues of A associated
with the eigentensor 119880119894
Such a representation is called the spectral decompositionofAThe trace and the determinant of the fourth-order tensorare defined respectively as
tr (A) =
1198992
sum
119894=1
120582119894 det (A) =
1198992
prod
119894=1
120582119894 (14)
4 Chinese Journal of Engineering
In[8] =indexrule2to1 = 1 1 -gt 1 2 2 -gt 2 3 3 -gt 3 2 3 -gt
4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt
5 2 1 -gt 6
indexrule1to2 = Map[Rule[[[2]] [[1]] ] amp indexrule2to1]
In[9] =Index6[1] = Range[ 6] indexrule1to2
Algorithm 3
In[10]=HookeVto4[ myC ] =
Array[ myC[[ 1 2 indexrule2to1 3 4 indexrule2to1]] amp Array[3 amp 4]]
In[11]=(C2 = MakeTensor[C 2 6]) MatrixForm
In[12]=C2
In[13]=(C2to4 = HookeVto4[C2] ) MatrixForm
In[14]=C2to4
In[15]=Hooke4toV[ myC ] =
Apply[ Part[ myC ] amp
Array[ Join[ 1 indexrule1to2 2 indexrule1to2] amp
Array[6 amp 2]] 2]
In[16]=C2back = Hooke4toV[C2to4] ) MatrixForm
In[17]=(C4to2 = Hooke4toV[C4] ) MatrixForm
In[18]=(C4back = HookeVto4[C4to2]) MatrixForm
In[19]=Hooke2toV[myc2 ] = Table[
Which[
i lt= 3 ampamp j lt= 3 myc2[[i j]]
4 lt= i ampamp j lt= 3 myc2[[i j]] 2 and(12)
i lt= 3 ampamp 4 lt= j myc2[[i j]] 2 and(12)
4 lt= i ampamp 4 lt= j myc2[[i j]] 2
] i 6 j 6]
HookeVto2[mycV ] = Table[
Which[
i lt= 3 ampamp j lt= 3 mycV[[i j]]
4 lt= i ampamp j lt= 3 mycV[[i j]] lowast 2 and(12)
i lt= 3 ampamp 4 lt= j mycV[[i j]] lowast 2 and(12)
4 lt= i ampamp 4 lt= j mycV[[i j]] lowast 2
] i 6 j 6]
Hooke4to2[myC ] = HookeVto2[Hooke4toV[ myC ]]
Hooke2to4[myC ] = HookeVto4[Hooke2toV[ myC ]]
In[20]=(CV = Hooke2toV[ C2 ] ) MatrixForm
In[21]=(C2back = HookeVto2[ CV ] ) MatrixForm
In[22]=(CV = Hooke4to2[C4]) MatrixForm
In[23]=C4back = Hooke2to4[CV]) MatrixForm
Algorithm 4
212 Minor Symmetry The second type of symmetry iscalled ldquominor symmetryrdquo and is defined by
⟨UAV⟩ = ⟨U119879AV119879⟩ forallUV isin Lin (V) (15)
In component form 119860119894119895119896119897
= 119860119895119894119896119897
= 119860119894119895119897119896
1 le 119894 119895 119896 119897 le 119899Now the invariance under the exchange of first pair of
the indices is called the ldquofirst minor symmetryrdquo and theinvariance under the exchange of second pair of indices iscalled ldquosecond minor symmetryrdquo
The set of all 4th-order tensors that satisfy the minorsymmetry is denoted by
Sym (V) = A isin Lin (V) | A satisfies minor symmetry (16)
213 Total Symmetry A fourth-order tensorA is said to havetotal symmetry if in addition to satisfying the major andminor symmetries it also satisfies
A119906 otimes U otimes V = A119906 otimes U119879 otimes V forallU isin Lin (V) 119906 V isin V
(17)
Chinese Journal of Engineering 5
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 13
12 22 23
13 23 33
120590ij
Cijkl
120598ij
lowast=
Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862
2313= 119862
1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the
components having gray background expose symmetry
1
2
3
12
34
56
Column Row
12
34
56
Figure 2 Histogram showing average elasticity matrix of Quipo
2468101214
12
34
56
Column Row
12
34
56
Figure 3 Histogram showing average elasticity matrix of white
246810121416
12
34
56
Column Row
12
34
56
Figure 4 Histogram showing average elasticity matrix of Khaya
or in abstract index notations119860119894119895119896119897
= 119860120590(119894)120590(119895)120590(119895)120590(119897)
for somepermutation 120590 of [1 119899]
The set of the fourth-rank tensors bearing total symmetryis denoted by
Symtotal(V) = A isin Lin (V) | A satisfies total symmetry
(18)
Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886
The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]
119860119904
119894119895119896119897=
1
3(119860
119894119895119896119897+ 119860
119894119896119895119897+ 119860
119894119897119896119895)
119860119886
119894119895119896119897=
1
3(119860
119894119895119896119897minus 119860
119894119896119895119897minus 119860
119894119897119895119896)
(19)
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Submit your manuscripts athttpwwwhindawicom
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Shock and Vibration
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
4 Chinese Journal of Engineering
In[8] =indexrule2to1 = 1 1 -gt 1 2 2 -gt 2 3 3 -gt 3 2 3 -gt
4 3 1 -gt 5 1 2 -gt 6 3 2 -gt 4 1 3 -gt
5 2 1 -gt 6
indexrule1to2 = Map[Rule[[[2]] [[1]] ] amp indexrule2to1]
In[9] =Index6[1] = Range[ 6] indexrule1to2
Algorithm 3
In[10]=HookeVto4[ myC ] =
Array[ myC[[ 1 2 indexrule2to1 3 4 indexrule2to1]] amp Array[3 amp 4]]
In[11]=(C2 = MakeTensor[C 2 6]) MatrixForm
In[12]=C2
In[13]=(C2to4 = HookeVto4[C2] ) MatrixForm
In[14]=C2to4
In[15]=Hooke4toV[ myC ] =
Apply[ Part[ myC ] amp
Array[ Join[ 1 indexrule1to2 2 indexrule1to2] amp
Array[6 amp 2]] 2]
In[16]=C2back = Hooke4toV[C2to4] ) MatrixForm
In[17]=(C4to2 = Hooke4toV[C4] ) MatrixForm
In[18]=(C4back = HookeVto4[C4to2]) MatrixForm
In[19]=Hooke2toV[myc2 ] = Table[
Which[
i lt= 3 ampamp j lt= 3 myc2[[i j]]
4 lt= i ampamp j lt= 3 myc2[[i j]] 2 and(12)
i lt= 3 ampamp 4 lt= j myc2[[i j]] 2 and(12)
4 lt= i ampamp 4 lt= j myc2[[i j]] 2
] i 6 j 6]
HookeVto2[mycV ] = Table[
Which[
i lt= 3 ampamp j lt= 3 mycV[[i j]]
4 lt= i ampamp j lt= 3 mycV[[i j]] lowast 2 and(12)
i lt= 3 ampamp 4 lt= j mycV[[i j]] lowast 2 and(12)
4 lt= i ampamp 4 lt= j mycV[[i j]] lowast 2
] i 6 j 6]
Hooke4to2[myC ] = HookeVto2[Hooke4toV[ myC ]]
Hooke2to4[myC ] = HookeVto4[Hooke2toV[ myC ]]
In[20]=(CV = Hooke2toV[ C2 ] ) MatrixForm
In[21]=(C2back = HookeVto2[ CV ] ) MatrixForm
In[22]=(CV = Hooke4to2[C4]) MatrixForm
In[23]=C4back = Hooke2to4[CV]) MatrixForm
Algorithm 4
212 Minor Symmetry The second type of symmetry iscalled ldquominor symmetryrdquo and is defined by
⟨UAV⟩ = ⟨U119879AV119879⟩ forallUV isin Lin (V) (15)
In component form 119860119894119895119896119897
= 119860119895119894119896119897
= 119860119894119895119897119896
1 le 119894 119895 119896 119897 le 119899Now the invariance under the exchange of first pair of
the indices is called the ldquofirst minor symmetryrdquo and theinvariance under the exchange of second pair of indices iscalled ldquosecond minor symmetryrdquo
The set of all 4th-order tensors that satisfy the minorsymmetry is denoted by
Sym (V) = A isin Lin (V) | A satisfies minor symmetry (16)
213 Total Symmetry A fourth-order tensorA is said to havetotal symmetry if in addition to satisfying the major andminor symmetries it also satisfies
A119906 otimes U otimes V = A119906 otimes U119879 otimes V forallU isin Lin (V) 119906 V isin V
(17)
Chinese Journal of Engineering 5
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 13
12 22 23
13 23 33
120590ij
Cijkl
120598ij
lowast=
Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862
2313= 119862
1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the
components having gray background expose symmetry
1
2
3
12
34
56
Column Row
12
34
56
Figure 2 Histogram showing average elasticity matrix of Quipo
2468101214
12
34
56
Column Row
12
34
56
Figure 3 Histogram showing average elasticity matrix of white
246810121416
12
34
56
Column Row
12
34
56
Figure 4 Histogram showing average elasticity matrix of Khaya
or in abstract index notations119860119894119895119896119897
= 119860120590(119894)120590(119895)120590(119895)120590(119897)
for somepermutation 120590 of [1 119899]
The set of the fourth-rank tensors bearing total symmetryis denoted by
Symtotal(V) = A isin Lin (V) | A satisfies total symmetry
(18)
Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886
The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]
119860119904
119894119895119896119897=
1
3(119860
119894119895119896119897+ 119860
119894119896119895119897+ 119860
119894119897119896119895)
119860119886
119894119895119896119897=
1
3(119860
119894119895119896119897minus 119860
119894119896119895119897minus 119860
119894119897119895119896)
(19)
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 5
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 1312 22 2313 23 33
11 12 13
12 22 23
13 23 33
120590ij
Cijkl
120598ij
lowast=
Figure 1 Hookersquos law (reduction process of elastic coefficients) Here still there are 36 components seen in the stiffness table but due to thecomponents for example 119862
2313= 119862
1323 and so forth the counting of 36 components will be reduced up to 21 Also in the above figure the
components having gray background expose symmetry
1
2
3
12
34
56
Column Row
12
34
56
Figure 2 Histogram showing average elasticity matrix of Quipo
2468101214
12
34
56
Column Row
12
34
56
Figure 3 Histogram showing average elasticity matrix of white
246810121416
12
34
56
Column Row
12
34
56
Figure 4 Histogram showing average elasticity matrix of Khaya
or in abstract index notations119860119894119895119896119897
= 119860120590(119894)120590(119895)120590(119895)120590(119897)
for somepermutation 120590 of [1 119899]
The set of the fourth-rank tensors bearing total symmetryis denoted by
Symtotal(V) = A isin Lin (V) | A satisfies total symmetry
(18)
Thus any fourth-order tensor can be decomposed into itstotally symmetric part A119904 and its totally antisymmetric partA119886 that is A = A119904 + A119886
The components of totally symmetric and antisymmetricparts of a fourth-order tensor are described as [25]
119860119904
119894119895119896119897=
1
3(119860
119894119895119896119897+ 119860
119894119896119895119897+ 119860
119894119897119896119895)
119860119886
119894119895119896119897=
1
3(119860
119894119895119896119897minus 119860
119894119896119895119897minus 119860
119894119897119895119896)
(19)
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Active and Passive Electronic Components
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Chinese Journal of Engineering
Table 4 The stress strain elasticity and compliance tensorscomponents in Voigtrsquos Kelvinrsquos and MCN patterns
Voigtrsquos notations Kelvinrsquos notations MCN
Stress
12059011
1205901
1
12059022
1205902
2
12059033
1205903
3
12059023
1205904
4
12059013
1205905
5
12059012
1205905
6
Strain
12059811
1205981
1205981
12059822
1205982
1205982
12059833
1205983
1205983
12059823
1205984
1205984
12059813
12059815
1205985
12059812
1205986
1205986
Elasticity
1198621111
11986211
11986211
1198622222
11986222
11986222
1198623333
11986233
11986233
1198621122
11986212
11986212
1198621133
11986213
11986213
1198622233
11986223
11986223
1198622323
11986244
1
211986244
1198621313
11986255
1
211986255
1198621212
11986266
1
211986266
1198621323
11986254
1
211986254
1198621312
11986256
1
211986256
1198621223
11986264
1
411986264
1198622311
11986241
1
radic211986241
1198621311
11986251
1
radic211986251
1198621211
11986261
1
radic211986261
1198622322
11986242
1
radic211986242
1198621322
11986252
1
radic211986252
1198621222
11986262
1
radic211986262
1198622333
11986243
1
radic211986243
1198621333
11986253
1
radic211986253
1198621233
11986263
1
radic211986263
Table 4 Continued
Voigtrsquos notations Kelvinrsquos notations MCN
Compliance
1198701111
11987811
11987811
1198702222
11987822
11987822
1198703333
11987833
11987833
1198701122
11987812
11987812
1198701133
11987813
11987813
1198702233
11987823
11987823
1198702323
1
411987844
1
211987844
1198701313
1
411987855
1
211987855
1198701212
1
411987866
1
211987866
1198701323
1
411987854
1
211987854
1198701312
1
411987856
1
211987856
1198701223
1
411987864
1
211987864
1198702311
1
211987841
1
radic211987841
1198701311
1
211987851
1
radic211987851
1198701211
1
211987861
1
radic211987861
1198702322
1
211987842
1
radic211987842
1198701322
1
211987852
1
radic211987852
1198701222
1
211987862
1
radic211987862
1198702333
1
211987843
1
radic211987843
1198701333
1
211987853
1
radic211987853
1198701233
1
211987863
1
radic211987863
24681012141618
12
34
56
Column Row
12
34
56
Figure 5 Histogram showing average elasticity matrix ofMahogany
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Submit your manuscripts athttpwwwhindawicom
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 7
Table 5 The elastic constants data for hardwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Quipo 01 0045 0251 1075 0027 0033 0025 0226 0118 00782 Quipo 02 0159 0427 3446 0069 0131 0178 0430 0280 01443 White 038 0547 1192 10041 0399 0360 0555 1442 1344 00224 Khaya 044 0631 1381 10725 0389 0520 0662 1800 1196 04205 Mahogany 050 0952 1575 11996 0571 0682 0790 1960 1498 06386 Mahogany 053 0765 1538 13010 0655 0631 0841 1218 0938 03007 S Germ 054 0772 1772 12240 0558 0530 0871 2318 1582 05408 Maple 058 1451 2565 11492 1197 1267 1818 2460 2194 05849 Walnut 059 0927 1760 12432 0707 0936 1312 1922 1400 046010 Birch 062 0898 1623 17173 0671 0714 1075 2346 1816 037211 Y Birch 064 1084 1697 15288 0777 0883 1191 2120 1942 048012 Oak 067 1350 2983 16958 1007 1005 1463 2380 1532 078413 Ash 068 1135 2142 16958 0827 0917 1427 2684 1784 054014 Ash 080 1439 2439 17000 1037 1485 1968 1720 1218 050015 Beech 074 1659 3301 15437 1279 1433 2142 3216 2112 0912Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Table 6 The elastic constants data for softwoods
S no Species 120588 11986211
11986222
11986233
11986212
11986213
11986223
11986244
11986255
11986266
1 Blasa 02 0127 0360 6380 0086 0091 0154 0624 0406 00662 Spruce 039 0572 1030 11950 0262 0365 0506 1498 1442 00783 Spruce 043 0594 1106 14055 0346 0476 0686 1442 10 00644 Spruce 044 0443 0775 16286 0192 0321 0442 1234 152 00725 Spruce 050 0755 0963 17221 0333 0549 0548 125 1706 0076 Douglas Fir 045 0929 1173 16095 0409 0539 0539 1767 1766 01767 Douglas Fir 059 1226 1775 17004 0753 0747 0941 2348 1816 01608 Pine 054 0721 1405 16929 0454 0535 0857 3484 1344 0132Source this data is consulted from [47 48] The number of repetitions of the particular species in this table shows the number of measurements done for thatparticular wood species The units of 120588 (bulk or apparent density) are gcm3 and the units of elastic constants are GPa = 1010 dynescm2
Obviously a fourth-order tensor is totally symmetric if A =
A119904 or equivalently A119886 = O where O stands for a fourth-order null tensor Likewise A is antisymmetric if A = A119886 orequivalently A119904 = O
Now it is straightforward to show that if A is totallysymmetric and B is a symmetric tensor then tr(AB) = 0
Reference [25] has also shown that there is an isomor-phism (ie a 1-1 linear map) between totally symmetricfourth-rank tensor and a homogeneous polynomial of degreefour Also if a totally symmetric fourth-order tensor istraceless that is tr(A) = 0 then it is isomorphic toa harmonic polynomial of degree four For this reason atotally symmetric and traceless tensor is called harmonictensor
To meet the objectives of proposed research let us turnour attention to the flattening (sometimes called unfolding)of the fourth-order 3-dimensional tensor withminor symme-tries as it is customary in the 3-dimensional elasticity theory
In the following section we discuss the notion of9-dimensional representation of a 3-dimensional fourth-order tensor and then particularize this representation to
a fourth-order elasticity tensor havingminor symmetries dueto the well-known anisotropic Hookersquos law
3 A 9-Dimensional Exposition ofa 3-Dimensional 4th-Order Tensorand the Anisotropic Hookersquos Law
In accordance with the evolution of tensor theory the algebraof the 3-dimensional fourth- order tensor is still not fullydeveloped For instance the techniques for calculating thelatent roots an latent tensors of a tensor like119862119894119895119896119897 or the com-putation of its inverse (which is called a compliance 119878119894119895119896119897) arestill not widely available There are also some certain psycho-logical constraints that is we are trained to tackle matricesbut not trained to deal with the multiway arrays (sometimescalled matrix of matrices) Though recently Tamra [26ndash29]has developed a tensor toolbox that runs under MATLAB[30] in my view this tool still needs some enhancementsto handle symbolic tensor data Moreover Constantinescuand Korsunsky [31] have developed some crucial packages
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Electrical and Computer Engineering
Journal of
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
8 Chinese Journal of Engineering
Table 7 The elastic constants data for the three specimens of cancellous bone
Elastic constants Specimen 1 Specimen 2 Specimen 311986211
986 7912 698311986212
2147 3609 281611986213
2773 3186 284511986214
minus0162 11 8011986215
minus293 54 minus0711986216
minus0821 minus06 11411986222
935 8529 896711986223
2346 3281 322711986224
minus1204 09 minus162311986225
minus0936 minus25 minus3211986226
0445 minus42 16011986233
5952 14730 916011986234
minus1309 minus24 minus162211986235
minus1791 169 minus7211986236
054 06 2411986244
561 3587 348911986245
0798 05 10511986246
minus2159 51 minus9811986255
6032 3448 242611986256
minus0335 minus07 minus64411986266
7425 2304 2378Source the data for cancellous bone has been taken from [2 11]
24681012141618
12
34
56
Column Row
12
34
56
Figure 6 Histogram showing average elasticity matrix of S Germ
namely ldquoTensor2Analysismrdquo ldquoIntegrateStrainmrdquo ldquoParamet-ricMeshmrdquo and ldquoVectorAnalysismrdquo which are compatiblewith Mathematica and Mathematica programming usingthese packages enables one to compute symbolic computa-tions regarding elasticity tensors Anyways we shall comeback to these CAS assisted issues in the subsequent sectionswhere we shall demonstrate the proposed objectives usingCAS
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 7 Histogram showing average elasticity matrix of maple
In order to study material symmetries and anisotropicHookersquos law it is customary to transform the 3-dimensional4th-order tensor as a 9 times 9 matrix that is a 9-dimensionalrepresentation of a 3-dimensional fourth-order tensor Forthis purpose there is a basic notion of representinga fourth-order tensor as a second-order tensor [22] Accord-ing to [22] if 1 le 119894 119895 le 119899 then we set
119890120595(119894119895)
= 119890119894otimes 119890
119895 (20)
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 9
Table 8 The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Quipo
[[[[[[[[[
[
1076866026
004067345638
02534605191
02260000000
01180000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
003276733390 09918780499 minus01228992897 00 00 00
003131140851 minus01239237163 minus09917976143 00 00 00
09989724208 minus002865041271 003511774899 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Quipo
[[[[[[[[[
[
3461963876
01399776173
04300584948
04300000000
02800000000
01440000000
]]]]]]]]]
]
[[[[[[[[[
[
004079957804 09756137601 minus02156691559 00 00 00
005942480245 minus02178361212 minus09741745817 00 00 00
09973986606 minus002692981555 006686327745 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 White
[[[[[[[[[
[
1009116301
03556701722
1333166815
1442000000
1344000000
002200000000
]]]]]]]]]
]
[[[[[[[[[
[
004028666644 09048796003 minus04237568827 00 00 00
006399313350 minus04255671399 minus09026613361 00 00 00
09971368323 minus0009247686096 007505076253 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Khaya
[[[[[[[[[
[
1080102614
04606834511
1475290392
1800000000
1196000000
04200000000
]]]]]]]]]
]
[[[[[[[[[
[
005368516527 09266208315 minus03721447810 00 00 00
007220768570 minus03753089731 minus09240829102 00 00 00
09959437502 minus002273783078 008705767064 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Mahogany
[[[[[[[[[
[
1210253321
06098853915
1810581412
1960000000
1498000000
06380000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006484967920 minus08666704601 minus04946481887 00 00 00
minus007817061387 04985803653 minus08633116316 00 00 00
minus09948285648 001731853005 01000809391 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Mahogany
[[[[[[[[[
[
1310854783
03895295651
1814922632
1218000000
09380000000
03000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus005490172356 minus08720473797 minus04863323643 00 00 00
minus007547523076 04892979419 minus08688446422 00 00 00
minus09956351187 001099502014 009268127742 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 S Germ
[[[[[[[[[
[
1234053410
05212570563
1922208822
2318000000
1582000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
004967528691 09161715971 minus03976958243 00 00 00
008463937788 minus04006166011 minus09123280736 00 00 00
09951726186 minus001165943224 009744493938 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Maple
[[[[[[[[[
[
1205449768
06868279555
2766674361
2460000000
2194000000
05840000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01387533797 minus08476774112 minus05120454135 00 00 00
minus02031928413 05304148448 minus08230265872 00 00 00
minus09692575344 001015375725 02458388325 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
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International Journal of
10 Chinese Journal of Engineering
Table 9 In continuation to Table 8
S no Hardwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
9 Walnut
[[[[[[[[[
[
1267869546
05204134969
1919891015
1922000000
1400000000
04600000000
]]]]]]]]]
]
[[[[[[[[[
[
008621257414 08755641188 minus04753471023 00 00 00
01243595697 minus04828494241 minus08668282006 00 00 00
09884847438 minus001561753067 01505124700 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
10 Birch
[[[[[[[[[
[
3168908680
04747157903
05946755301
2346000000
1816000000
03720000000
]]]]]]]]]
]
[[[[[[[[[
[
03961520086 09160614623 006240981414 00 00 00
06338768023 minus02236797319 minus07403833993 00 00 00
06642768888 minus03328645006 06692812840 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
11 Y Birch
[[[[[[[[[
[
1545414040
05549651499
2059894445
2120000000
1942000000
04800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus006591789198 minus08289381135 minus05554425577 00 00 00
minus008975775227 05593224991 minus08240763851 00 00 00
minus09937798435 0004466103673 01112729817 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Oak
[[[[[[[[[
[
1718666431
08648974218
3239438239
2380000000
1532000000
07840000000
]]]]]]]]]
]
[[[[[[[[[
[
006975010637 09079000793 minus04133429180 00 00 00
01071019008 minus04187725641 minus09017531389 00 00 00
09917984199 minus001862756541 01264472547 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
13 Ash
[[[[[[[[[
[
1715567967
06696042476
2409716064
2684000000
1784000000
05400000000
]]]]]]]]]
]
[[[[[[[[[
[
006190313535 08746549514 minus04807772027 00 00 00
009781749768 minus04846986500 minus08691944279 00 00 00
09932772717 minus0006777439445 01155609239 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
14 Ash
[[[[[[[[[
[
1742363268
07844815431
2669885744
1720000000
1218000000
05000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus01004077048 minus08554189669 minus05081108976 00 00 00
minus01363859604 05177044741 minus08446188165 00 00 00
minus09855542417 001550704374 01686486565 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
15 Beech
[[[[[[[[[
[
1599002755
09558575345
3451114905
3216000000
2112000000
09120000000
]]]]]]]]]
]
[[[[[[[[[
[
01135176976 08848520771 minus04518302069 00 00 00
01764907831 minus04654963442 minus08672739791 00 00 00
09777344911 minus001870707734 02090103088 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
Thus 119890120572otimes 119890
1205731le120572120573le119899
2 is an orthonormal basis of Lin(V) equiv
Lin(V)Therefore any fourth-rank tensor A isin Lin(V) can
be assumed as an 119899-dimensional fourth rank tensor A =
119860119894119895119896119897
119890119894otimes 119890
119895otimes 119890
119896otimes 119890
119897 or as an 119899
2-dimensional second-ordertensor A harr 119860 = 119860
120572120573119890120572otimes 119890
120573 where 119860
120595(119894119895)120595119896119897= 119860
119894119895119896119897
1 le 119894 119895 119896 119897 le 119899
Again we have from (8) that
tr (A119879B) = ⟨AB⟩ = tr (119860119879119861119879) A =1003817100381710038171003817100381711986010038171003817100381710038171003817 (21)
hence the two-way map A harr 119860 is an isometryLet us apply this concept to anisotropic Hookersquos law by
assuming that the anisotropic Hookersquos law given below is
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 11
Table 10 The eigenvalues and eigenvectors for 8 softwoods species calculated using MAPLE Λ[119894] stand for the eigenvalues of 119894th softwoodspecies and 119881[119894] stand for the corresponding eigenvectors
S no Softwood species Eigenvalues Λ[119894] Eigenvectors 119881[119894]
1 Blasa
[[[[[[[[[
[
6385324208
009845837744
03832174064
06240000000
04060000000
006600000000
]]]]]]]]]
]
[[[[[[[[[
[
001488818113 09510211778 minus03087669978 00 00 00
002575997614 minus03090635474 minus09506924583 00 00 00
09995572851 minus0006200252135 002909967665 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
2 Spruce
[[[[[[[[[
[
1198583568
04516442734
1114520065
1498000000
1442000000
007800000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003300264531 minus09144925828 minus04032544375 00 00 00
minus004689865645 04044467539 minus09133582749 00 00 00
minus09983543167 001123114838 005623628701 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
3 Spruce
[[[[[[[[[
[
1410927768
04185382987
1227183961
1442000000
10
006400000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003651783317 minus08968065074 minus04409132956 00 00 00
minus005361649666 04423303564 minus08952480817 00 00 00
minus09978936411 0009052295016 006423657043 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
4 Spruce
[[[[[[[[[
[
1630529953
03544288592
08442715844
1234000000
1520000000
007200000000
]]]]]]]]]
]
[[[[[[[[[
[
minus002057139660 minus09124371483 minus04086994866 00 00 00
minus002869707087 04091564350 minus09120128765 00 00 00
minus09993764538 0007032901380 003460119282 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
5 Spruce
[[[[[[[[[
[
1725845208
05092652753
1171282625
1250000000
1706000000
007000000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003391880763 minus08104111857 minus05848788055 00 00 00
minus003428302033 05858146064 minus08097196604 00 00 00
minus09988364173 0007413313465 004765347716 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
6 Douglas Fir
[[[[[[[[[
[
1613459769
06233733771
1439028943
1767000000
1766000000
01760000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003639420861 minus08057068065 minus05911954018 00 00 00
minus003697194691 05922678885 minus08048924305 00 00 00
minus09986533619 0007435777607 005134366998 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
7 Douglas Fir
[[[[[[[[[
[
1710150156
06986859431
2204812526
2348000000
1816000000
01600000000
]]]]]]]]]
]
[[[[[[[[[
[
minus004991838192 minus08212998538 minus05683086323 00 00 00
minus006364825251 05704773676 minus08188433771 00 00 00
minus09967231590 0004703484281 008075160717 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
8 Pine
[[[[[[[[[
[
1699539133
04940019684
1565606760
3484000000
1344000000
01320000000
]]]]]]]]]
]
[[[[[[[[[
[
minus003436105770 minus08973128643 minus04400556096 00 00 00
minus005585205149 04413516031 minus08955943895 00 00 00
minus09978476167 0006195562437 006528207193 00 00 00
00 00 00 10 00 00
00 00 00 00 10 00
00 00 00 00 00 10
]]]]]]]]]
]
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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DistributedSensor Networks
International Journal of
12 Chinese Journal of Engineering
Table11Th
eeigenvalues
andeigenvectorsfor3
specim
enso
fcancello
usbo
necalculated
usingMAPL
EΛ[119894]sta
ndforthe
eigenvalueso
f119894th
specim
enof
cancellous
boneand
119881[119894]sta
ndfor
thec
orrespon
ding
eigenvectors
Cancellous
bone
EigenvaluesΛ
[119894]
Eigenvectors
119881[119894]
Specim
en1
[ [ [ [ [ [ [ [
1334869880
minus02071167473
8644210768
6837676373
3768806080
4123724855
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03316115964
08831238953
02356441950
02242668491
005354786914
003787816350
06475092156
minus01099136965
01556396475
minus06940063479
minus01278953723
02154647417
04800906963
minus02832250652
007648827487
02494278793
06410471445
minus04585742261
minus02737532692
minus006354940263
04518342532
minus006214920576
05836952694
06101669758
minus03706582070
03314035780
minus01276606259
minus06337030234
03639432801
minus04499474345
01671829224
01180163925
minus08330343502
002008124471
03109325968
04087706408
] ] ] ] ] ] ] ]
Specim
en2
[ [ [ [ [ [ [ [
1754634171
7773848354
1550688791
2301790452
3445174379
3589156342
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
minus03057290719
minus03005337771
minus09030341401
001584500766
minus002175914011
minus0003741933916
minus04311831740
minus08021570593
04130146809
00001362645283
minus0003854432451
0005396441669
minus08488209390
05154657137
01152135966
minus0004741957786
002229739182
minus0002068264586
00009402467441
minus0005358584867
0003987518000
minus003989381331
003796849613
minus09984595017
minus001058157817
002100470582
002092543658
0006230946613
minus09987520802
minus003826770492
00009823402498
0006977574167
001484146906
09990475909
0008196718814
minus003958286493
] ] ] ] ] ] ] ]
Specim
en3
[ [ [ [ [ [ [ [
1418542670
5838388363
9959058231
3433818493
3024968125
1757492470
] ] ] ] ] ] ] ]
[ [ [ [ [ [ [ [
03244960223
minus0004831643783
minus08451713783
04062092722
01164524794
minus004239310198
06461152835
minus07125334236
02668138654
004131783689
minus002431040234
003665187401
06621315621
07009969173
02633438279
002836289512
minus0001711614840
0005269111592
minus01962401199
0009159859564
03740839141
08850746598
01904577552
004284654369
minus0008597323304
minus0002526763452
001897677365
01738799329
minus06977939665
minus06945566457
001533190599
minus002803230310
006961922985
minus01374446216
06801869735
minus07159517722
] ] ] ] ] ] ] ]
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Active and Passive Electronic Components
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 13
24681012141618
12
34
56
Column Row
12
34
56
Figure 8 Histogram showing average elasticity matrix of Walnut
12345678
12
34
56
Column Row
12
34
56
Figure 9 Histogram showing average elasticity matrix of Birch
generalized one and also valid in the case where stress andstrain tensors are not necessarily symmetric
The anisotropic Hookersquos law in abstract index notations isoften depicted as
120590119894119895= 119862
119894119895119896119897120598119896119897 or in index free notations 120590 = C120598 (22)
where 119862119894119895119896119897
are the components of an elastic tensor Writingthe stress strain and elastic tensors in usual tensor bases wehave
120590 = 120590119894119895119890119894otimes 119890
119895 120598 = 120598
119896119897119890119896otimes 119890
119897
C = 119862119894119895119896119897
119890119894otimes 119890
119896otimes 119890
119896otimes 119890
119897
(23)
5
10
15
20
12
34
56
Column Row
12
34
56
Figure 10 Histogram showing average elasticity matrix of Y Birch
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 11 Histogram showing average elasticity matrix of Oak
Now when working with orthonormal basis one needs tointroduce a new basis which should be composed of the threediagonal elements
E1equiv 119890
1otimes 119890
1
E2equiv 119890
2otimes 119890
2
E3equiv 119890
3otimes 119890
3
(24)
the three symmetric elements
E4equiv
1
radic2(1198901otimes 119890
2+ 119890
2otimes 119890
1)
E5equiv
1
radic2(1198902otimes 119890
3+ 119890
3otimes 119890
2)
E6equiv
1
radic2(1198903otimes 119890
1+ 119890
1otimes 119890
3)
(25)
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
14 Chinese Journal of Engineering
and the three asymmetric elements
E7equiv
1
radic2(1198901otimes 119890
2minus 119890
2otimes 119890
1)
E8equiv
1
radic2(1198902otimes 119890
3minus 119890
3otimes 119890
2)
E9equiv
1
radic2(1198903otimes 119890
1minus 119890
1otimes 119890
3)
(26)
In this new system of bases the components of stress strainand elastic tensors respectively are defined as
120590 = 119878119860E
119860
120598 = 119864119860E
119860
C = 119862119860119861
119890119860otimes 119890
119861
(27)
where all the implicit sums concerning the indices 119860 119861
range from 1 to 9 and 119878 is the compliance tensorThus for Hookersquos law and for eigenstiffness-eigenstrain
equations one can have the following equivalences
120590119894119895= 119862
119894119895119896119897120598119896119897
lArrrArr 119878119860= 119862
119860119861E119861
119862119894119895119896119897
120598119896119897
= 120582120598119894119895lArrrArr 119862
119860119861E119861= 120582119864
119860
(28)
Using elementary algebra we can have the components of thestress and strain tensors in two bases
((((((((
(
1198781
1198782
1198783
1198784
1198785
1198786
1198787
1198788
1198789
))))))))
)
=
((((((((
(
12059011
12059022
12059033
120590(12)
120590(23)
120590(31)
120590[12]
120590[23]
120590[31]
))))))))
)
((((((((
(
1198641
1198642
1198643
1198644
1198645
1198646
1198647
1198648
1198649
))))))))
)
=
((((((((
(
12059811
12059822
12059833
120598(12)
120598(23)
120598(31)
120598[12]
120598[23]
120598[31]
))))))))
)
(29)
where we have used the following notations
120579(119894119895)
equiv1
radic2(120579119894119895+ 120579
119895119894) 120579
[119894119895]equiv
1
radic2(120579119894119895minus 120579
119895119894) (30)
Now in the new basis 119864119860 the new components of stiffness
tensor C are
((((((((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986217
11986218
11986219
11986221
11986222
11986223
11986224
11986225
11986226
11986227
11986228
11986229
11986231
11986232
11986233
11986234
11986235
11986236
11986237
11986238
11986239
11986241
11986242
11986243
11986244
11986245
11986246
11986247
11986248
11986249
11986251
11986252
11986253
11986254
11986255
11986256
11986257
11986258
11986259
11986261
11986262
11986263
11986264
11986265
11986266
11986267
11986268
11986269
11986271
11986272
11986273
11986274
11986275
11986276
11986277
11986278
11986279
11986281
11986282
11986283
11986284
11986285
11986286
11986287
11986288
11986289
11986291
11986292
11986293
11986294
11986295
11986296
11986297
11986298
11986299
))))))))
)
=
1198621111
1198621122
1198621133
1198622211
1198622222
1198623333
1198623311
1198623322
1198623333
11986211(12)
11986211(23)
11986211(31)
11986222(12)
11986222(23)
11986222(31)
11986233(12)
11986233(23)
11986233(31)
11986211[12]
11986211[23]
11986211[31]
11986222[12]
11986222[23]
11986222[31]
11986233[12]
11986233[23]
11986233[31]
119862(12)11
119862(12)22
119862(12)33
119862(23)11
119862(23)22
119862(23)33
119862(31)11
119862(31)22
119862(31)33
119862(1212)
119862(1223)
119862(1231)
119862(2312)
119862(2323)
119862(2331)
119862(3112)
119862(3123)
119862(3131)
119862(12)[12]
119862(12)[23]
119862(12)[31]
119862(23)[12]
119862(23)[23]
119862(23)[31]
119862(31)[12]
119862(31)[23]
119862(31)[31]
119862[12]11
119862[12]22
119862[12]33
119862[23]11
119862[23]22
119862[123]33
119862[21]11
119862[31]22
119862[31]33
119862[12](12)
119862[12](23)
119862[12](21)
119862[23](12)
119862[23](23)
119862[23](21)
119862[31](12)
119862[31](23)
119862[31](21)
119862[1212]
119862[1223]
119862[1231]
119862[2312]
119862[2323]
119862[2331]
119862[3112]
119862[3123]
119862[3131]
(31)
where
119862119894119895(119896119897)
equiv1
radic2(119862
119894119895119896119897+ 119862
119894119895119897119896) 119862
119894119895[119896119897]equiv
1
radic2(119862
119894119895119896119897minus 119862
119894119895119897119896)
119862(119894119895)119896119897
equiv1
radic2(119862
119894119895119896119897+ 119862
119895119894119896119897) 119862
[119894119895]119896119897equiv
1
radic2(119862
119894119895119896119897minus 119862
119895119894119896119897)
119862(119894119895119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896+ 119862
119895119894119896119897+ 119862
119895119894119897119896)
119862(119894119895)[119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119894119895119897119896+ 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895](119896119897)
equiv1
2(119862
119894119895119896119897+ 119862
119894119895119897119896minus 119862
119895119894119896119897minus 119862
119895119894119897119896)
119862[119894119895119896119897]
equiv1
2(119862
119894119895119896119897minus 119862
119895119894119896119897minus 119862
119894119895119897119896+ 119862
119895119894119897119896)
(32)
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 15
12
34
56
ColumnRow
12
34
56
600005000040000300002000010000
Figure 12 Histogram showing average elasticity matrix of Ash
5
10
15
20
25
12
34
56
Column Row
12
34
56
Figure 13 Histogram showing average elasticity matrix of Beech
In case if we impose symmetries of stress and strain tensorslet us see what happens with generalized Hookersquos law (22)
In generalized Hookersquos law when the stress-strain sym-metries do not affect the stiffness tensor C the number ofcomponents of stiffness tensor in 3-dimensional space isequal to 3
4= 81 Now if we impose the symmetries of stress-
strain tensors that is 120590119894119895= 120590
119895119894and 120598
119896119897= 120598
119897119896 the 119862
119894119895119896119897will be
like 119862119894119895119896119897
= 119862119895119894119896119897
= 119862119894119895119897119896
Moreover imposing symmetrical connection that is
119862119894119895119896119897
= 119862119896119897119894119895
we would have only 21 significant componentsout of 81 Thus if we flatten the stiffness tensor under thenotion of Hookersquos law we will definitely have a 6 times 6 matrixhaving only 21 independent elastic coefficients instead of 9 times
9 matrix having 81 elastic coefficientsHere we depict a figure (see Figure 1) which delineates
the component reduction process for the stiffness tensor
Now with 21 significant independent components thestiffness tensor 119862
119894119895119896119897can be mapped on a symmetric 6 times 6
matrixAs the elasticity of a material is described by a fourth-
order tensor with 21 independent components as shownin (Figure 1) and the mathematical description of elasticitytensor appears in Hookersquos law now how to map this 3-dimensional fourth order tensor on a 6 times 6 matrix
We are fortunate to have a long series of research papersconcerning this issue For instance [2 3 32ndash39] and manymore
The very first approach to map 21 significant componentsof the elasticity tensor on a symmetric 6 times 6 matrix wasintroduced by Voigt [32] and after that various successors ofVoigt have used his fabulous notions for flattening of fourth-order tensors But Lord Kelvin [40] found the Voigt notationsare inadequate from the perspective of tensorial nature andthen introduced his own advanced notations now known asKelvinrsquos mapping More recently [39] have introduced somesophisticated methodology to unfold a fourth-rank tensorcalled ldquoMCNrdquo (Mehrabadi and Cowinrsquos notations)
Let us briefly go through these three notions of tensorflattening one by one
31 The Voigt Six-Dimensional Notations for Unfolding anElasticity Tensor It is well known that the Voigt mappingpreserves the elastic energy density of thematerial and elasticstiffness and is given by
2 sdot 119864energy = 120590119894119895120598119894119895= 120590
119901120598119901 (33)
TheVoigtmapping receives this relation by themapping rules
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(34)
Using this rule we have 120590119894119895= 120590
119901 119862
119894119895119896119897= 119862
119901119902 and 120598
119902= (2 minus
120575119896119897)120598119896119897
This Voigt mapping can be visualized as shown in Table 1Thus in accordance with Voigtrsquos mappings Hookersquos law
(22) can be represented in matrix form as follows
(
(
1205901
1205902
1205903
1205904
1205905
1205906
)
)
= (
(
11986211
11986212
11986213
11986214
11986215
11986216
11986221
11986222
11986223
11986224
11986225
11986226
11986231
11986232
11986233
11986234
11986235
11986236
11986241
11986242
11986243
11986244
11986245
11986246
11986251
11986252
11986253
11986254
11986255
11986256
11986261
11986262
11986263
11986264
11986265
11986266
)
)
(
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(35)
where the simple index conversion rule of Voigt (see Table 2)is applied
But in the Voigt notations many disadvantages werenoticed For instance
(1) the 120590119894119895and 120598
119896119897are treated differently
(2) the norms of 120590119894119895 120598119896119897 and 119862
119894119895119896119897are not preserved
(3) the entries in all the three Voigt arrays (see (35)) arenot the tensor or the vector components and thus
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
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DistributedSensor Networks
International Journal of
16 Chinese Journal of Engineering
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
(a)
01
02
03
04
05
06
07
08
09
01 02 03 04 05 06 07 08
(b)
Figure 14The graphics placed in left as well as right positions showing graphs of I and II eigenvalues of all 15 hardwood species against theirapparent densities
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 15 The graphics placed in left as well as right positions showing graphs of III and IV eigenvalues of all 15 hardwood species againsttheir apparent densities
one may not be able to have advantages of tensoralgebra like tensor law of transformation rotation ofcoordinate system and so forth
32 Kelvinrsquos Mapping Rules Likewise Voigtrsquos mapping rulesKelvinrsquosmapping rules also preserve the elastic energy densityof material under the following methodology
119901 = 119894120575119894119895+ (1 minus 120575
119894119895) (9 minus 119894 minus 119895)
119902 = 119896120575119896119897+ (1 minus 120575
119896119897) (9 minus 119896 minus 119897)
(36)
In this way
120590119901= (120575
119894119895+ radic2 (1 minus 120575
119894119895)) 120598
119894119895
120598119902= (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 120598
119896119897
119862119901119902
= (120575119894119895+ radic2 (1 minus 120575
119894119895)) (120575
119896119897+ radic2 (1 minus 120575
119896119897)) 119862
119894119895119896119897
(37)
Naturally Kelvinrsquos mapping preserves the norms of the threetensors and the stress and strain are treated identically Alsothe mappings of stress strain and elasticity tensors have allthe properties of tensor of first- and second-rank tensorsrespectively in 6-dimensional space
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
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Shock and Vibration
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Electrical and Computer Engineering
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 17
But the only disadvantage of this mapping is that thevalues of stiffness components are changed However thereis a simple tool for conversion between Voigtrsquos and Kelvinrsquosnotations For this purpose one just needed a single array (seeTable 3) Thus
120590kelvin
= 120590viogt
120585 120590voigt
=120590kelvin
120585
120598kelvin
= 120598voigt
120585 120598voigt
=120598kelvin
120585
(38)
33 Mehrabadi-Cowin Notations (MCN) To preserve thetensor properties of Hookersquos law during the unfolding offourth-order elasticity tensor [39] have introduced a newnotion which is nothing but a conversion of Voigtrsquos notationsinto Kelvin ones This conversion is done using the followingrelation
119862kelvin
=((
(
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
1 1 1 radic2 radic2 radic2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
radic2 radic2 radic2 2 2 2
))
)
119862voigt
(39)
Exploiting this new notion (35) becomes
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
= (
1198621111 1198621122 1198621133radic21198621123
radic21198621113radic21198621112
1198622211 1198622222 1198622333radic21198622223
radic21198622213radic21198622212
1198623311 1198623322 1198623333radic21198623323
radic21198623313radic21198623312
radic21198622311radic21198622322
radic21198622333 21198622323 21198622313 21198622312
radic21198621311radic21198621322
radic21198621333 21198621323 21198621313 21198621312
radic21198621211radic21198621222
radic21198621233 21198621223 21198621213 21198621212
)
lowast(
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(40)
Further to involve the features of tensor in the matrix rep-resentation (40) [39] have evoked that (40) can be rewrittenas
= C (41)
where the new six-dimensional stress and strain vectorsdenoted by and respectively are multiplied by the factors
radic2 Also C is a new six-by-six matrix [39] The matrix formof (40) is given as
(
(
12059011
12059022
12059033
radic212059023
radic212059013
radic212059012
)
)
=((
(
11986211
11986212
11986213
radic211986214
radic211986215
radic211986216
11986212
11986222
11986223
radic211986224
radic211986225
radic211986226
11986213
11986223
11986233
radic211986234
radic211986235
radic211986236
radic211986214
radic211986224
radic211986234
211986244
211986245
211986246
radic211986215
radic211986225
radic211986235
211986245
211986255
211986256
radic211986216
radic211986226
radic211986236
211986246
211986256
211986266
))
)
lowast (
(
12059811
12059822
12059833
radic212059823
radic212059813
radic212059812
)
)
(42)
The representation (42) is further produced in MCN patternlike below
(
(
1
2
3
4
5
6
)
)
=((
(
11986211
11986212
11986213
11986214
11986215
11986216
11986212
11986222
11986223
11986224
11986225
11986226
11986213
11986223
11986233
11986234
11986235
11986236
11986214
11986224
11986234
11986244
11986245
11986246
11986215
11986225
11986235
11986245
11986255
11986256
11986216
11986226
11986236
11986246
11986256
11986266
))
)
lowast (
(
1205981
1205982
1205983
1205984
1205985
1205986
)
)
(43)
To deal with all the above representations of a fourth-orderelasticity tensor as a six-by-six matrix a simple conversiontable has been proposed by [39] which is depicted as inTable 4
Even though the foregoing detailed description is inter-esting and important from the viewpoint of those who arebeginners in the field of mathematical elasticity the modernscenario of the literature of mathematical elasticity nowinvolves the assistance of various computer algebraic systems(CAS) like MATLAB [30] Maple [41] Mathematica [42]wxMaxima [43] and some peculiar packages like TensorToolbox [26ndash29] GRTensor [44] Ricci [45] and so forth tohandle particular problems of the scientific literature
However in the following section we shall delineate aCAS approach for Section 3
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
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RotatingMachinery
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
18 Chinese Journal of Engineering
4 Unfolding the Anisotropic Hookersquos Lawwith the Aid of lsquolsquoMathematicarsquorsquo
ldquoMathematicardquo is a general computing environment inti-mated with organizing algorithmic visualization and user-friendly interface capabilities Moreover many mathematicalalgorithms encapsulated by ldquoMathematicardquo make the compu-tation easy and fast [42 46]
A fabulous book entitled ldquoElasticity with Mathematicardquohas been produced by [31] This book is equipped with somegreat Mathematica packages namely VectorAnalysismDisplacementm IntegrateStrainm ParametricMeshm andTensor2Analysism
Overall the effort of [31] is greatly appreciated for pro-ducing amasterpiecewithMathematica packages notebooksand worked examples Moreover all the aforementionedpackages notebooks and worked examples are freely avail-able to readers and can be downloaded from the publisherrsquoswebsite httpwwwcambridgeorg9780521842013
We consider the following code that easily meets thepurpose discussed in Section 3
Here kindly note that only the input Mathematica codesare being exposed For considering the entire processingan Electronic (see Appendix A in Supplementary Materialavailable online at httpdxdoiorg1011552014487314) isbeing proposed to readers
First of all install the ldquoTensor2Analysisrdquo package inMathematica using the following command
In [1]= ltlt Tensor2AnalysislsquoWe have loaded the above package like Loading the
package ldquoTensor2Analysismrdquo by setting the following pathSetDirectory[CProgram FilesWolframResearch
Mathematica80AddOnsPackages]Next is concerning the creation of tensor The code in
Algorithm 1 generates the stress and strain tensors using theindex rules specified by [31]
Use theMakeTensor command to create tensorial expres-sions having components with respect to symmetric condi-tions This command combines all the previous commandsto do so (see Algorithm 2)
Now the next code deals with the index rules that is ableto handle Hookersquos tensor conversion from the fourth-orderto the second-order tensor or second-order Voigt form usingindexrule (see Algorithm 3)
Afterwards we have a crucial stage that concerns withthe transformation of Voigtrsquos notations into a fourth-ranktensor and vice versa This stage includes the codes likeHookeVto4 Hooke4toV Also the codes HookeVto2 andHook2toV transform the Voigt notations into the second-order tensor notations and vice versa Finally the code inAlgorithm 4 provides us a splendid form of fourth-orderelasticity tensor as a six-by-six matrix in MCN notations
We have presented here a minimal code of mathematicadeveloped by [31] However a numerical demonstration ofthis code has also been presented by [31] and the reader(s)interested in understanding this code in full are requested torefer to the website already mentioned above
Let us now proceed to Section 5 which in our viewis the soul of the present work In this section we shall
deal with the eigenvalues eigenvectors the nominal averageof eigenvectors and the average eigenvectors and so forthfor different species of softwoods hardwoods and somespecimen of cancellous bone
5 Analysis of Known Values of ElasticConstants of Woods and CancellousBone Using MAPLE
Of course ldquoMAPLErdquo is a sophisticated CAS and producedunder the results of over 30 years of cutting-edge researchand development which assists us in analyzing exploringvisualizing and manipulating almost every mathematicalproblem Having more than 500 functions this CAS offersbroadness depth and performance to handle every phe-nomenon of mathematics In a nutshell it is a CAS that offershigh performance mathematics capabilities with integratednumeric and symbolic computations
However in the present study we attempt to explorehow this CAS handles complicated analysis regarding elasticconstant data
The elastic constant data of our interest for hardwoodsand softwoods as calculated by Hearmon [47 48] aretabulated in Tables 5 and 6
For cancellous bone we have the elastic constants dataavailable for three specimens [2 11] These data are tabulatedin Table 7
Precisely speaking to meet the requirements of proposedresearch a MAPLE code consisting of almost 81 steps hasbeen developed This MAPLE code (in its minimal form) isappended to Appendix A while its entire working mecha-nism is included in an electronic appendix (Appendix B) andcan be accessed from httpdxdoiorg1011552014487314Particularly as far as our intention concerns analysis ofelastic constants data regarding hardwoods softwoods andcancellous bone using MAPLE is consisting of the followingsteps
(i) Computating the eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computing the nominal averages of eigenvectors andthe average eigenvectors for all the 15 hardwoodspecies
(iii) Computing the average eigenvalues for hardwoodspecies
(iv) Computing the average elasticity matrices for all the15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphs for I II III IV V and VI eigen-values of 15 hardwood species against their apparentdensities (see Figures 14ndash16 also see Figure 17 for acombined view)
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 19
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(a)
01 02 03 04 05 06 07 08
05
1
15
2
25
3
(b)
Figure 16 The graphics placed in left as well as right positions showing graphs of V and VI eigenvalues of all 15 hardwood species againsttheir apparent densities
01 02 03 04 05 06 07 08
2
4
6
8
10
12
14
16
Figure 17 A combined view of Figures 14 15 and 16
51 Computing the Eigenvalues and Eigenvectors for Hard-woods Species Softwood Species and Cancellous Bone UsingMAPLE Here we present the outcomes of MAPLE code(which is mentioned in Appendix A) regarding the computa-tion of eigenvalues and eigenvectors of the stiffness matricesC concerning 15 hardwood species (see Table 5)
It is well known that the eigenvalues and eigenvectors of astiffness matrix C or its compliance matrix S are determinedfrom the equations [3]
(C minus ΛI) N = 0 or (S minus1
ΛI) N = 0 (44)
where I is the 6 times 6 identity matrix and N representsthe normalized eigenvectors of C or S Naturally C or Sbeing positive definite therefore there would be six positiveeigenvalues and these values are known as Kelvinrsquos moduliThese Kelvinrsquos moduli are denoted by Λ
119894 119894 = 1 2 6 and
if possible are ordered in such a way that Λ1ge Λ
2ge sdot sdot sdot ge
Λ6gt 0 Also the eigenvalues of S can simply de computed by
taking the inverse of the eigenvalues of CA MAPLE code mentioned in Step 16 of Appendix A
enables us to simultaneously compute the eigenvalues andeigenvectors for all the 15 hardwood species Also the resultsof this MAPLE code are summarized in Tables 8 and 9Further by simply substituting the elasticity constants fromTable 6 in Step 2 to Step 11 of Appendix A and performingStep 16 again one can have the eigenvalues and eigenvectorsfor the 8 softwood speciesThe computed results for softwoodspecies are summarized in Table 10 Similarly substitution ofelastic constants data for cancellous bone from Table 7 intoStep 2 to Step 11 and repeating Step 16 of the MAPLE codeyields the desired results tabulated in Table 11
52 Computing the Average Elasticity Tensors for Woodsand Cancellous Bone Using MAPLE In order to computeaverage elasticity tensors for the hardwoods softwoods andcancellous bone let us go through a brief mathematicaldescription regarding this issue [3] as this notion helpsus developing an appropriate MAPLE code to have desiredresults
Suppose we have 119872 measurements of the elasticconstants data for some particular species or materialC119868 C119868119868 C119872 and we want to construct a tensor CAVG
representing an average elasticity tensor for that particular
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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International Journal of
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DistributedSensor Networks
International Journal of
20 Chinese Journal of Engineering
material Then to do so we need the following key algebra[3]
(i) The eigenvalues Λ119884119896 119896 = 1 2 6 119884 = 1 2 119872
and the eigenvectors N119884119896 119896 = 1 2 6 119884 =
1 2 119872 for each of the 119884th measurement of theparticular species or material
(ii) The nominal average (NA) NNA119896
of the eigenvectorsN119884119896associated with the particular species The nomi-
nal average is given by
NNA119896
equiv1
119872
119872
sum
119884=1
N119884119896 (45)
(iii) The average NAVG119896
which is obtained by the followingformula
NAVG119896
equiv JNANNA119896
(46)
where JNA stands for the inverse square root of thepositive definite tensor | sum6
119896=1NNA119896
otimes NNA119896
| that is
JNA equiv (
6
sum
119896=1
NNA119896
otimes NNA119896
)
minus12
(47)
(iv) Now we proceed to average the eigenvalue Forthis we need to transform each set of eigenvaluescorresponding to each eigenvector to the averageeigenvector NAVG
119896 that is
ΛAVG119902
equiv1
119872
119872
sum
119884=1
6
sum
119896=1
Λ119884
119896(N119884
119896sdot NAVG
119902)2
(48)
It is obvious that ΛAVG119896
gt 0 for all 119896 = 1 2 6
and ΛAVG119896
= ΛNA119896 where Λ
NA119896
is the nominal averageof eigenvalues of material and is defined by
ΛNA119896
equiv1
119872
119872
sum
119884=1
Λ119884
119896 (49)
(v) Eventually we can now calculate the average elasticitymatrix CAVG in such a way that
CAVG=
6
sum
119896=1
ΛAVG119896
NAVG119896
otimes NAVG119896
(50)
Taking care of the above outlined steps we have developedsome MAPLE codes (See Step 17 to Step 81 of AppendixA) that enable us to calculate the nominal averages averageeigenvectors average eigenvalues and the average elasticitymatrices for each of the 15 hardwood species tabulated inTable 5 In addition to this Appendix A also retains few stepsfor example Steps 21 26 31 36 41 46 51 56 61 66 73 and80 which are particularly developed to plot histograms ofaverage elasticity matrices of each of the hardwood species aswell as graphs between I II III IV V and VI eigenvalues ofall hardwood species and the apparent densities (see Table 5)
We summarize the above claimed consequences of eachof the hardwood species one by one ss follows
521 MAPLE Evaluations for Quipo
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 1 and 2 of Table 5) of Quipoare
[[[[[[[
[
00367834559700000
00453681054800000
0998185540700000
00
00
00
]]]]]]]
]
[[[[[[[
[
0983745905000000
minus0170879918750000
minus00277901141300000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0169284222800000
minus0982986098000000
00509905132200000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(51)
(2) The average eigenvectors for Quipo are
[[[[[[[
[
003679134179
004537783172
09983995367
00
00
00
]]]]]]]
]
[[[[[[[
[
09859858256
minus01712690003
minus002785339027
00
00
00
]]]]]]]
]
[[[[[[[
[
minus01697052873
minus09854311019
005111734310
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(52)
(3) The averages eigenvalue for the twomeasurements (Snos 1 and 2 of Table 5) of Quipo is 3339500000
(4) The average elasticity matrix for Quipo is
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 21
[[[
[
334725276837330 0000112116752981933 000198542359441849 00 00 00
0000112116752981933 334773746006714 minus0000991802322788501 00 00 00
000198542359441849 minus0000991802322788501 334013593769726 00 00 00
00 00 00 333950000000000 00 00
00 00 00 00 333950000000000 00
00 00 00 00 00 333950000000000
]]]
]
(53)
(5) The histogram of the average elasticity matrix forQuipo is shown in Figure 2
522 MAPLE Evaluations for White
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 3 of Table 5) of White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048796003
minus04255671399
minus0009247686096
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(54)
(2) The average eigenvectors for White are
[[[[[[[
[
004028666644
006399313350
09971368323
00
00
00
]]]]]]]
]
[[[[[[[
[
09048795994
minus04255671395
minus0009247686087
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04237568827
minus09026613361
007505076253
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(55)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 3 of Table 5) of White is 1458799998
(4) The average elasticity matrix for White is given as
[[[
[
1458799998 000000001785571215 minus000000001009489597 00 00 00000000001785571215 1458799997 0000000002407020197 00 00 00minus000000001009489597 0000000002407020197 1458799997 00 00 00
00 00 00 1458799998 00 0000 00 00 00 1458799998 0000 00 00 00 00 1458799998
]]]
]
(56)
(5) The histogram of the average elasticity matrix forWhite is shown in Figure 3
523 MAPLE Evaluations for Khaya
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 4 of Table 5) of Khaya are
[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]
]
[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]
]
[[[[[
[
minus03721447810
minus09240829102
008705767064
00
00
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
(57)
(2) The average eigenvectors for Khaya are
[[[[[[[[[
[
005368516527
007220768570
09959437502
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
09266208315
minus03753089731
minus002273783078
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
minus03721447806
minus09240829093
008705767055
00
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]
]
[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]
]
(58)
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
22 Chinese Journal of Engineering
(3) Theaverage eigenvalue for the singlemeasurement (Sno 4 of Table 5) of Khaya is 1615299998
(4) The average elasticity matrix for Khaya is given as
[[[
[
1615299998 0000000005508173090 minus0000000008722620038 00 00 00
0000000005508173090 1615299996 minus0000000004345156817 00 00 00
minus0000000008722620038 minus0000000004345156817 1615300000 00 00 00
00 00 00 1615299998 00 00
00 00 00 00 1615299998 00
00 00 00 00 00 1615299998
]]]
]
(59)
(5) The histogram of the average elasticity matrix forKhaya is shown in Figure 4
524 MAPLE Evaluations for Mahogany
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 5 and 6 of Table 5) ofMahogany are
[[[[[[[
[
minus00598757013800000
minus00768229223150000
minus0995231841750000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0869358919900000
0493939153600000
00141567750950000
00
00
00
]]]]]]]
]
[[[[[[[
[
minus0490490276500000
minus0866078136900000
00963811082600000
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(60)
(2) The average eigenvectors for Mahogany are
[[[[[[[[[[[
[
minus005987730132
minus007682497510
minus09952584354
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
minus08693926232
04939583026
001415732393
00
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
10
00
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
10
00
]]]]]]]]]]]
]
[[[[[[[[[[[
[
00
00
00
00
00
10
]]]]]]]]]]]
]
(61)
(3) The average eigenvalue for the two measurements (Snos 5 and 6 of Table 5) of Mahogany is 1819400002
(4) The average elasticity matrix forMahogany is given as
[[[
[
1819451173 minus00001438038661 00001358556898 00 00 00minus00001438038661 1819484018 minus00004764690574 00 00 0000001358556898 minus00004764690574 1819454255 00 00 00
00 00 00 1819400002 1819400002 0000 00 00 00 00 0000 00 00 00 00 1819400002
]]]
]
(62)
(5) The histogram of the average elasticity matrix forMahogany is shown in Figure 5
525 MAPLE Evaluations for S Germ
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 7 of Table 5) of S Gem are
[[[[[[[
[
004967528691
008463937788
09951726186
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715971
minus04006166011
minus001165943224
00
00
00
]]]]]]]
]
[[[[[
[
minus03976958243
minus09123280736
009744493938
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(63)
(2) The average eigenvectors for S Germ are
[[[[[[[
[
004967528696
008463937796
09951726196
00
00
00
]]]]]]]
]
[[[[[[[
[
09161715980
minus04006166015
minus001165943225
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 23
[[[[[[[
[
minus03976958247
minus09123280745
009744493948
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(64)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 7 of Table 5) of S Germ is 1922399998
(4) The average elasticity matrix for S Germ is givenas
[[[
[
1922399998 minus000000001176508811 minus000000001537920006 00 00 00
minus000000001176508811 1922400000 minus0000000007631928036 00 00 00
minus000000001537920006 minus0000000007631928036 1922400000 00 00 00
00 00 00 1922399998 1922399998 00
00 00 00 00 00 00
00 00 00 00 00 1922399998
]]]
]
(65)
(5) The histogram of the average elasticity matrix for SGerm is shown in Figure 6
526 MAPLE Evaluations for maple
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 8 of Table 5) of maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533797
minus02031928413
minus09692575344
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(66)
(2) The average eigenvectors for maple are
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
[[[[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
(67)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 8 of Table 5) of maple is 2074600000
(4) The average elasticity matrix for maple is given as
[[[
[
2074599999 0000000004356660361 000000003402343994 00 00 00
0000000004356660361 2074600004 000000002232269567 00 00 00
000000003402343994 000000002232269567 2074600000 00 00 00
00 00 00 2074600000 2074600000 00
00 00 00 00 00 00
00 00 00 00 00 2074600000
]]]
]
(68)
(5) The histogram of the average elasticity matrix forMAPLE is shown in Figure 7
527 MAPLE Evaluations for Walnut
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 9 of Table 5) of Walnut are
[[[[[[[
[
008621257414
01243595697
09884847438
00
00
00
]]]]]]]
]
[[[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]]]
]
[[[[[
[
minus04753471023
minus08668282006
01505124700
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(69)
(2) The average eigenvectors for Walnut are
[[[[[
[
008621257423
01243595698
09884847448
00
00
00
]]]]]
]
[[[[[
[
08755641188
minus04828494241
minus001561753067
00
00
00
]]]]]
]
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
24 Chinese Journal of Engineering
[[[[[[[
[
minus04753471018
minus08668281997
01505124698
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(70)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 9 of Table 5) of walnut is 1890099997
(4) The average elasticity matrix for Walnut is givenas
[[[
[
1890099999 000000001209663993 minus000000002532733996 00 00 00000000001209663993 1890099991 0000000001701090138 00 00 00minus000000002532733996 0000000001701090138 1890100001 00 00 00
00 00 00 1890099997 1890099997 0000 00 00 00 00 0000 00 00 00 00 1890099997
]]]
]
(71)
(5) The histogram of the average elasticity matrix forWalnut is shown in Figure 8
528 MAPLE Evaluations for Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 10 of Table 5) of Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614623
minus02236797319
minus03328645006
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981414
minus07403833993
06692812840
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(72)
(2) The average eigenvectors for Birch are
[[[[[[[
[
03961520086
06338768023
06642768888
00
00
00
]]]]]]]
]
[[[[[[[
[
09160614614
minus02236797317
minus03328645003
00
00
00
]]]]]]]
]
[[[[[[[
[
006240981426
minus07403834008
06692812853
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(73)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 10 of Table 5) of Birch is 8772299998
(4) The average elasticity matrix for Birch is given as
[[[
[
8772299997 minus000000003535236899 000000003421196978 00 00 00minus000000003535236899 8772300024 minus000000001649192439 00 00 00000000003421196978 minus000000001649192439 8772299994 00 00 00
00 00 00 8772299998 8772299998 0000 00 00 00 00 0000 00 00 00 00 8772299998
]]]
]
(74)
(5) The histogram of the average elasticity matrix forBirch is shown in Figure 9
529 MAPLE Evaluations for Y Birch
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 11 of Table 5) of Y Birch are
[[[[[[[
[
minus006591789198
minus008975775227
minus09937798435
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08289381135
05593224991
0004466103673
00
00
00
]]]]]]]
]
[[[[[
[
minus05554425577
minus08240763851
01112729817
00
00
00
]]]]]
]
[[[[[
[
00
00
00
10
00
00
]]]]]
]
[[[[[
[
00
00
00
00
10
00
]]]]]
]
[[[[[
[
00
00
00
00
00
10
]]]]]
]
(75)(2) The average eigenvectors for Y Birch are
[[[[[[[
[
minus01387533798
minus02031928415
minus09692575354
00
00
00
]]]]]]]
]
[[[[[[[
[
minus08476774112
05304148448
001015375725
00
00
00
]]]]]]]
]
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 25
[[[[
[
minus05120454135
minus08230265872
02458388325
00
00
00
]]]]
]
[[[[
[
00
00
00
10
00
00
]]]]
]
[[[[
[
00
00
00
00
10
00
]]]]
]
[[[[
[
00
00
00
00
00
10
]]]]
]
(76)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 11 of Table 5) of Y Birch is 2228057495
(4) The average elasticity matrix for Y Birch is given as
[[[
[
2228057494 0000000004678921127 000000003654014285 00 00 000000000004678921127 2228057499 000000002397389829 00 00 00000000003654014285 000000002397389829 2228057495 00 00 00
00 00 00 2228057495 2228057495 0000 00 00 00 00 0000 00 00 00 00 2228057495
]]]
]
(77)
(5) The histogram of the average elasticity matrix for YBirch is shown in Figure 10
5210 MAPLE Evaluations for Oak
(1) The nominal averages of eigenvectors for the singlemeasurement (S no 12 of Table 5) of Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429180
minus09017531389
01264472547
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(78)
(2) The average eigenvectors for Oak are
[[[[[[[
[
006975010637
01071019008
09917984199
00
00
00
]]]]]]]
]
[[[[[[[
[
09079000793
minus04187725641
minus001862756541
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04133429184
minus09017531398
01264472548
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(79)
(3) Theaverage eigenvalue for the singlemeasurement (Sno 12 of Table 5) of Oak is 2598699998
(4) The average elasticity matrix for Oak is given as
[[[
[
2598699997 minus000000001889254939 minus0000000003638179937 00 00 00minus000000001889254939 2598700006 0000000008575709980 00 00 00minus0000000003638179937 0000000008575709980 2598699998 00 00 00
00 00 00 2598699998 2598699998 0000 00 00 00 00 0000 00 00 00 00 2598699998
]]]
]
(80)
(5) Thehistogram of the average elasticity matrix for Oakis shown in Figure 11
5211 MAPLE Evaluations for Ash
(1) The nominal averages of eigenvectors for the twomeasurements (S nos 13 and 14 of Table 5) of Ash are
[[[[[
[
minus00192522847250000
minus00192842313600000
000386151499999998
00
00
00
]]]]]
]
[[[[[
[
000961799224999998
00165029120500000
000436480214750000
00
00
00
]]]]]
]
[[[[[[
[
minus0494444050150000
minus0856906622200000
0142104790200000
00
00
00
]]]]]]
]
[[[[[[
[
00
00
00
10
00
00
]]]]]]
]
[[[[[[
[
00
00
00
00
10
00
]]]]]]
]
[[[[[[
[
00
00
00
00
00
10
]]]]]]
]
(81)
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
26 Chinese Journal of Engineering
(2) The average eigenvectors for Ash are
[[[[[[[[[[
[
minus2541745843
minus2545963537
5098090872
00
00
00
]]]]]]]]]]
]
[[[[[[[[[[
[
2505315863
4298714977
1136953303
00
00
00
]]]]]]]]]]
]
[[[[[[[
[
minus04949599718
minus08578007511
01422530677
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(82)
(3) The average eigenvalue for the two measurements (Snos 13 and 14 of Table 5) of Ash is 2477950014
(4) The average elasticity matrix for Ash is given as
[[[
[
315679169478799 427324383657779 384557503470365 00 00 00427324383657779 618700481877479 889153535276175 00 00 00384557503470365 889153535276175 384768762708609 00 00 00
00 00 00 247795001400000 00 0000 00 00 00 247795001400000 0000 00 00 00 00 247795001400000
]]]
]
(83)
(5) The histogram of the average elasticity matrix for Ashis shown in Figure 12
5212 MAPLE Evaluations for Beech
(1) The nominal averages of eigenvectors for the singlemeasurements (S no 15 of Table 5) of Beech are
[[[[[[[
[
01135176976
01764907831
09777344911
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(84)
(2) The average eigenvectors for Beech are
[[[[[[[
[
01135176977
01764907833
09777344921
00
00
00
]]]]]]]
]
[[[[[[[
[
08848520771
minus04654963442
minus001870707734
00
00
00
]]]]]]]
]
[[[[[[[
[
minus04518302069
minus08672739791
02090103088
00
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
10
00
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
10
00
]]]]]]]
]
[[[[[[[
[
00
00
00
00
00
10
]]]]]]]
]
(85)
(3) The average eigenvalue for the single measurements(S no 15 of Table 5) of Beech is 2663699998
(4) The average elasticity matrix for Beech is given as
[[[
[
2663700003 000000004768023095 000000003169803038 00 00 00000000004768023095 2663699992 0000000008310743498 00 00 00000000003169803038 0000000008310743498 2663700001 00 00 00
00 00 00 2663699998 2663699998 0000 00 00 00 00 0000 00 00 00 00 2663699998
]]]
]
(86)
(5) The histogram of the average elasticity matrix forBeech is shown in Figure 13
53 MAPLE Plotted Graphics for I II III IV V and VIEigenvalues of 15 Hardwood Species against Their ApparentDensities This subsection is dedicated to the expositionof the extreme quality of MAPLE which can enable one
to simultaneously sketch up graphics for I II III IV Vand VI eigenvalues of each of the 15 hardwood species(See Tables 8 and 9) against the corresponding apparentdensities 120588 (See Table 5 for apparent density) The MAPLEcode regarding this issue has been mentioned in Step 81 ofAppendix A
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 27
6 Conclusions
In this paper we have tried to develop a CAS environmentthat suits best to numerically analyze the theory of anisotropicHookersquos law for different species ofwood and cancellous boneParticularly the two fabulous CAS namely Mathematicaand MAPLE have been used in relation to our proposedresearch work More precisely a Mathematica package ldquoTen-sor2Analysismrdquo has been employed to rectify the issueregarding unfolding the tensorial form of anisotropicHookersquoslaw whereas a MAPLE code consisting of almost 81 steps hasbeen developed to dig out the following numerical analysis
(i) Computation of eigenvalues and eigenvectors for allthe 15 hardwood species 8 softwood species and 3specimens of cancellous bone using MAPLE
(ii) Computation of the nominal averages of eigenvectorsand the average eigenvectors for all the 15 hardwoodspecies
(iii) Computation of the average eigenvalues for all 15hardwood species
(iv) Computation of the average elasticity matrices for allthe 15 hardwood species
(v) Plotting the histograms for elasticity matrices of allthe 15 hardwood species
(vi) Plotting the graphics for I II III IV V and VIeigenvalues of 15 hardwood species against theirapparent densities
It is also claimed that the developedMAPLE code cannotonly simultaneously tackle the 6 times 6 matrices relevant to thedifferent wood species and cancellous bone specimen butalso bears the capacity of handling 100 times 100 matrix whoseentries vary with respect to some parameter Thus from theviewpoint of author the MAPLE code developed by him isof great interest in those fields where higher order matrixalgebra is required
Disclosure
This is to bring in the kind knowledge of Editor(s) andReader(s) that due to the excessive length of present researchwe have ceased some computations concerning cancellousbone and softwood species These remaining computationswill soon be presented in the next series of this paper
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author expresses his hearty thanks and gratefulnessto all those scientists whose masterpieces have been con-sulted during the preparation of the present research articleSpecial thanks are due to the developer(s) of Mathematicaand MAPLE who have developed such versatile Computer
Algebraic Systems and provided 24-hour online support forall the scientific community
References
[1] S C Cowin and S B Doty Tissue Mechanics Springer NewYork NY USA 2007
[2] G Yang J Kabel B Van Rietbergen A Odgaard R Huiskesand S C Cowin ldquoAnisotropic Hookersquos law for cancellous boneand woodrdquo Journal of Elasticity vol 53 no 2 pp 125ndash146 1998
[3] S C Cowin andGYang ldquoAveraging anisotropic elastic constantdatardquo Journal of Elasticity vol 46 no 2 pp 151ndash180 1997
[4] Y J Yoon and S C Cowin ldquoEstimation of the effectivetransversely isotropic elastic constants of a material fromknown values of the materialrsquos orthotropic elastic constantsrdquoBiomechan Model Mechanobiol vol 1 pp 83ndash93 2002
[5] C Wang W Zhang and G S Kassab ldquoThe validation ofa generalized Hookersquos law for coronary arteriesrdquo AmericanJournal of Physiology Heart andCirculatory Physiology vol 294no 1 pp H66ndashH73 2008
[6] J Kabel B Van Rietbergen A Odgaard and R Huiskes ldquoCon-stitutive relationships of fabric density and elastic properties incancellous bone architecturerdquo Bone vol 25 no 4 pp 481ndash4861999
[7] C Dinckal ldquoAnalysis of elastic anisotropy of wood materialfor engineering applicationsrdquo Journal of Innovative Research inEngineering and Science vol 2 no 2 pp 67ndash80 2011
[8] Y P Arramon M M Mehrabadi D W Martin and SC Cowin ldquoA multidimensional anisotropic strength criterionbased on Kelvin modesrdquo International Journal of Solids andStructures vol 37 no 21 pp 2915ndash2935 2000
[9] P J Basser and S Pajevic ldquoSpectral decomposition of a 4th-order covariance tensor applications to diffusion tensor MRIrdquoSignal Processing vol 87 no 2 pp 220ndash236 2007
[10] P J Basser and S Pajevic ldquoA normal distribution for tensor-valued random variables applications to diffusion tensor MRIrdquoIEEE Transactions on Medical Imaging vol 22 no 7 pp 785ndash794 2003
[11] B Van Rietbergen A Odgaard J Kabel and RHuiskes ldquoDirectmechanics assessment of elastic symmetries and properties oftrabecular bone architecturerdquo Journal of Biomechanics vol 29no 12 pp 1653ndash1657 1996
[12] B Van Rietbergen A Odgaard J Kabel and R HuiskesldquoRelationships between bone morphology and bone elasticproperties can be accurately quantified using high-resolutioncomputer reconstructionsrdquo Journal of Orthopaedic Researchvol 16 no 1 pp 23ndash28 1998
[13] H Follet F Peyrin E Vidal-Salle A Bonnassie C Rumelhartand P J Meunier ldquoIntrinsicmechanical properties of trabecularcalcaneus determined by finite-element models using 3D syn-chrotron microtomographyrdquo Journal of Biomechanics vol 40no 10 pp 2174ndash2183 2007
[14] D R Carte and W C Hayes ldquoThe compressive behavior ofbone as a two-phase porous structurerdquo Journal of Bone and JointSurgery A vol 59 no 7 pp 954ndash962 1977
[15] F Linde I Hvid and B Pongsoipetch ldquoEnergy absorptiveproperties of human trabecular bone specimens during axialcompressionrdquo Journal of Orthopaedic Research vol 7 no 3 pp432ndash439 1989
[16] J C Rice S C Cowin and J A Bowman ldquoOn the dependenceof the elasticity and strength of cancellous bone on apparentdensityrdquo Journal of Biomechanics vol 21 no 2 pp 155ndash168 1988
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
28 Chinese Journal of Engineering
[17] R W Goulet S A Goldstein M J Ciarelli J L Kuhn MB Brown and L A Feldkamp ldquoThe relationship between thestructural and orthogonal compressive properties of trabecularbonerdquo Journal of Biomechanics vol 27 no 4 pp 375ndash389 1994
[18] R Hodgskinson and J D Currey ldquoEffect of variation instructure on the Youngrsquos modulus of cancellous bone A com-parison of human and non-human materialrdquo Proceedings of theInstitution ofMechanical EngineersH vol 204 no 2 pp 115ndash1211990
[19] R Hodgskinson and J D Currey ldquoEffects of structural variationon Youngrsquos modulus of non-human cancellous bonerdquo Proceed-ings of the Institution of Mechanical Engineers H vol 204 no 1pp 43ndash52 1990
[20] B D Snyder E J Cheal J A Hipp and W C HayesldquoAnisotropic structure-property relations for trabecular bonerdquoTransactions of the Orthopedic Research Society vol 35 p 2651989
[21] C H Turner S C Cowin J Young Rho R B Ashman andJ C Rice ldquoThe fabric dependence of the orthotropic elasticconstants of cancellous bonerdquo Journal of Biomechanics vol 23no 6 pp 549ndash561 1990
[22] D H Laidlaw and J Weickert Visualization and Processingof Tensor Fields Advances and Perspectives Springer BerlinGermany 2009
[23] J T Browaeys and S Chevrot ldquoDecomposition of the elastictensor and geophysical applicationsrdquo Geophysical Journal Inter-national vol 159 no 2 pp 667ndash678 2004
[24] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover New York NY USA 4th edition 1944
[25] G Backus ldquoA geometric picture of anisotropic elastic tensorsrdquoReviews of Geophysics and Space Physics vol 8 no 3 pp 633ndash671 1970
[26] T G Kolda and B W Bader ldquoTensor decompositions andapplicationsrdquo SIAM Review vol 51 no 3 pp 455ndash500 2009
[27] B W Bader and T G Kolda ldquoAlgorithm 862 MATLAB tensorclasses for fast algorithm prototypingrdquo ACM Transactions onMathematical Software vol 32 no 4 pp 635ndash653 2006
[28] M D Daniel T G Kolda and W P Kegelmeyer ldquoMultilinearalgebra for analyzing data with multiple linkagesrdquo SANDIAReport Sandia National Laboratories Albuquerque NM USA2006
[29] W B Brett and T G Kolda ldquoEffcient MATLAB computationswith sparse and factored tensorsrdquo SANDIA Report SandiaNational Laboratories Albuquerque NM USA 2006
[30] R H Brian L L Ronald and M R Jonathan A Guideto MATLAB for Beginners and Experienced Users CambridgeUniversity Press Cambridge UK 2nd edition 2006
[31] A Constantinescu and A Korsunsky Elasticity with Mathe-matica An Introduction to Continuum Mechanics and LinearElasticity Cambridge University Press Cambridge UK 2007
[32] WVoigt LehrbuchDer Kristallphysik JohnsonReprint LeipzigGermany
[33] J Dellinger D Vasicek and C Sondergeld ldquoKelvin notationfor stabilizing elastic-constant inversionrdquo Revue de lrsquoInstitutFrancais du Petrole vol 53 no 5 pp 709ndash719 1998
[34] B A AuldAcoustic Fields andWaves in Solids vol 1 JohnWileyamp Sons 1973
[35] S C Cowin and M M Mehrabadi ldquoOn the identification ofmaterial symmetry for anisotropic elastic materialsrdquo QuarterlyJournal of Mechanics and Applied Mathematics vol 40 pp 451ndash476 1987
[36] S C Cowin BoneMechanics CRC Press Boca Raton Fla USA1989
[37] S C Cowin and M M Mehrabadi ldquoAnisotropic symmetries oflinear elasticityrdquo Applied Mechanics Reviews vol 48 no 5 pp247ndash285 1995
[38] M M Mehrabadi S C Cowin and J Jaric ldquoSix-dimensionalorthogonal tensorrepresentation of the rotation about an axis inthree dimensionsrdquo International Journal of Solids and Structuresvol 32 no 3-4 pp 439ndash449 1995
[39] M M Mehrabadi and S C Cowin ldquoEigentensors of linearanisotropic elastic materialsrdquo Quarterly Journal of Mechanicsand Applied Mathematics vol 43 no 1 pp 15ndash41 1990
[40] W K Thomson ldquoElements of a mathematical theory of elastic-ityrdquo Philosophical Transactions of the Royal Society of Londonvol 166 pp 481ndash498 1856
[41] YW Frank Physics withMapleThe Computer Algebra Resourcefor Mathematical Methods in Physics Wiley-VCH 2005
[42] R HeikkiMathematica Navigator Mathematics Statistics andGraphics Academic Press 2009
[43] T Viktor ldquoTensor manipulation in GPL maximardquo httpgrten-sorphyqueensucaindexhtml
[44] httpgrtensorphyqueensucaindexhtml[45] J M Lee D Lear J Roth J Coskey Nave and L Ricci A
Mathematica Package For Doing Tensor Calculations in Differ-ential Geometry User Manual Department of MathematicsUniversity of Washington Seattle Wash USA Version 132
[46] httpwwwwolframcom[47] R F S Hearmon ldquoThe elastic constants of anisotropic materi-
alsrdquo Reviews ofModern Physics vol 18 no 3 pp 409ndash440 1946[48] R F S Hearmon ldquoThe elasticity of wood and plywoodrdquo Forest
Products Research Special Report 9 Department of Scientificand Industrial Research HMSO London UK 1948
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
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Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of