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J Elasticity (2006) 85: 120DOI 10.1007/s10659-006-9065-1

Revised Bounds on the Elastic Moduli of

Two-Dimensional Random Polycrystals

Duc Chinh Pham

Received: 26 December 2005 / Accepted: 15 March 2006 /

Published online: 24 May 2006 Springer Science+Business Media B.V. 2006

Abstract Our earlier derived bounds on the elastic moduli of two-dimensionalrandom polycrystals [1,2] involve a geometric restriction through an assumption onthe form of an isotropic eight-rank tensor. The general form of the tensor is usedin this study to reconstruct the bounds, which are expected to approach the scatterrange for the moduli of the irregular aggregate.

Mathematics Subject Classifications (2000) 74Q20

Key words boundselastic modulitwo-dimensional polycrystalsrandomaggregatestatistical isotropystatistical symmetry

1. Introduction

Macroscopic (effective) properties of randomly inhomogeneous materials, includingrandom polycrystals, depend upon the properties of their constituents as well asmaterial microstructure. Because of the irregular nature of the microgeometry of therandom aggregates, generally, there do not exist unique formulae for the effectiveproperties, which may scatter over some intervals. Hence, the purpose is to find upperand lower limits for those possible scatter intervals. General variational approachhas been developed to construct upper and lower bounds on the effective properties[114] (i.e. to approach the limits from outside). Alternatively, one could alsoapproach the scatter limits from inside by constructing realizable geometric models,e.g., as has been done in[15,16]. In[1,11,13] we developed a variational approach

based on minimum energy principles and used polarization trial fields, which are

D. C. Pham (B)Vietnamese Academy of Science and Technology, Vien Co hoc, 264 Doi Can, Hanoi, Vietname-mail: pdchinh@im01.ac.vn

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2 J Elasticity (2006) 85: 120

similar to those of Hashin and Shtrikman[4], to bound the effective properties of thecomposites, and the results agree well with those from the general perturbation seriesexpansion approach [510, 17]. The advantage of the former approach is its simplicityand effectiveness, especially in dealing with random polycrystals. The bounds on the

elastic moduli of two-dimensional random polycrystals derived in Pham [1]based onthe hypothesis that the shape and crystalline orientations of the constituent grains ina random aggregate are not correlated (consult also [4,68,14,18]). To evaluate thebounds in Pham [1,2] we suggested an assumption about an isotropic eighth-ranktensor. In this work, that geometric restriction is to be substituted by the generalform of the isotropic tensor. In consequence, we obtain the new, more-sophisticatedand less-restricted, but more general bounds on the macroscopic elastic moduli of theaggregate.

2. Formal Bounds

A representative volume element of the two-dimensional random polycrystal oc-cupies circular region V of Euclidean space R2, the center of which is also theorigin of the Cartesian system of coordinates {x1,x2}. The representative volumeelement V consists of N components occupying regions V V of volume v, =1, ...,N (the volume of V is assumed to be the unity); each component Vis composed of the grains of the same crystal orientation, the elastic property ofwhich is described by the fourth-rank tensor C(x)= C , x V . In particular, a

random uniphase polycrystal is supposed to be represented by such N-componentconfiguration when N , v =v0 =

1N

0, with the grains of all possible crystalorientations allowed to be presented equally. The contact between the grains isperfect (i.e., without any slip or cleavage during deformation of the aggregate). Theaverage value of tensor C(x) on V (or of C over all possible orientations ) isdefined as

C =

V

C(x)dx= C =

N=1

vC . (1)

The effective elastic moduli Ce of the polycrystal, which relate the average stressand strain tensors onV, can be determined by the minimum energy principle

0 :Ce : 0 = inf

=0

V

:C : dx , (2)

where the strain field (x)expressed through a displacement field w

ij= 1

2(wi,j+wj,i) , (3)

0 is a constant strain tensor; a Latin index after comma designates the differentiation

with respect to the respective Cartesian coordinate. Once V is assumed to bestatistically isotropic, the effective elastic tensorCe is a fourth-rank isotropic tensor

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J Elasticity (2006) 85: 120 3

and can be expressed through the effective elastic bulk ( Ke), shear (e) moduli, andthe isotropic fourth-rank tensor functionT(K, )

Ce =T(Ke, e) , (4)

Tijkl(k, )= ( K)ijkl+ (ikjl+iljk) , (5)

ij is usual Kronecker symbol. Instead of finding the exact solution of the difficultproblem (2) and (3) with irregular random geometry of V, one can substitute anappropriately constructed trial field into (2) to get upper bounds on the effectivemoduliKe, e that is the approach we follow. In the process we use the Green func-tion of the elastic equilibrium problem, which involves the harmonic and biharmonicpotentials

(x) = 12

V

ln| xy| dy , (x)= 18

V

|xy|2 ln| xy| dy,

2 (x) = 4 (x)= , x V . (6)

As the shape and crystalline orientations of the constituents in the random aggregateare supposed to be uncorrelated, statistically isotropic and symmetric hypotheses forthem can be adopted. Statistically isotropic distribution of the components Vin thecircleVimplies

1

v

V

,ijdx = 1

2 ij ,

1

v

V

,ijkldx = 1

8(ijkl+iljk+ ikjl) . (7)

Let us introduce microgeometrical parameters A ,B (,, =1, ...,N):

A =

V

ij

ij dx ,

ij =

,ij 1

v

V

,ijdx,

B =

V

ijkl

ijkldx,

ijkl=

,ijkl 1

v

V

,ijkldx. (8)

Clearly, for allx Vand all, =1, ...,N,

ijkk =

ij ,

ii =0 , (9)

where conventional summation on repeating Latin indices is assumed. Statisticallysymmetry hypotheses for the aggregate implies [1,12]( = = =)

A =vv v(f1 f2) , A =v(1v )[(1v)f1+ vf2],

A =vv [(v1)f1 vf2], A =v v [(1v )f2+ v f1] ;

B =v v v(g1 g2) , B =v (1v)[(1v )g1+ vg2] ,

B =v v [(v 1)g1 vg2] , B =v v [(1v )g2+ vg1]; (10)

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where f1, f2,g1,g2 are microgeometry-dependent positive parameters, specificallyf1/f2 =g1/g2 =0 for the subclass of circular cell materials. Referring to the uniphasepolycrystal, the statistical symmetry hypothesis reflects the observation that aninterchange of the materials between any two sets of the grains of different crystal

orientations (i.e., two components V and V ) within the random polycrystal (therepresentative volume element V) should not alter the overall characteristics of theaggregate. Circular cell polycrystals are idealistic polydispersed random aggregatescomposed exclusively of the grains of circular form. The symmetric cell materialshave been firstly considered by Miller [6], and Williemse and Caspers [8], whodefined them as: The space is completely covered by cells; cells are distributedin a manner such that the material is statistically homogeneous and isotropic; thematerial properties of a cell are statistically independent of those of any other cells;the material properties are statistically independent of the geometrical distribution(shape, size, orientation and arrangement) of cells. Examples include Voronoi or

Delaunay tessellations of the space from a set of center-points thrown randomly intothe material space, where the cells are assigned the crystal orientations randomly.The microgeometry-dependent parameters A ,B

, f1, f2,g1,g2correspond to the

respective ones of[6, 8] by the relations specified in [13, 19]. They contain three-pointcorrelation information about the microgeometry of the aggregate. The geometricparameters A ,B

defined in (8) are not independent. Here, in particular, we

define

Eijkl=ijklS(

ij k l+

k lij+

ik jl+

jl ik+

il jk+

jkil), (11)

whereS is a scalar parameter. From the obvious inequality

0

V

EijklEijkldx= B

A

12S(13S), (12)

taking S= 16

, we see B A . Since the positive geometric parameters f1,

f2,g1,g2in (10) describe a specific microstructure of the symmetric cell material anddo not depend on the volume fractionv , we deduce

g1 f1 , g2 f2 . (13)

We come back to the minimum energy principle (2). Following the procedure ofPham[1,11], we make use of Hashin and Shtrikman trial polarization field (K0and0 any positive values; Greek indices under the sign of sum run on natural numbersfrom 1 to N)

ij= K0

0(K0+ 0)

pk l,ijkl

1

20

(pm i,jm+ p

mj

,im) , (14)

where

p

= D

:(C0

+C

): 0

,D = E(C +C)1 :

v (C +C)1

1,

E = T(1

2,

1

2) , C0 =T(K0, 0) ,

C = T(K, ) , K =0 , = K00

K0+ 20, (15)

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J Elasticity (2006) 85: 120 5

taking into account the statistical isotropy hypothesis (7), to get from the energyprinciple (2)the general formal upper bounds on the effective moduli [1]

0 :Ce : 0 0 :PC(C,K0, 0):

0 + U() , (16)

where

PC(C,K0, 0) =

v (C +C)1

1C

= T

PK(C,K0, 0),P(C,K0, 0)

,

PK(C,K0, 0) =

(C+C)1iijj

1K ,

P(C,K0, 0) =

(C+C)1ijij 12(C+C)1iijj

1 , (17)

U() =

V

:(C C0): dx

= 1

20

,,

C0ijkl

2pm np

p q

V

ijmn

k l p qdx+ p

imp

k n

V

jm

ln dx

2pm npk p

V

ijmn lp dx

, (18)

=

1

v

V

dx , C0 =C C0 , = K0

K0+ 0.

Similarly, the formal lower bounds on the elastic moduli are constructed fromminimum complementary energy principle [1].

3. Evaluation of the Bounds

To construct the best possible upper bounds we need to evaluate U in (18), whichcontains integral expressions formed with the harmonic and biharmonic potentials,with the help of statistically isotropic and symmetry hypotheses. These integrals arecertain high rank tensors. According to statistical hypothesis about polycrystallinerandom microgeometry, such tensors are the isotropic ones. General forms of theisotropic tensors are given by the theory of algebraic invariants[2023]. Some low-rank isotropic tensors have already been given in (7). Consider the isotropic fourth-rank tensor

V

jm

ln dx, which is symmetric on indices jandm, and onland n. The

general form of that tensor, involving three isomers, is

V

jm

ln dx= K

1jmln + M

1(jlm n+ jnm l) , (19)

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whereK1 andM

1 are some scalar invariants, which can be related to the geometricparameters defined in(8), using (9):

0 = V

jj

ll dx= 4 K

1 +4M

1 ,

A =

V

jm

jm dx= 2 K

1 +6M

1 . (20)

Hence V

jm

ln dx= 1

4A (jlm n+ jnm l jmln) . (21)

Next, consider the isotropic sixth-rank tensorV

ijmn

lp dx, which is symmetric on

indicesi, j, m, n, and onl,p. The general form of that tensor, involving 15 isomers, is

V

ijmn

lp dx = K

2ijmnlp

+M

2 (iljm n p+ jlimn p+ m lijn p+ nlijm p), (22)

where

ijmn =ijm n+ imjn+ injm . (23)

The scalar invariants K2 and M

2 can be found with the help of (8)and (9):

0 =

V

iinn

ll dx= 16 K

2 +32M

2 ,

A =

V

ijnn

ij dx= 8 K

2 +40M

2 . (24)

Hence V

ijmn

lp dx = 1

12A ijmnlp

+ 1

24A (iljm n p+ jlimn p+ m lijn p+ nlijm p). (25)

Lastly, consider the isotropic eighth-rank tensorV

ijmn

k l p qdx, which is symmetricon indicesi, j, m, n, and on k, l,p, q. The general form of that tensor, involving 105isomers, is

V

ijmn

klpqdx= K

3ijmnk l p q+ M

3ijmn

k l p q+ L

3 ijmn

k l p q , (26)

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J Elasticity (2006) 85: 120 7

where

ijmn

k l p q = ikjm n

lp q +iljm n

k p q+ ipjm n

k lq +iqjm n

k lp ,

jm n

lp q = jlm np q +jp

m nl q +jq

m nlp ,

m np q = m p n q+ m qn p ,

ijmn

k l p q = ijm nk l p q+ im

jn

k l p q+ in

jm

k l p q+ jmink l p q+ jn

imklpq+ m n

ij

k l p q ,

m nklpq = 1

2(m n klpq +

m n lk p q+

m n p

k lq +m nq

k lp ) . (27)

We put the coefficient 1/2 in the last equality (27), because as one can check everyisomer inside the brackets appear twice. The form (26) and (27) is similar to that ofWeaver[25]. The scalar invariants K3, M

3 and L

3 can be found with the help of(8) and (9):

0=

V

iin n

kk p pdx= 64 K

3 +64M

3 +288L

3 ,

A =

V

ijnn

ijp pdx= 32 K

3 +80M

3 +300L

3 ,

B =

V

ijpq

ijpqdx= 24 K

3 +120M

3 +264L

3 . (28)

Hence

V

ijmn

k l p qdx =

5

624B

23

624A

ijmnk l p q

+

1

48B

1

48A

ijmn

klpq+

1

78A

1

156B

ijmn

k l p q .

(29)

In Pham [1, 24], to calculate bounds on the effective elastic moduli of thecomposites, we presumed a hypothesis about the isotropic eighth-rank tensor, whichis just a specific form of (26) without the last term involving L3 . That is equivalentto imposing the restriction B =2A

, which has been stated in the mentioned

references, hence should be valid only for the restricted microgeometries. Here wesee that the most general form of the isotropic tensor is (26), which leads to (29).

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8 J Elasticity (2006) 85: 120

Substituting (21),(25) and (29) into (18), we obtain

U = 1

20

,,

C0iikkp

jjp

ll

52624

B 232

624A

+ C0iik kp

jlp

jl

239

A 2

78B

+C0ijijp

k kp

ll

239

A 2

78B

+ C0ijijp

k lp

k l

212

B 2

12A

+C0iiklp

k lp

jj

6

52

52

A +

2

156B

+C0iiklp

k jp

l j

8239

6

A

42

39B

+C0ijkjp

ikp

ll

8239

6

A

42

39B

+C0ijkjp

ilp

k l1

4

6

52

39 A +

32

13B

+C0ijklp

ijp

k l

3

1

4

32

13

A +

32

26B

+C0ijklp

ikp

jl

14

3+

82

39

A

42

39B

. (30)

Making use of(10), one may transform (30) into

U = 1

20

0 :T

(K0+ K)2UK, (0+ )

2UM

: 0 , (31)

whereUK =UK f1 f1+ UK f2 f2+ UKg1 g1+ UKg2 g2 ,

UM=UM f1 f1+ UM f2 f2+ UMg1 g1+ UMg2 g2 , (32)

Particular expressions ofUK f1, UK f2, UKg1, UKg2, UM f1, UM f2, UMg1, UMg2are givenin Apendix [see (A1A7)].

For specific circular cell polycrystals f1 = g1 =0. Now if we take K0 = KV, 0 =V, then from (A6) we get also UK f2 =UKg2 =0, hence from (32) UK=0, andfrom(16), (17) and (31)the simple upper bound for Ke:

Ke Kuc = PK(C,KV, V) . (33)

Similarly, the upper bound on the shear modulus of circular cell polycrystals reads

e uc = P(C,KV, V) . (34)

For general shape-unspecified upper bounds, the positive geometric parametersf1,g1, f2,g2are not specified. Using restriction (13)and taking K0 KV, 0 V,with (A6)we get

UK f2 f2+ UKg2 g2 (UK f2+ UKg2) f2

=

D

jjp p D

llqq

(KVK0)

1

12

2

392

+(V0)

6

132

5

6+

1

2

+

Dd1

klp p Dd1

klqq

(KVK0)

6

132

5

6+

1

2

+(V0)

1

3

1

392

f2 0,

(35)

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J Elasticity (2006) 85: 120 9

keeping in mind that0 1 as can be seen from (18). Also,

UK f1 f1+ UKg1 g1 (UK f1+ UKg1) f1 if UKg1 0 . (36)

Hence, if we take any K0, 0such that

K0 KV , 0 V , UKg1 0 , UK f1+ UKg1 0 , (37)

then from (32)we get UK0, and from (16), (17) and (31)an upper bound KePK(C,K0, 0)(by neglecting the negative UKto strengthen the inequality). The bestpossible shape-unspecified upper bound (for all possible positive microgeometricparameters f1,g1, f2,g2) on the effective bulk modulus is

Ke K

u

= infK0,0{PK(C,K0, 0)| K0 KV, 0 V, UKg1 0, UK f1+ UKg1 0},

(38)

keeping in mind thatPKisdefinedin(17),KV, V in (A5), and UK f1, UK g1 in (A1)and (A2). Similarly, the best possible shape-unspecified upper bound on the effectiveshear modulus reads

e u = inf

K0,0{P(C,K0, 0)| K0 KV, 0 V, UMg1 0, UM f1+ UMg1 0}.

(39)

Analogously, the shape-unspecified lower bounds on the effective moduli are con-structed from the minimum complementary energy principle and have the forms

Ke Kl = sup

K0, 0

{PK(C,K0, 0)| K10 K

1R ,

10

1R ,

UKg1 0, UK f1+ UKg1 0},

e

l = supK0, 0

{P

(C

,K0

, 0

)|K

1

0

K

1

R , 1

0 1

R ,

UMg1 0, UM f1+ UMg1 0}, (40)

where

KR = [(C )1iijj]

1 , R = [(C )1ijij

1

2(C )1iijj]

1 (41)

are Reuss harmonic averages. UK f1, UKg1, UM f1, UMg1 have similar forms as those

of UK f1, UKg1, UM f1, UMg1, with C0 and D taking the places of C0 and D ,respectively,

C0

= C0 : [(C)1 (C0)1] :C0 ,

D

= C0 :

E [(C )1 +(C)1]1 : [(C )1 +(C)1]11

. (42)

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The lower bounds for the specific circular cell polycrystals are

Ke Klc = PK(C,KR, R) ,

e l

c = P(C,KR, R) . (43)

The simple bounds for specific circular cell polycrystals (33), (34) and (43) areidentical to those derived in Pham [1], however the general ones (38)(40) are not.

4. Calculation of the Bounds

As all the sums on repeating Latin indices involving Cijkl and Dijkl in our final

formulae (A1) and(A2),... are tensor invariants (scalars), they do not depend onspecific orientation ; hence all the constantsCijkland D

ijklthere can be taken in the

basic crystal reference and denoted as Cijkland Dijkl, respectively. In the literaturethe convenient two-index notationCijfor crystal elastic constants are used. A planarcrystal of general anisotropy has six independent elastic constants, which in two-indexnotation are given as C11, C22, C33, C12, C13, C23. The correspondence between theusual fourth-rank elasticity tensor and those in two-index notation is

C1111 =C11 , C2222 =C22 , C1212 =C33 ,

C1122 =C12 , C1112 =C13 , C2212 =C23 . (44)

Voight and Reuss averages are

KV= 1

4(C11+ C22+ 2C12) , V=

1

8(C11+ C22 2C12+ 4C33) ,

KR = C11C22C33+ 2C12C13C23 C

212C33 C

223C11 C

213C22

C11C33+ C22C33 C213 C

223+ 2C13C23 2C12C33

,

R = 2(C11C22C33+ 2C12C13C23 C

212C33 C

223C11 C

213C22)

C11C33+ C22C33 C213 C

223 2C13C23+ 2C12C33+ C11C22 C

212

.(45)

The property functions PKand Pfrom (17) take the particular forms

PK(C,K0, 0) = 0

+C+11C

+22C

+33 +2C

+12C13C23 (C

+12)

2C+33 C223C

+11 C

213C

+22

C+11C+33 + C

+22C

+33 C

213 C

223+ 2C13C23 2C

+12C

+33

,

P(C,K0, 0) =

+ 2[C+11C

+22C

+33+2C

+12C13C23(C

+12)

2C+33C223C

+11C

213C

+22]

C+11C+33+C

+22C

+33C

213C

2232C13C23+2C

+12C

+33+C

+11C

+22(C

+12)

2,

(46)

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Table 1 The elastic constants (in GPa) of some monoclinic crystals in their plane of symmetry

C11 C22 C33 C12 C13 C23 SK S

S 20.5 48.3 4.3 15.9 0 0 0.0059 0.0056

U 198.6 267.1 124.4 107.6 0 0 0.0003 0.0013

SiO 2 160.8 231.6 73.3 102.9 36.2 39.3 0.0049 0.0034

SnF2 47.9 33.6 12.9 5.3 5.1 6.5 0.0007 0.0030

C14 H14 9.45 7.20 2.55 4.15 2.4 0.7 0.0112 0.0042

KAlSi3 O8 66 122 23.6 26 3 13 0.0020 0.0023

NaFeSi2 O6 186 234 51.0 71 10 21 0.0003 0.0004

NaAlSi3 O8 74 128 29.6 39.4 6.6 20.00 0.0029 0.0023

C12 H10 5.95 14.6 2.26 2.88 0.40 2.02 0.0068 0.0073

K HC2 O4 15.6 87.7 3.60 8.24 1.50 3.09 0.0537 0.0889

SK=( Ku Kl)/(Ku +Kl) and S =(u l)/(u +l) are the scatter deviation mea-

sures for the effective bulk and shear moduli of the random polycrystals (Ku

,Kl,

u,

l

fromTable2).

where

C+11 =C11+ 0+ , C+22 =C22+ 0+ , C

+33 =C33+ ,

C+12 =C12+ 0 , = K00

K0

+ 20

. (47)

The bounds for circular cell polycrystals are calculated according to (33), (34) and(43). Expansive expressions of the terms UK f1, UK g1, ... needed for calculations ofthe general shape-unspecified bounds in (38)(40) are given in Appendix (A8A19).

For numerical applications, we consider some monoclinic crystals, whose elasticconstants in their plane of symmetry are taken from [26] and collected in Table1.In the Table2,the new general (shape-unspecified) bounds Ku,Kl.u, l from (38)to (40), the bounds for circular cell polycrystals Kuc ,K

lc

uc,

lc from (33),(34) and

Table 2 The upper and lower bounds on the polycrystalline effective moduli: Ku,Kl, u, l - thegeneral shape-unspecified bounds; Kuc ,K

lc,

uc ,

lc- the bounds for specific circular cell polycrystals

(all in GPa)

Ku Kuc Klc K

l u uc lc

l

S 21.61 21.61 21.36 21.36 5.750 5.735 5.686 5.686

U 167.5 167.5 167.4 167.4 85.77 85.77 85.54 85.54

SiO 2 132.5 132.5 131.6 131.2 54.56 54.52 54.30 54.19

SnF2

22.37 22.37 22.34 22.33 13.32 13.32 13.24 13.24

C14 H14 5.603 5.603 5.564 5.556 2.001 1.996 1.978 1.956

KAlSi3 O 8 54.83 54.83 54.65 54.61 26.57 26.53 26.46 26.45

NaFeSi2 O6 136.4 136.4 136.4 136.4 58.09 58.08 58.05 58.04

NaAlSi 3 O 8 63.44 63.44 63.18 63.07 27.89 27.87 27.79 27.76

C12 H10 5.481 5.481 5.408 5.406 2.527 2.517 2.490 2.490

K HC2 O4 18.53 18.53 16.64 16.64 7.035 6.833 5.886 5.886

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(43) are presented. The shape-unspecified bounds generally are broader than theones derived for restricted geometries in Pham [1, 2], but remain coincide with, or notfar from the much simpler (and tighter) bounds for the subclass of circular cell poly-crystals, which are supposed to approximate practical equi-axed cell aggregates. The

scatter deviation measures SK= (Ku

Kl

)/(Ku

+Kl

)and S =(u

l

)/(u

+l

)are included in Table 1, which indicate that the uncertainty is mainly a fewthousandths.

5. Closure

The bounds derived in Pham [1] are deficit because the form of the isotropic eight

rank tensor used there is not correct for the general random polycrystal, whichshould have the form (26). As already noted, the assumption in Pham [1]means anadditional geometric restriction on the aggregate microgeometry, hence the boundstherein should be more restrictive than the ones derived in this new work. Because ofthe change, the fluctuation terms in the expressions of the bounds here are morecomplicated and contain four positive geometric parameters f1,g1, f2,g2, whichare subjected to the additional restrictions (13)(in Pham[1], the fluctuation termscontain just two geometric parameters). Allowing these geometric parameters totake all possible values, we obtain the shape-unspecified bounds given. The bounds

for specific circular cell polycrystals remain simple and exactly as those derived inPham[1].Our bounds on the elastic moduli of random polycrystals do not require a

particular irregular microgeometry be specified, but just overall statistical isotropyand symmetry assumptions, and are partly third order in expansion series in powersof the crystal anisotropy contrast, which are definitely tighter than the first orderVoigtReussHill bounds and second order HashinShtrikman ones. Meanwhile thegeneral perturbation approach reveals that one cannot go to the third or higherorder bounds without specifying certain shape and arrangement information aboutthe grains, in addition to the statistical isotropy and symmetry assumptions, but

such information is unlikely to be definite for random polycrystalline materials withirregular microgeometry. Hence our bounds might be close to the best possibleones in predicting the scatter ranges for the polycrystals effective moduli. Shouldthe effective properties of random polycrystals (and more generally randomlyinhomogeneous materials) scatter, or should they be unique as widely assumed?We do not have mathematical or experimental proofs for either claims. Unique-ness (in the large volume representative element limit) is generally presumed forrigorous mathematical treatment of a homogenization problem. Uncertainty of themacroscopic properties related to the finite size effect of the representative volume

element (relative to the sizes of the constituent inhomogeneities) should decreaseas the size of the representative volume element increases [27, 28]. However theuncertainty may not decrease to zero, as the size of the representative volumeelement increases to infinity, but to approach asymptotically certain scatter intervalfor a particular random system. There are specific random systems with uniqueeffective properties, as evidenced analytically or numerically, which include randomhard spheres (circles), random overlapping spheres (circles), multicoated sphere

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(circle) assemblages...[29,30], however our views is that the uncertainty generallyprevails for configurations with irregular microgeometries. It is imperative to performnumerical simulations to access the possible uniqueness or uncertainty of the elasticmoduli of the Voronoi or Delaunay random aggregates compared to our bounds

established in this study. This is a formidable computing work, however the problemfor two-dimensional random polycrystals is greatly simpler than the respective forthe three-dimensional aggregates. It might be even simpler to treat the conductivityproblem for the two-dimensional polycrystals. However the effective conductivityof the macroscopically isotropic aggregates has been found analytically and uniquedue to the specific mathematical structure of the two-dimensional conductivityproblem [30], hence the case is not typical for the general randomly inhomogeneousmedia. Closely related to the problem considered here, the numerical simulationsfor random Voronoi cellular solid in [31,32] reveal that the effective moduli of thematerial scatter over an interval as large as about few percents of the averaged moduli

in agreement with empirical observations. Still much more efforts are needed toachieve the convincing quantitative results: Larger representative volume elements,broader and more systematic procedures are to be developed until a clear asymptotictendency toward an uncertainty or uniqueness can be established. Another relatedinteresting but difficult subject is the random irregular network problem. Up tothe present, in the literature, the numerical simulations have been carried out justfor random regular networks, for which the uniqueness is evidenced, however theeffective properties are distinctly different for different particular kinds of the base(regular) skeletons (see, e.g.,[29]and the references therein).

If the effective moduli of a randomly inhomogeneous material should scatter andnot unique, the homogeneity and isotropy hypotheses for it would not be exactanymore and could be considered only as approximate. Hence, the macroscopicmoduli tensor Ce determined for a particular representative polycrystalline samplemay be slightly anisotropic, and the bounds should be given more generally as

0 :T(Kl, l): 0 0 :Ce : 0 0 :T(Ku, u): 0 , for all symmetric 0 . (48)

Moreover all the mathematical bounds based on those hypotheses become notrigorous and should be understood in asymptotic sense (roughly speaking, the boundsmay be violated by amounts asymptotically small compared to the intervals betweenthe bounds)[33]. In that sense, the scatter measures Sin Table1should also containcertain asymptotic error, which is asymptotically small compared to S. Here we donot consider size effects. If the sizes of the grains become significant compared tomacroscopic dimensions of interest, the uncertainty of the macroscopic behaviorsshould be larger [28,29]. Here we consider random aggregates where the shape and

crystalline orientations of the grains are uncorrelated. If local texture developmentor artificial arrangements of the grains are allowed in all possible combinations, thepossible range for the effective properties also should be much larger [15,34].

High-accuracy numerical simulations (most simple to be performed for the two-dimensional aggregates) and experiments should test the possible uniqueness oruncertainty of the effective moduli of random aggregates and practical validity ofour bounds.

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J Elasticity (2006) 85: 120 15

The terms UK f2, UKg2, UM f2, UMg2 contain the sums of the type C0ijkl, whichcould be calculated beforehand:

C0ijkl =Tijkl(KVK0, V0) , (A4)

where

KV = 1

4Ciijj, V=

1

4Cijij

1

8Ciijj (A5)

are the Voigts arithmetic averages. Hence, the terms can be simplified considerably:

UK f2 =

D

jjpp D

llqq

(KVK0)

1

12

4

392

+(V0)

11

262

5

6+

1

2

+

Dd1

klpp Dd1

klqq

(KVK0)11

26 2

5

6 +

1

2

+(V0)1

3

41

39 2

,

UK g2 =

D

jjpp D

llqq

(KVK0)

2

39+(V0)

1

26

2

+

Dd1

klpp Dd1

klqq

(KVK0)

1

26+(V0)

40

39

2 , (A6)

where

D

d1

klpp= D

klpp

1

2 k lD

nnpp . (A7)

Expansive expressions ofUK f1, UKg1in two-index-crystal notation are:

UK f1 =

(D 11+ D 12)2 +( D 31+ D 32)

2

C011 + C

022

14

3

2

26

+ C012

202

39

3

C0

331

2

232

39

3

+(D 11+ D 12)2

2C033

3

1

4

82

39

C011 + C

012

239

2C012

3

1

4

32

13

+(D 31+ D 32)2

C033

2

3

1

2

202

39

+2C012

1

4

3+

82

39

(D 11+ D 12)(D 31+ D 32)(C013 C

023)

42

39, (A8)

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16 J Elasticity (2006) 85: 120

UKg1 =

(D 11+ D 12)2 +( D 31+ D 32)

2

(C011 + C022)

7

26 C012

10

39+ C033

31

39

2

+(D 11+ D 12)2(C011 + C012)

1

78

C0123

13

+ C0338

392

+(D 31+ D 32)2

C033

10

39 C012

8

39

2

+(D 11+ D 12)(D 31+ D 32)(C013 C

023)

22

39, (A9)

where

D 11 = D 1111 =1 K+(S+

12

+S+

11

) ++(S+

12

S+

11

) ,

D 22 = D 2222 =1 K+(S+12 +S

+22) +

+(S+12 S+22) ,

D 12 = D 1122 = K+(S+12 +S

+11)

+(S+12 S+11) ,

D 21 = D 2211 = K+(S+12 +S

+22)

+(S+12 S+22) ,

D 33 = D 1212 = 1

22+ S+33 , D 13 = D 1112 = 2

+ S+13 ,

D 31 = D 1211 = K+(S+13 +S

+23)

+(S+13 S+23) ,

D 23 = D 2212 = 2+

S+

23 ,

D 32 = D 1222 = K+(S+13 +S

+23) +

+(S+13 S+23) , (A10)

K+ = PK(C,K0, 0)+ 0 , + = P(C,K0, 0)+ , (A11)

S+11 = C+

22 C+

33 C223

+c, S+22 = C

+

11C+

33 C213

+c,

S+12 = C13C23 C

+12C

+33

+c, S+33 =

C+11C+22 (C

+12)

2

4+c,

S+13 = C+12C23 C

+22C13

2+c, S+23 =

C+12C13 C+11C23

2+c,

+c =C+11C

+22C

+33 +2C

+12C13C23 (C

+12)

2C+33 C223C

+11 C

213C

+22; (A12)

and

C011 =C01111 =C11 K0 0 , C

022 =C

02222 =C22 K0 0 ,

C012 =C01122 =C12 K0+ 0 , C

033 =C

01212 =C33 0 ,

C013 =C01112 =C13 , C

023 =C

02212 =C23 . (A13)

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J Elasticity (2006) 85: 120 17

For calculations ofUM f1, UMg1we need

C0iikk Djjpq D

llpq = (C

011 +C

022 +2C

012)

(D 11 + D21)

2 +( D12 + D22)2 +2(D13 + D23)

2

,

C0

iikk

D

jlp q

D

jlp q

= (C0

11

+ C0

22

+2C0

12

)D211

+ D2

22

+ D2

12

+ D 2

21

+2D213+ 2 D223+ 2 D

231+ 2D

232+ 4D

233

,

C0ijijDk k p q D

llpq = (C

011 + C

022 +2C

033)

(D 11+ D21)

2 +( D13+ D23)2

2,

C0ijijDk l p q D

k l p q = (C

011 + C

022 +2C

033)

D211+ D

222+ D

212+ D

221

+2D213+ 2 D223+ 2 D

231+ 2D

232+ 4D

233

,

C0

iiklD

klpqD

jjp q= (C0

11+C0

12)D

11(D

11+ D

21)+ D

12(D

12+ D

22)+2D

13(D

13+D

23)

+(C012 +C022)

D21(D11 + D21)+ D22(D12 + D22)+2D23(D13 + D23)

+2(C013+C023)

D31(D11+D21)+D32(D12+D22)+2D33(D13+D23)

,

C0iiklDkjpq D

ljpq = (C

011 + C

012)

D211+ D

212+ 2D

213+ D

231+ D

232+ 2 D

233

+(C012 + C022)

D221+ D

222+ 2 D

223+ D

231+ D

232+ 2 D

233

+2(C013

+C023

)D11

D31

+D12

D32

+2D13

D33

+ D31

D21

+ D32

D22

+2D33

D23,

C0ijk jDikpq D

llpq = (C

011 +C

033)

D11(D11 + D21)+ D12(D12 + D22)+2D13(D13 + D23)

+(C022 + C033)

D21(D11+ D21)+ D22(D12+ D22)+2D23(D13+ D23)

+2(C013 + C023)

D31(D11+ D 21)+ D32(D12+ D 22)+2D33(D13+ D23)

,

C0ijk jDilpq D

klpq = (C

011 + C

033)

D211+ D

212+ 2 D

213+ D

231+ D

232+ 2 D

233

+(C

0

22 + C

0

33)

D

2

21+ D

2

22+ 2D

2

23+ D

2

31+ D

2

32+ 2 D

2

33

+2(C013 +C023)

D11 D31 + D12 D32 +2D13 D33 + D31 D21 + D32 D22 +2D33 D23

,

C0ijklDijpq D

klpq = C

011

D211+ D

212+ 2 D

213

+ C022

D221+ D

222+ 2 D

223

+2C012

D11 D21 + D12 D22 +2D13 D23

+4C033

D231 + D

232+ 2 D

233

+4C013

D11 D31 + D12 D32 +2D13 D33

+4C023

D21 D31 + D22 D32 +2D23 D33

,

C0ijkl

Di k p q

Djlp q

= C011D2

11+ D 2

12+ 2 D2

13+ C0

22D2

21+ D2

22+ 2 D2

23

+2C012

D231+ D

232+ 2D

233

+2C033

D11 D21+ D12 D22+ 2 D13 D23

+D231+ D232+ 2D

233

+4C013

D11 D31+ D 12 D32+ 2D13 D33

+4C023

D21 D31+ D22 D32+ 2 D23 D33

, (A14)

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18 J Elasticity (2006) 85: 120

As already said after Eq. (41),UK f1, UKg1, UM f1, UMg1have similar forms as thoseofUK f1, UKg1, UM f1, UMg1, with

D11 = (K0+ 0

) K+(K0+

0

)C+

11

+2K0C+

12

+(K0 0

)C+

22

++

(K0+ 0)C+

11 +20C+

12 +(K0 0)C+

22

,

D22 = (K0+ 0) K+

(K0+ 0)C+

22 +2K0C+

12 +(K0 0)C+

11

++

(K0+ 0)C+

22 +20C+

12 +(K0 0)C+

11

,

D12 = (K0 0) K+

(K0+ 0)C+

11 +2K0C+

12 +(K0 0)C+

22

++

(K0+ 0)C+11 20C+12 (K0 0)C

+22

,

D21 = (K0 0) K+

(K0+ 0)C+

22 +2K0C+

12 +(K0 0)C+

11

++

(K0+ 0)C+

22 20C+

12 (K0 0)C+

11

,

D33 = 0 40+C

+

33 ,

D13 = 2+

(K0+ 0)C

+

13 +(K0 0)C+

23 ,

D23 = 2+

(K0+ 0)C+

23 +(K0 0)C+

13

,

D31 = 20 K+

(C+

13 + C+

23) 20+(C

+

13 C+

23),

D32 = 20 K+

(C+

13 + C+

23) 20+(C

+

23 C+

13), (A15)

where

C+11 = S

+

22S

+

33 S

2

23

+

c

, C+22 = S

+

11 S

+

33 S

2

13

+

c

,

C+

12 = S13 S23 S

+

12 S+

33

+

c

, C+

33 = S

+

11 S+

22 (S+

12)2

4+

c

,

C+

13 = S

+

12 S23 S+

22S13

2+

c

, C+

23 = S

+

12 S13 S+

11 S23

2+

c

,

+

c = S+

11 S+

22S+

33 +2S+

12 S13 S23 (S+

12)2 S

+

33 S223 S

+

11 S213 S

+

22 , (A16)

K+

=(C+

11 + C+

22 +2C+

12)1 ,

+ =2(C+

11 + C+

22 2C+

12 +4C+

33)1 , (A17)

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J Elasticity (2006) 85: 120 19

S+

11 = C22C33 C

223

c+

1

40+

1

4, S

+

22 = C11C33 C

213

c+

1

40+

1

4,

S+

12 = C13C23 C12C33

c

+ 1

40

1

4

, S+

33 = C11C22 C

212

4c

+ 1

2

,

S13 = C12C23 C22C13

2c, S23 =

C12C13 C11C23

2c,

c = C11C22C33+ 2C12C13C23 C33C212 C11C

223 C22C

213 ; (A18)

and

C0

11 = 1

c

(K0 r0)

2(C11C33 C213+ C22C33 C

223+ 2C13C23 2C12C33)+4

20

(C22C33 C2

23)+40(K0 0)(C22C33 C2

23 +C13C23 C12C33)

K0 0,

C0

22 = 1

c

(K0 0)

2(C11C33 C213+ C22C33 C

223+ 2C13C23 2C12C33)+4

20

(C11C33 C213)+40(K0 0)(C11C33 C

213 +C13C23 C12C33)

K0 0,

C0

12 = 1

c

(K

2

0 20)(C11C33 C

213+ C22C33 C

223+ 2C13C23 2C12C33)

+420(C13C23 C12C33) K0+ 0 ,C

0

33 = 1

c20(C11C22 C

212)0 ,

C0

13 = 1

cK00(C12C23 C22C13)+0(K0 0)(C12C13 C11C23)],

C0

23 = 1

cK00(C12C13 C11C23)+0(K0 0)(C12C23 C22C13)]. (A19)

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