Biomechanics of Hard Tissues (Modeling, Testing, and Materials) || Constitutive Modeling of the Mechanical Behavior of Trabecular Bone - Continuum Mechanical Approaches

159

6Constitutive Modeling of the Mechanical Behavior of TrabecularBone – Continuum Mechanical ApproachesAndreas Ochsner and Seyed Mohammad Hossein Hosseini

6.1Introduction

Typical uniaxial stress–strain curves for different types of bones are shown inFigure 6.1 for the compressive regime. For all the curves, an initial linear elasticbehavior can be observed. The common approach is to describe this elasticpart on the basis of Hooke’s law, cf. Sections 6.3.1–6.3.4. This elastic range isfollowed by a strong nonlinear behavior of almost constant stress (so-called stressplateau). At higher strains, some curves show a strong increase in the stress wheredensification begins.

These macroscopic stress–strain curves are similar to the behavior known fromcompletely different types of materials such as cellular polymers and metals oreven concrete. Although the deformation mechanism on the microlevel can becompletely different, a common approach is to use the constitutive equations ofmetal plasticity to describe the nonelastic behavior, cf. [1–3]. As we see in thischapter, bones have some kind of cellular or porous structure, and the classicalequations of full dense metals (e.g., von Mises or Tresca) must be extended by atleast the hydrostatic pressure to account for the fact that such a material is evenin the plastic range compressible. This general theory of a yield or failure surfacebased on stress invariants is introduced in Section 6.3.5. Many extensions of thistheory are known for bones. However, the main focus is to thoroughly introducethe concept of a yield and limit surface so that possible extensions (e.g., by damagevariables or the consideration of anisotropy) are easier to incorporate.

Many different approaches to derive new constitutive equations are known.Nowadays, the finite element method is the standard tool in computational engi-neering and advanced analysis tools (e.g., µCT) allow an extremely detailed imagingof bone structure. More and more powerful computer hardware (RAM and CPU)enables and supports this trend. However, there are approaches based on sim-plified model structures that reveal some advantages compared to these highlycomputerized approaches. Thus, some classical model structures are presented inthe second part of the chapter. These simpler models are, in many cases, able toconsider the major physical effect and may finally yield a mathematical equation to

Biomechanics of Hard Tissues: Modeling, Testing, and Materials.Edited by Andreas Ochsner and Waqar AhmedCopyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32431-6

160 6 Continuum Mechanical Approaches in Trabecular Bone Modeling

Str

ess

Str

ess

(a) (b)

0.5

0.4

0.3

Strain Strain

Figure 6.1 Uniaxial stress–strain curves in compression:(a) trabecular bone with different relative densities (after [4]);(b) compact bone (after [5]).

syy

syzszy

szz

szx

sxy sm sm

sm

sxx

syy − sm

szz − smsxx − sm

syzszy

syxsxy

szx sxzsxz

Total state Hydrostatic part(change in volume)

Deviatoric part(change in shape)

(a) (b) (c)

syx

Figure 6.2 Decomposition of the stress tensor into itsspherical and deviatoric parts. (a) totale state, (b) hydro-static part (change in volume), and (c) deviatoric part(change in shape).

describe the material behavior. However, incorporated material parameters shouldbe obtained from well-defined experimental investigations.

6.2Summary of Elasticity Theory and Continuum Mechanics

6.2.1Stress Tensor and Decomposition

It is of great importance in the framework of limit or failure surfaces of isotropicmaterials to decompose the stress tensor σij into a pure volume changing (sphericalor hydrostatic) tensor σ o

ij and a pure shape changing (deviatoric) stress tensor sij

(cf. Figure 6.2)1):

σij = σ oij + sij = σmδij + sij (6.1)

1) It should be noted that in the case ofanisotropic materials, a hydrostatic stressstate may result in a shape change, [6].

6.2 Summary of Elasticity Theory and Continuum Mechanics 161

In Eq. (6.1), σm = 13 (σxx + σyy + σzz) denotes the mean normal stress2) and δij the

Kronecker tensor (δij is equal to 1 if i = j, and 0 if i �= j). Furthermore, Einstein’ssummation convention was used [7].

Equation (6.1) can be written with components in the following way asσxx σxy σxz

σxy σyy σyz

σxz σyz σzz

︸︷︷︸Stress tensor σij

=σm 0 0

0 σm 00 0 σm

︸︷︷︸Hydrostatic tensor σo

ij

+sxx sxy sxz

sxy syy syz

sxz syz szz

︸︷︷︸Deviatoric tensor sij

(6.2)

It can be seen that the elements outside the main diagonal, that is, the shearstresses, are the same for the stress and the deviatoric stress tensors:

sij = σij for i �= j (6.3)

sij = σij − σm for i = j (6.4)

The hydrostatic part of σij has in the case of metallic materials (full dense materials),for temperatures approximately under 0.3 · Tkf (Tkf : melting temperature), nearlyno influence on the occurrence of inelastic strains since dislocations slip onlyunder the influence of shear stresses. On the other hand, the hydrostatic stress hasa considerable influence on the failure or yielding behavior in the case of cellularmaterials, in soil or damage mechanics.

6.2.2Invariants

To ensure the independence of a stress-based description of physical phenomenafrom the chosen coordinate system (objectiveness), it is meaningful to use a setof independent tensor invariants instead of the stress tensor components σij.These invariants are independent of the orientation of the coordinate system andrepresent the physical content of the stress tensor. The so-called characteristicequation3)

det(σij − λδij

) = 0 (6.5)

or in components∣∣∣∣∣∣σxx − λ σxy σxz

σxy σyy − λ σyz

σxz σyz σzz − λ

∣∣∣∣∣∣ = 0 (6.6)

leads to the cubic equation

λ3 − I1(σij)λ2 + I2(σij)λ − I3(σij) = 0 (6.7)

2) Also called the hydrostatic stress. In thecontext of soil mechanics, also the pressurep = −σm is used. Be aware that some finiteelement codes use this definition.

3) det(. . . ) denotes the determinant of a matrixor tensor.


for the definition of the three scalar principal stress invariants I1, I2 and I3.The three roots λi of Eq. (6.7) are the principal stresses: σI = max(λ1, λ2, λ3),σIII = min(λ1, λ2, λ3), and σII = (I1 − σI − σIII). It may be noted here that theprincipal stress state is the state that has no shear components, that is, only thestress components are the principal stresses σi (i = I, II, III) on the main diagonalof the stress tensor. Another interpretation of the principal invariants is given by

• I1 = trace4) of σij:

I1 = σxx + σyy + σzz (6.8)

• I2 = sum of the two-row main subdeterminants of σij:

I2 =∣∣∣∣σxx σxy

σxy σyy

∣∣∣∣ +∣∣∣∣σyy σyz

σyz σzz

∣∣∣∣ +∣∣∣∣σxx σxz

σxz σzz

∣∣∣∣ (6.9)

• I3 = determinant of σij:

I3 =∣∣∣∣∣∣σxx σxy σxz

σxy σyy σyz

σxz σyz σzz

∣∣∣∣∣∣ (6.10)

In addition to these principal invariants, there is also often another set of invariantsused. This set is included in the principal invariants Ii and called basic invariants Ji:

J1 = I1 (6.11)

J2 = 12 I2

1 − I2 (6.12)

J3 = 13 I3

1 − I1I2 + I3 (6.13)

The definition of both principal and basic invariants is summarized and comparedin Table 6.1.

It can be seen from Table 6.1 that the spherical tensor σ oij is completely

characterized by its first invariant, because the second and third invariants areits powers. The stress deviator tensor sij is completely characterized by its secondand third invariants. Therefore, the physical contents of the stress state σij can becompletely described either by the three invariants or, if we use the decompositionin its spherical and deviatoric parts, by the first invariant of the spherical tensor andthe second and third invariants of the stress deviator tensor. It should be noted herethat this statement is valid for both, that is, the principal and basic invariants. Inthe following, we will only use the basic invariants, and thus, the physical contentof stress state will be described by the following set of invariants:

σij → Jo1, J′

2, J′3 (6.14)

Table 6.2 summarizes formulae based on the general stress components σij for thepractical calculation of the basic stress invariants.

Finally, it should be mentioned here that it is also possible to derive the samesets of invariants from the strain tensor εij.

4) The trace of a tensor is the sum of thediagonal terms.

6.2 Summary of Elasticity Theory and Continuum Mechanics 163

Table 6.1 Definition of principal and basic invariants.

Stress tensorσij σ o

ij sij

Principal invariantsI1, I2, I3 Io

1, Io2, Io

3 I′1, I′

2, I′3

I1 = σii Io1 = σii I′

1 = 0I2 = 1

2 (σiiσjj − σijσji) Io2 = 1

3 (σii)2 I′2 = − 1

2 sijsji

I3 = 13

( 12 σiiσjjσkk + σijσjkσki Io

3 = 127 (σii)3 I′

3 = 13 sijsjkski

− 32 σijσjiσkk

)⇒ I1, I2, I3 ⇒ Io

1 ⇒ I′2, I′

3

Basic invariantsJ1, J2, J3 Jo

1 , Jo2 , Jo

3 J′1, J′

2, J′3

J1 = σii Jo1 = σii J′

1 = 0J2 = 1

2 σijσji Jo2 = 1

6 (σii)2 J′2 = 1

2 sijsji

J3 = 13 σijσjkσki Jo

3 = 19 (σii)3 J′

3 = 13 sijsjkski

⇒ J1, J2, J3 ⇒ Jo1 ⇒ J′

2, J′3

Table 6.2 Basic invariants in terms of σij.

Invariants General stress values

Stress tensorJ1 σxx + σyy + σzz

J212

(σ 2

xx + σ 2yy + σ 2

zz

)+ σ 2

xy + σ 2xz + σ 2

yz

J313

(σ 3

xx + σ 3yy + σ 3

zz + 3σ 2xyσxx +

+ 3σ 2xyσyy + 3σ 2

xzσxx + 3σ 2xzσzz +

+ 3σ 2yzσyy + 3σ 2

yzσzz + 6σxyσxzσyz)

Spherical tensorJo

1 σxx + σyy + σzz

Jo2

16

(σxx + σyy + σzz

)2

Jo3

19

(σxx + σyy + σzz

)3

Stress deviator tensorJ′

1 0J′

216

[(σxx − σyy)2 + (σyy − σzz)2

+ (σzz − σxx)2] + σ 2

xy + σ 2yz + σ 2

zx

J′3 sxxsyyszz + 2σxyσyzσzx

−sxxσ2yz − syyσ

2zx − szzσ

2xy

with sxx = 13 (2σxx − σyy − σzz)

syy = 13 (−σxx + 2σyy − σzz)

szz = 13 (−σxx − σyy + 2σzz)


6.3Constitutive Equations

Figure 6.3 shows the emplacement of the constitutive equation in the frameworkof solid mechanical modeling of a material particle. The constitutive equationcombines the kinematic and equilibrium equations by indicating the dependenceof strains on stresses.

According to the level of description, the constitutive equations are distinguishedas microscopic, stochastic, and macroscopic [8]. Microscopic constitutive equationsdescribe the material behavior based on variables (e.g., dislocation density) takenfrom materials physics. A transfer of the microscopic material behavior on themacroscopic level has not been successful till now. Stochastic constitutive equationsalso describe the material behavior on the micro level. For that purpose, proba-bilistic processes are applied. The transfer on the macroscopic level is due to itspossible mathematical structure. Macroscopic constitutive equations are also calledphenomenological models. Their field of application is the continuum mechanicalmodeling of components and structures. The material is idealized as a homoge-neous continuum, which constitution is described by a few variables such as strain,temperature, and occasionally by additional so-called internal variables. In the caseof multiphase or inhomogeneous materials, such as cellular or fiber-reinforced ma-terials, this assumption does not hold in the first place. By averaging the differentmaterial properties, an approximate homogeneity can be assumed, cf. Figure 6.4.

External forces

Equilibrium

Stresses

Measure forloading

Continuum mechanicalmodeling

Constitutiveequation

Displacements

Kinematic

Strains

Measure fordeformation

Figure 6.3 Solid mechanical modeling of engineering materials.

Multiphase material Equivalent material

Homogenization

Figure 6.4 Averaging the materials properties to obtain an equivalent material.

6.3 Constitutive Equations 165

6.3.1Linear Elastic Behavior: Generalized Hooke’s Law for Isotropic Materials

In general Cauchy elasticity, the stress tensor σij is a nonlinear tensor function ofthe strain tensor εkl:

σij = fij(εkl) (6.15)

whereas in linear elasticity of homogeneous materials the Cauchy stress tensor is,in extension to Hooke’s law (σ = Eε) of the year 1678, proportional to the Cauchystrain tensor through the linear transformation:

σij = Cijklεkl (6.16)

It may be noted here that Eq. (6.16) is the simplest generalization of the lineardependence of stress and strain observed by Hooke in a simple tension test, andconsequently Eq. (6.16) is referred to as the generalized Hooke’s law. The fourth-ordertensor Cijkl has 81 independent components in total and is called the elasticity tensor.In the case of linear elasticity, the components of the elasticity tensor are constantvalues. Owing to symmetry of σij and εkl, the pairs of indices ij and kl in Cijkl can bepermutated. Hence, the number of independent constants of the elasticity tensoris reduced to 36. A further reduction of constants is obtained if we assume theexistence of a strain energy potential w(εij) from which the stresses are derived bydifferentiation as

σij = ∂w

∂εkl(6.17)

After partial differentiation of Eq. (6.16) with respect to εij and consideration ofEq. (6.17), one obtains the components of the elasticity tensor as

Cijkl = ∂σij

∂εkl= ∂2w

∂εij∂εkl(6.18)

There is an additional symmetry due to the commutation of the order of the partialderivative with respect to εij and εkl, and the number of independent constantsis reduced to 21. A material for which such a constitutive relation is assumed isalso called Green elastic material [9]. By considering the material symmetries, (e.g.,crystal systems), a further reduction in the number of constants can be achieved.For an isotropic material, the mechanical properties are the same for all directionsand it can be shown [10] that the components of the elasticity tensor are determinedby two independent constants, the so-called Lame’s constants λ and µ and thestress–strain relationship is given for this case by

σij = 2µεij + λεkkδij (6.19)

The characterization of metallic materials is in general based on the engineeringconstants, Young’s modulus E and Poisson’s ratio ν. On the other hand, thedescription of isotropic materials in soil or rock mechanics is based on the shearmodulus G and the bulk modulus K rather than E and ν. The conversion of theelastic constants can be done with the aid of the relationships given in Table 6.3.


Table 6.3 Conversion of elastic constants.

λ, µ E, ν µ, ν E, µ K, ν G, ν K, G

λ λ νE(1+ν)(1−2ν)

2µν

1−2ν

µ(E−2µ)3µ−E

3Kν1+ν

2Gν1−2ν

K − 2G3

µ µ E2(1+ν) µ µ

3K(1−2ν)2(1+ν) µ µ

K λ + 23 µ E

3(1−2ν)2µ(1+ν)3(1−2ν)

µE3(3µ−E) K 2G(1+ν)

3(1−2ν) K

E µ(3λ+2µ)λ+µ

E 2µ(1 + ν) E 3K(1 − 2ν) 2G(1 + ν) 9KG3K+G

ν λ2(λ+µ) ν ν E

2µ− 1 ν ν 3K−2G

2(3K+G)

G µ E2(1+ν) µ G 3K(1−2ν)

2(1+ν) G G

Table 6.4 Reference values of typical engineering materials(all values are given near room temperature).

Material E (GPa) ν in – G (GPa) Density (g cm−3)

Aluminum 70 0.35 26 2.70Iron 211 0.29 82 7.874Magnesium 45 0.290 17 1.738Titanium 116 0.32 44 4.506Human bone 0.186 0.25

(trabecular)...

.

.

.

0.724 0.50

Mechanical and physical reference values5) of some typical engineering materialsare summarized in Table 6.4. The measured and predicted values for trabecularbone comprise a wide range and are influenced by many factors (e.g., age, wet ordry etc.). In order to provide some typical values for trabecular bone and relatethem to these classical engineering materials, the power law by Hodgskinson et al.[11] was applied, which describes Young’s modulus over a large density range.

In the following description, the tensor notation is abandoned and the engineer-ing notation (so-called Voigt notation) is consistently introduced and observed, thatis, the components of the second-order strain tensor εij and the stress tensor σij arearranged into column vectors

εij = εx εxy εxz

εyx εy εyz

εzx εzy εz

→ ε = {εx εy εz 2εxy 2εyz 2εxz}T (6.20)

5) Conversion from GPa to MPa: value times1000; conversion from g cm−3 to kg m−3:value times 1000; 1 Pa = 1 N m−2.


y

dy dy

dxdxx

∂u/∂y

∂u/∂x

x

gxy

y

(b)(a)

Figure 6.5 Definition of shear strain: (a) tensor definitionas the average εxy = εyx = (∂v/∂x + ∂u/∂y)/2; (b) engineer-ing definition as the total γxy = ∂v/∂x + ∂u/∂y.

and

σij = σx σxy σxz

σyx σy σyz

σzx σzy σz

→ σ = {σx σy σz σxy σyz σxz}T (6.21)

and the fourth-order elasticity tensor Cijkl is represented by a square (6 × 6)-matrixC. This formalism is closer to actual computer implementations than the tensorialnotation [12]. Furthermore, we use the unit vector 1 = {1 1 1 0 0 0}T and the diagonalmatrix L = �1 1 1 2 2 2�, which is not a unit matrix because the strain vector isexpressed using the engineering definition of shear strain (e. g., γxy = 2εxy) ratherthan the tensor definition, cf. Figure 6.5.

Now, the generalized Hooke’s law for isotropic materials based on Lame’sconstants can be written in matrix form as

σ = (2µL−1 + λ1 ⊗ 1)ε (6.22)

or in explicit form with all components as

σx

σy

σz

σxy

σyz

σxz

=

λ + 2µ λ λ 0 0 0λ λ + 2µ λ 0 0 0λ λ λ + 2µ 0 0 00 0 0 µ 0 00 0 0 0 µ 00 0 0 0 0 µ

·

εx

εy

εz

2 εxy

2 εyz

2 εxz

(6.23)

The dyadic product ⊗ used in Eq. (6.22) is in general defined for two vectors a andb by

a ⊗ b =

a1

a2

...

an

⊗

b1

b2

...

bn

=

a1b1 a1b2 . . . a1bn

a2b1 a2b2 . . . a2bn

......

. . ....

anb1

... . . . anbn

(6.24)

where the same result can be obtained by the matrix multiplication a · bT.


Alternatively, Eq. (6.22) can be written in the form of Eq. (6.19) as

σ = 2µL−1ε+ λ1 ⊗ 1 (εm1 + e) = 2µL−1ε+ λεm 11T1︸︷︷︸31

+λ 11Te︸︷︷︸0

= 2µL−1ε+ λ(3εm)1 (6.25)

Substituting for λ and µ in terms of E and ν (cf. Table 6.3) gives the alternativeformulation of Eq. (6.23) based on the engineering constants as

σx

σy

σz

σxy

σyz

σxz

= E

(1 + ν)(1 − 2ν)

1 − ν ν ν 0 0 0ν 1 − ν ν 0 0 0ν ν 1 − ν 0 0 00 0 0 1−2ν

2 0 00 0 0 0 1−2ν

2 00 0 0 0 0 1−2ν

2

·

εx

εy

εz

2 εxy

2 εyz

2 εxz

(6.26)

In a simple tension test, the only nonzero stress component σx causes axial strainεx and transverse strains εy = εz. Thus, Eq. (6.26) yields

εx = σx

Eand εy = −νεx = −νσx

E(6.27)

By using Eq. (6.27), one can calculate the elastic constants, Young’s modulus Eand Poisson’s ratio ν, from a uniaxial tension or compression test. Introducing theshear modulus G and the bulk modulus K according to Table 6.3 yields a furtherformulation of Hooke’s generalized law as

σx

σy

σz

σxy

σyz

σxz

=

K + 43 G K − 2

3 G K − 23 G 0 0 0

K − 23 G K + 4

3 G K − 23 G 0 0 0

K − 23 G K − 2

3 G K + 43 G 0 0 0

0 0 0 G 0 00 0 0 0 G 00 0 0 0 0 G

·

εx

εy

εz

2 εxy

2 εyz

2 εxz

(6.28)

The quantity K + 43 G in Eq. (6.28) is also known as the constraint modulus [13].

Decomposing the stress and strain vectors into spherical and deviatoric componentsdecouples the volumetric response from the distortional response, and Hooke’s lawcan be expressed in terms of the volumetric and deviatoric strains in the followingform:6)

σ = σo + s = σm1 + s = 3Kεm · 1 + 2GL−1e (6.29)

From the point of view of continuum mechanics, a state under pure shear stress(result: shear modulus G) and a state under pure hydrostatic stress (result: bulkmodulus K) should aim to determine the elastic constants, since the constantsare independent of each other in this case. However, the experimental determi-nation of the elastic constants is mostly based on a simple realizable tension or

6) This form is also called the canonical form[14].


Table 6.5 Formulations of generalized isotropic Hooke’s lawbased on different elastic constants (elastic stiffness form).

Hooke’s law

σ = C ε | σij = Cijkl εkl

Formulation based on Lame’s constants

σ = (λ1 ⊗ 1 + 2µL−1)ε | σij = λεkkδij + 2µεij

Formulation based on Young’s modulus and Poisson’s ratio

σ = E1+ν

· (ν

1−2ν1 ⊗ 1 + L−1

)ε

∣∣ σij = E1+ν

· (εij + ν1−2ν

δijεkk)

Formulation based on bulk modulus and shear modulus

σ = (K1 ⊗ 1 + 2G(L−1 − 1

3 1 ⊗ 1))ε

∣∣ σij = 2G · (εij + 3K−2G6G δijεkk

)Decomposition in volumetric and deviatoric parts

σ = 3Kεm1︸︷︷︸Volumetric response

+ 2GL−1e︸︷︷︸Deviatoric response

∣∣∣∣∣∣∣∣ σij = Kεkkδij︸︷︷︸Volumetric response

+ 2Geij︸︷︷︸Deviatoric response

compression test, from which Young’s modulus E and Poisson’s ratio ν can beobtained.

A summary of the different formulations of Hooke’s law in matrix form is givenin Table 6.5 and compared with the corresponding tensor form given in commonliterature, for example, [2, 10, 15, 16].

The decomposition of the stress vector into spherical and deviatoric componentscan also be performed based on projection tensors. The deviatoric stress vector s isobtained by subtracting the spherical state of stress from the actual state of stress.Thus, we can write

s = σ − σm1 = σ − 1

311Tσ =

(I − 1

311T

)σ =

(I − 1

31 ⊗ 1

)σ = I′σ (6.30)

and define the deviatoric projection tensor

I′ = I − 1

31 ⊗ 1 (6.31)

to transform the actual stress state in its deviatoric part. Respecting 1T1 = 3, wecan indicate the following properties of the deviatoric projection tensor:


•

I′1 = 0 (6.32)

Proof:

I′1 = (I − 13 11T)1 = I1 − 1

3 1 1T1︸︷︷︸3

= 1 − 1 = 0

•

I′I′ = I′ or in general(I′)n = I′ for any n ∈ IN (6.33)

Proof:

I′I′ = (I − 1

31 ⊗ 1)(I − 1

31 ⊗ 1)

= II︸︷︷︸I

− 13 I11T︸︷︷︸

11T

− 13 11TI︸︷︷︸

11T

+ 19 1 1T1︸︷︷︸

3

1T

= I − 23 11T + 1

3 11T

= I − 13 11T

This relationship expresses the fact that the deviatoric part of the stress cannot bechanged by any other deviatoric projection and that the spherical part is equal tozero. A corresponding projection tensor can be derived for the spherical part of theactual state of stress:

Io = I − I′ = 1

31 ⊗ 1 = 1

311T (6.34)

Thus, we can rewrite Eq. (6.29) based on the transformation tensors as

σ = σo + s = Ioσ + I′σ (6.35)

In the following, the strain energy per unit volume w will be derived in its canonicalform of two decoupled contributions, that is, its spherical and deviatoric parts:

w = 1

2σε = 1

2

(σm1Tεm1 + sTε

)(6.36)

By using Eq. (6.29), we can substitute σm and s and express the strain energy as

w = 1

2

(3Kεm1Tεm1 + 2GL−1eTe

) = 1

2

(9Kε2

m + 2GeTL−1e)

(6.37)

In Eq. (6.37), the positive strain energy argument delimits the range of possiblevalues of Poisson’s ratio to −1 ≤ ν ≤ 0.5 (cf. Table 6.3).

In the last part of this section, we will provide relationships between the strainvector and the stress vector. Equation (6.16) can be written in the matrix form as

ε = Dσ (6.38)

where the elastic compliance matrix D is given by the inverse of the elasticitymatrix C:

D = C−1 (6.39)


Table 6.6 Formulations of generalized isotropic Hooke’slaw based on different elastic constants (elastic complianceform).

Hooke’s law

ε = C−1 σ∣∣∣ εij = C−1

ijkl σkl

Formulation based on Lame’s constants

ε = 12µ

·(

L − λ2µ+3λ

1 ⊗ 1)σ

∣∣∣ εij = 12µ

·(σij − λ

2µ+3λσkkδij

)Formulation based on Young’s modulus and Poisson’s ratio

ε = 1+νE · (

L − ν1+ν

1 ⊗ 1)σ

∣∣ εij = 1+νE · (

σij − ν1+ν

σkkδij)

Formulation based on bulk modulus and shear modulus

ε = 12G · (L − 3K−2G

9K 1 ⊗ 1)σ

∣∣ εij = 12G · (σij − 3K−2G

9K σkkδij)

Decomposition in volumetric and deviatoric parts

ε = 13K σm1 + 1

2G Ls∣∣ εij = 1

3K σmδij + 12G sij

This inversion can be done using Sherman–Morrison formula [17, 18] given ingeneral form for a matrix a and two vectors u and v (α, β scalars) as

(αA + β(u ⊗ v)

)−1 = 1

α

(A−1 − β

(A−1u

) ⊗ (vT · A−1

)T

α + βvT · A−1u

)(6.40)

as long as β

αvT · A−1u �= −1

A summary of the different formulations of Hooke’s law in compliance formis given in Table 6.6 and compared with the corresponding tensor form given incommon literature, for example, [2, 10, 15, 16].

6.3.2Linear Elastic Behavior: Generalized Hooke’s Law for Orthotropic Materials

Although most materials can be treated as approximately isotropic, strictly speaking,all materials are anisotropic to some extent and the material properties are not thesame in every direction. In the following, we consider the important case of anorthotropic material, that is, a material with three principal, mutually orthogonalaxes. These axes are also called the material principal axes. Let us assume in thefollowing that the principal axes are coincident with the global coordinate system


(x, y, z). We will start with the compliance form of Hooke’s law:

εx

εy

εz

2εxy

2εyz

2εxz

=

1Ex

− νyxEy

− νzxEz

0 0 0

− νxyEx

1Ey

− νzyEz

0 0 0

− νxzEx

− νyzEy

1Ez

0 0 0

0 0 0 1Gxy

0 0

0 0 0 0 1Gyz

0

0 0 0 0 0 1Gxz

·

σx

σy

σz

σxy

σyz

σxz

(6.41)

Here, Ex , Ey, and Ez are Young’s moduli for the principal axes, and νij are Poisson’sratios for these axes. The Poisson’s ratio νxy, for example, represents the ratiobetween strains εx and εy when the material is subjected to uniaxial stress in they-direction, that is

νxy = −εx

εy

∣∣∣σy = σ

(6.42)

The coefficients Gxy, Gyz, and Gxz represent the shear moduli for the x-y, y-z, andx-z planes, respectively. The compliance matrix is symmetric, and therefore thefollowing relations must be satisfied (observe the fact that Poisson’s ratios are notsymmetric):

νxy

Ex= νyx

Ey,

νyz

Ey= νzy

Ez,

νxz

Ex= νzx

Ez(6.43)

From the previous equation, it follows that the compliance matrix comprises nineindependent material constants and that the following equation holds7):

νxyνyzνzx = νxzνyxνzy (6.44)

The stiffness form of Hooke’s law is obtained by inverting the compliance matrix as

σx

σy

σz

σxy

σyz

σxz

=

1−νyzνzyEyEzD

νyx+νzxνyzEyEzD

νzx+νyxνzyEyEzD 0 0 0

νyx+νzxνyzEyEzD

1−νxzνzxExEzD

νzy+νxyνzxExEzD 0 0 0

νzx+νyxνzyEyEzD

νzy+νxyνzxExEzD

1−νxyνyxExEyD 0 0 0

0 0 0 Gxy 0 0

0 0 0 0 Gyz 0

0 0 0 0 0 Gxz

·

εx

εy

εz

2εxy

2εyz

2εxz

(6.45)

where

D = 1 − νxyνyx − νxzνzx − νyzνzy − 2νyxνxzνzy

ExEyEz(6.46)

is the determinant of the compliance matrix in Eq. (6.41) multiplied by GxyGyzGxz.

7) Used to simplify the expression for the de-terminant of Eq. (6.41).


6.3.3Linear Elastic Behavior: Generalized Hooke’s Law for Orthotropic Materials withCubic Structure

If the properties of an orthotropic material are identical in all three directions (x, y,and z), the material is said to have a cubic structure. A huge number of materialshave cubic symmetry, for example, all the face centered cubic (FCC) and bodycentered cubic (BCC) metals. Thus, if we have

Ex = Ey = Ez = E (6.47)

Gxy = Gyz = Gxz = G (6.48)

νxy = νyx = · · · = ν (6.49)

the compliance form of Hooke’s law for an orthotropic material with cubic structureis reduced to

εx

εy

εz

2εxy

2εyz

2εxz

= 1

E·

1 −ν −ν 0 0 0

−ν 1 −ν 0 0 0

−ν −ν 1 0 0 0

0 0 0 EG 0 0

0 0 0 0 EG 0

0 0 0 0 0 EG

·

σx

σy

σz

σxy

σyz

σxz

(6.50)

where E, ν, and G are the three independent material constants. The stiffness formof Hooke’s law is obtained by inverting the compliance matrix as

σx

σy

σz

σxy

σyz

σxz

=

E(ν−1)

2 ν2+ν−1− Eν

2 ν2+ν−1− Eν

2 ν2+ν−10 0 0

− Eν

2 ν2+ν−1E(ν−1)

2 ν2+ν−1− Eν

2 ν2+ν−10 0 0

− Eν

2 ν2+ν−1− Eν

2 ν2+ν−1E(ν−1)

2 ν2+ν−10 0 0

0 0 0 G 0 0

0 0 0 0 G 0

0 0 0 0 0 G

·

εx

εy

εz

2εxy

2εyz

2εxz

(6.51)

6.3.4Linear Elastic Behavior: Generalized Hooke’s Law for Transverse Isotropic Materials

Special classes of orthotropic materials are those that have the same properties inone plane (e.g., the x–y plane) and different properties in the direction normal tothis plane (e.g., the z-axis). This implies that the solid can be rotated with respectto the loading direction about one axis without measurable effect on the solid’sresponse. Such materials are called transverse isotropic, and they are describedby five independent elastic constants, instead of nine for fully orthotropic ones.


Examples of transversely isotropic materials include hexagonal close-packed crys-tals, some piezoelectric materials (e.g. PZT-4, barium titanate), and fiber-reinforcedcomposites, where all fibers are in parallel.

By convention, the five elastic constants in transverse isotropic constitutiveequations are Young’s modulus and Poisson’s ratio in the x–y symmetry plane(index ‘p’), Ep and νp, Young’s modulus and Poisson’s ratio in the z-direction, Epz

and νpz, and the shear modulus in the z-direction, Gzp.The compliance form of Hooke’s law takes the form

εx

εy

εz

2εxy

2εyz

2εxz

=

1Ep

− νpEp

− νzpEz

0 0 0

− νpEp

1Ep

− νzpEz

0 0 0

− νpzEp

− νpzEp

1Ez

0 0 0

0 0 01+νp

Ep0 0

0 0 0 0 1Gpz

0

0 0 0 0 0 1Gpz

·

σx

σy

σz

σxy

σyz

σxz

(6.52)

where Poisson’s ratios are not symmetric, but satisfyνpz

Ep= νzp

Ez(6.53)

The stiffness form of Hooke’s law is obtained by inverting the compliance matrixas

σx

σy

σz

σxy

σyz

σxz

=

1−νpzνzpEpEzD

νp+νzpνpzEpEzD

νzp+νpνzpEpEzD 0 0 0

νp+νzpνpzEpEzD

1−νpzνzpEpEzD

νzp+νpνzpEpEzD 0 0 0

νzp+νpνzpEpEzD

νzp+νpνzpEpEzD

1−ν2p

E2p D

0 0 0

0 0 0Ep

1+νp0 0

0 0 0 0 Gpz 0

0 0 0 0 0 Gpz

·

εx

εy

εz

2εxy

2εyz

2εxz

(6.54)

where

D = (1 − νp)(1 − νp − 2νpzνzp)

E2pEz

(6.55)

Table 6.7 summarizes the different formulations of Hooke’s law and the assignednumber of independent variables.

6.3.5Plastic Behavior, Failure, and Limit Surface

The three essential ingredients of plastic analysis are the yield criterion, theflow rule, and the hardening rule, cf. [19]. The yield criterion relates the stateof stress to the onset of yielding. The flow rule relates the state of stress σij tothe corresponding increments of plastic strain dε

pij when an increment of plastic


Table 6.7 Generalized linear elastic Hooke’s law andindependent material constants.

Type Number constants

Anisotropic 21Orthotropic 9Transverse isotropic 5Orthotropic, cubic structure 3Isotropic 2

flow occurs. The hardening rule describes how the yield criterion is modifiedby straining beyond the initial yield. In the following section, the mathematicaland graphical representations of the initial yield criterion are discussed in detail.The initial yield criterion can generally be expressed for isotropic and anisotropicmaterials as

F = F(σij) (6.56)

where the state of stress σij can be split into its spherical part σ oij and deviatoric

part σ ′ij. For an isotropic material, the stress state can then be expressed in terms

of combinations of three independent stress invariants. In the following, the set ofthe so-called basic invariants, Jo

1, J′2, and J′

3, is used, where Jo1 is the first invariant of

the spherical stress tensor (σ oij ) and J′

2 and J′3 are the second and third invariants

of the deviatoric tensor (σ ′ij), [20, 21]. Further sets of independent invariants can be

found, for example, in Altenbach et al. [22].Thus, one can replace Eq. (6.56) for an isotropic material by

F = F(Jo1, J′

2, J′3) (6.57)

The yield condition F = 0 represents a hypersurface in the n-dimensional stressspace (in the case of isotropic materials, n is equal to the six independentstress tensor components) and is also called the yield surface. A direct graphicalrepresentation of this yield surface in a Cartesian coordinate system with threecoordinates is not possible due to its dimension. However, a reduction of thedimensions is possible if a principal axis transformation is applied to the argumentσij. The components of the stress tensor are then uniquely reduced for isotropicmaterials to the principal stresses σI, σII, and σIII on the principal diagonal of thestress tensor. In such a principal stress space, it is now possible to graphicallyrepresent the yield condition as a three-dimensional surface. A hydrostatic stressstate (σI = σII = σIII) lies in such a principal stress system on the space diagonal(hydrostatic axis). Any plane perpendicular to the hydrostatic axis is called anoctahedral plane. The particular octahedral plane passing through the origin isthe deviatoric plane or π -plane [20]. On the basis of the dependency of the yieldcondition on the invariants, a descriptive classification can be performed. Yieldconditions independent of hydrostatic stress can be represented by the invariantsJ′

2 and J′3. Stress states with J′

2 = const. lie on a circle around the hydrostatic axis in


sI sI

sIII

P (sI, sII, sIII)

sIIsII sIII

O

2J ′2 Hydrostaticaxis

N (sm, sm, sm)

Octahedral plane

31 J o

1

332

(a) (b)

y

P

q

cos 3q =J ′3

(J ′2)3/22π3 N

x

Figure 6.6 Geometrical interpretation of basic stress invari-ants in relation to an arbitrary stress state P: (a) principalstress space; (b) octahedral plane, Jo

1 = const. (view alongthe hydrostatic axis).

an octahedral plane. A dependency of the yield condition on J′3 results in a deviation

from the circle shape. A dependency on Jo1 denotes a size change of the cross

section of the yield surface along the hydrostatic axis. However, the shape of thecross section remains similar in the mathematical sense. Therefore, a dependencyon Jo

1 can be represented by sectional views through planes along the hydrostaticaxis. The geometrical interpretation of stress invariants [13] is given in Figure 6.6.

It can be seen that an arbitrary stress state P can be expressed by its positionalong the hydrostatic axis ( 1√

3Jo

1) = ξ and its polar coordinates (√

2J′2 = ρ, θ (J′

2, J′3))

in the octahedral plane through P. For the set of polar coordinates, the so-calledstress Lode angle θ is defined in the range 0 ≤ θ ≤ 60◦ as [23]

cos(3θ ) = 3√

3

2· J′

3

(J′2)3/2

(6.58)

The set of coordinates (ξ , ρ, cos(3θ )) is known as the Haigh–Westergaard coordi-nates.

To investigate the shape of the yield surface, multiaxial stress states must besimulated and the shape within the corresponding octahedral plane has to be drawnaccording to the following transformation, which projects a principal stress statefirst in the octahedral plane (angle of transformation: cos ϑ = 1/

√3) and then to

the Cartesian coordinate system (x, y) shown in Figure 6.6(b).

y = 2√6

· (σI − 0.5 · (σII + σIII)

)(6.59)

x = 1√2

· (σIII − σII) (6.60)

In such a case, it is essential whether the plastic behavior is pressure sensitive,that is, depends on Jo

1, or not. If there is no dependency on Jo1, that is, a constant


Table 6.8 Values of the stress Lode angle for basic tests.

Case Component θ according to Eq. (6.58) θ as given in Figure 6.6

Uniaxial tension σI = σ or 0◦ 0◦

σII = σ or 0◦ 120◦ (−60◦)σIII = σ 0◦ 240◦ (60◦)

Uniaxial compression σI = −σ or 60◦ 0◦ (180◦)σII = −σ or 60◦ −60◦ (120◦)σIII = −σ 60◦ 60◦

Biaxial tension σI = σII = σ or 60◦ 60◦

σI = σIII = σ or 60◦ −60◦

σII = σIII = σ 60◦ 180◦

Biaxial compression σI = σII = −σ or 0◦ 240◦

σI = σIII = −σ or 0◦ 120◦

σII = σIII = −σ 0◦ 0◦

Triaxial tension σI = σII = σIII = σ − −Triaxial compression σI = σII = σIII = −σ − −Pur shear σI = τ ; σII = −τ 30◦ −30◦

(σxy = τ )σI = τ ; σIII = −τ 30◦ 30◦

(σxz = τ )σII = τ ; σIII = −τ 30◦ 90◦

(σyz = τ )

shape along the hydrostatic axis, then all evaluated points can be drawn in a singleoctahedral plane. However, if one expects a dependency, then only stress states withthe same hydrostatic stress are allowed to be represented in the same octahedralplane where Jo

1 = const. holds. As a result, for example, uniaxial tensile (Jo1 = σI)

and pure shear tests (Jo1 = 0) cannot be represented in the same octahedral

plane. In order to draw the shape of the yield surface for a pressure-sensitivematerial, the following differing multiaxial stress states with Jo

1 = 0, for example,σI = −σII (σIII = 0), or σI = −2σII = −2σIII (σI > 0 ∨ σI < 0), are a possibility toobtain values in the same deviatoric plane. It should be noted here again that theshape changes only its size along the hydrostatic axes but remains similar in themathematical sense. Typical Lode angles for basic experiments are summarized inTable 6.8, where the evaluation of Eq. (6.58) and the graphical representation inthe octahedral plane are given.

For an isotropic material, the labels I, II, III attached to the principal coordi-nate axes are arbitrary. It follows that the yield condition must have threefoldsymmetry and it is only required to investigate the sector θ = 0◦ to 60◦, cf.Figure 6.7. The other sectors follow directly from the symmetry. If the uniaxialyield stress is in addition the same in tension and compression, a sixfold sym-metry is obtained and only the sector θ = 0◦ to 30◦ needs to be investigated, cf.Figure 6.7.


sI sI

sII sIII sIIIsII

30° 60°

(a) (b)

Figure 6.7 Schematic representation of periodic segmentsin the octahedral plane: (a) 30◦ symmetry segment; (b) 60◦

symmetry segment.

sII sIII sII sIII

sI sI

A k k

C

C

30° 30°

O O

Outerbound

(a) (b)

AInner bound

Figure 6.8 Schematic representation of outer (a) and inner(b) bounds of a yield condition in the octahedral space.

Further restrictions for the shape of the yield condition in the octahedral planeare based on the requirement of a convex shape8), cf. Figure 6.8. The inner boundcorresponds to the Tresca yield criterion. An influence of the third invariant wouldresult in a deviation from the circular shape (e.g., von Mises).

The maximum distance between the circle and the outer bound is obtained forθ = 30◦ as (cf. Figure 6.8(b))

AC

k= AO − k

k= 1

cos 30◦ − 1 = 2√3

− 1 = 15.47 % (6.61)

of the circle diameter. The maximum distance between the circle and the innerbound is obtained under the same angle as (cf. Fig. 6.8(b))

AC

k= k − AO

k= 1 − cos 30◦ = 1 −

√3

2= 13.40 % (6.62)

8) The convexity of the yield surface can bederived from Drucker’s stability postulate[6, 24].


Uniaxial compression

Biaxial compression

Triaxial compression

Pure shear

Uniaxial tension

Biaxial tension

Triaxial tension

J o1

3J ′2

Not defined

Figure 6.9 Schematic representation of basic tests in the Jo1 –

√3J′2 invariant space.

In the case that a material has the same uniaxial yield stress in tension andcompression, the influence of the third invariant is often disregarded since themaximum error is in the indicated range of the outer and inner bounds, cf. Eqs(6.61) and (6.62). In such a case, the mathematical description can be performedin a J◦

1 –√

3J′2 invariant space as indicated in Figure 6.9.

In the J◦1 –

√3J′

2 invariant space, basic material tests can be identified as linesthrough the origin as indicated in Figure 6.9. Table 6.9 summarizes the linearequations for these basic experiments. As one can see, a uniaxial tensile test isrepresented, for example, by the bisecting line in the first quadrant of the CartesianJ◦

1 –√

3J′2 coordinate system. Performing a uniaxial tensile test would mean to

‘‘walk’’ from the origin along this straight line (by monotonically increasing theload) until a material limit, for example, initial yield or failure, is reached. Thispoint (indicated in Figure 6.9 by ◦) makes part of the yield or limit surface and the

Table 6.9 Definition of basic tests in the Jo1 –

√3J′2 invariant space.

Case Jo1

√3J′2 Comment

Uniaxial tension (σ ) σ σ Slope: 1Uniaxial compression (−σ ) −σ σ Slope: −1Biaxial tension (σ ) 2σ σ Slope: 0.5Biaxial compression (−σ ) −2σ σ Slope: −0.5Triaxial tension (σ ) 3σ 0 Horizondal axisTriaxial compression (−σ ) −3σ 0 Horizondal axisPure shear (τ ) 0

√3 τ Vertical axis


3J ′2

3J ′2

3J ′2

(a)

(b)

(c)

Any q

Any q

q = 60°

q = 0°

J o1

J o1

J o1

sII

sI

sI

sII sIII

sI

sII sIII

sIII

q

q

q

Figure 6.10 Schematic representation of differentyield/failure conditions: (a) F = F( J′2); (b) F = F( Jo

1, J′2);(c) F = F( Jo

1, J′2, J′3).

connection of all such points obtained from different experiments (i.e., from walksalong different lines) give the final shape, which must be described in terms of thevariables to obtain the expression F(J◦

1, J′2) = 0.

Some typical representations of yield or limit surfaces are schematically shownin Figure 6.10. A representation that depends only on J′

2 is shown in part (a): aconstant diameter is obtained along the hydrostatic axis, and a circular shape in theoctahedral plane. Incorporating a dependency on the hydrostatic stress (part (b))changes the diameter along the hydrostatic axis but the octahedral plain still displaysa circle. Only the third invariant (part (c)) can deform the circle in the octahedralplane. For such a dependency, sectional views for θ = 0◦ (‘‘tensile meridian’’),θ = 60◦ (‘‘compression meridian’’), or θ = 30◦ (‘‘shear meridian’’) are commonrepresentations in the J◦

1 –√

3J′2 space. Along these meridians, a hydrostatic stress

state is superimposed on the respective stress states.

6.4 The Structure of Trabecular Bone and Modeling Approaches 181

The flow rule is in general stated in terms of a function Q , which is describedin units of stress, and is called a plastic potential. With dλ, a scalar called plasticmultiplier, the plastic strain increments are given by

dε = dλ · ∂Q

∂σ(6.63)

The flow rule given in Eq. (6.63) is said to be associated if Q = F, otherwise it isnonassociated. Hardening can be modeled as isotropic (i.e., initial yield surfaceexpands uniformly without distortion and translation as plastic flow occurs) oras kinematic (i.e., initial yield surface translates as a rigid body in stress space,maintaining its size, shape, and orientation), either separately or in combination.With the isotropic hardening parameter κ and the kinematic hardening parametersα = {αx, . . . , αxz}T, the hardening vector q = {κ ,α}T can be composed and theyield condition under consideration of hardening effects results as

F = F(σ, q) = 0 (6.64)

Equation (6.64) can be further extended, for example, by incorporating damageeffects or anisotropy (fabric-dependent criterion), [25, 26]. More details on modelingof plastic material behavior can be found in the standard books [22, 27–29].

6.4The Structure of Trabecular Bone and Modeling Approaches

At the first glance, the bone may look as a solid dense material. However, mostof the bones rather form some kind of composite material where the outer shellis made up of a dense layer, the so-called compact bone9) and a core made up of acellular structure, the so-called trabecular bone10), cf. Figure 6.11. These two types ofbones are distinguished based on their porosity11) and microstructure. Trabecularbone is much more porous with porosity ranging anywhere from approximately 50to 90% whereas the cortical bone is much denser with a porosity ranging between5 and 10%. According to Gibson and Ashby [30], any bone with a porosity largerthan 30% is classified as ‘‘cancelous.’’ A typical range for the density of trabecularbone is between 94 kg m−3 (extremely low density) and 780 kg m−3 (moderatelyhigh), [11]. Compact bone has a density of about 1800 . . . 1900 kg m−3, [11, 31].

9) Also known as cortical bone; Latin name:substantia corticalis.

10) Also known as cancellous or spongy bone;Latin name: substantia spongiosa or sub-stantia spongiosa ossium.

11) The porosity is defined as the fraction ofthe volume of void-space and the total or

bulk volume of material, including the solidand void components. The range is be-tween 0 and 1, or as a percentage between 0and 100%.


Figure 6.11 Cross-sectional view of thehead of a femur.

A typical structure of a vertebral cancellous bone (µCT image) is shown inFigure 6.12 for a healthy state (a) and osteoporotic12) state (b). It can be seen that thestructure is open and made up of a network of beams or rods (so-called trabeculae).In the case of the diseased structure, a much higher porosity characterized by muchbigger holes and spaces compared to the healthy bone can be observed. This lossof density or mass may result in the fracture of the bone even for normal, that is,every day, loads. The density of the trabeculae was investigated by Galante et al. [32]based on 63 specimens, and a mean value of 1820 kg m−3 was found. This valueis exactly in the range of compact bone and this is the reason why some authors[31, 33] assign the physical properties of the compact bone to the beams or rods oftrabecular bone in the scope of modeling approaches.

The structure of trabecular bone and its relation to mechanical properties haveseveral important applications in biomedical engineering. The loss of the normaldensity of bone (osteoporosis) results in a fragile bone. The knowledge about thefracture strength as a function of the bone density can help predict the risk offracture and recommend appropriate treatments and behavioral rules. The actualdensity of the bone can be determined by a safe and painless bone mineral density(BMD) test13).

12) Osteoporosis, or porous bone, is a diseasecharacterized by low bone mass andstructural deterioration of bone tissue,leading to bone fragility and an increasedsusceptibility to fractures, especially of thehip, spine, and wrist, although any bonecan be affected.

13) A routine X-ray can reveal osteoporosis ofthe bone, which appears much thinner and

lighter than normal bones. Unfortunately,by the time X-rays can detect osteoporosis,at least 30% of the bone has alreadybeen lost. Major medical organizationsare recommending a dual energy X-rayabsorptiometry (DXA, formerly known asDEXA) scan for diagnosing osteoporosis.


(a) (b)

Figure 6.12 Images (µCT) of human bones: (a) healthy ver-tebra; (b) osteoporotic vertebra, ( by SCANCO MedicalAG, Bruttisellen Switzerland).

Total hip joint replacement involves surgical removal of the diseased head(‘‘ball’’) of a thigh bone, the femur, and a ‘‘cup-shaped’’ bone of the pelvis called theacetabulum (‘‘socket’’) and replacing them with an artificial ball and stem insertedinto the femur bone and an artificial plastic cup socket. Knowing the mechanicalproperties of trabecular bone allows the design and optimization of artificial hipswith properties close to the natural complement, which is supposed to prevent theloosening of the implanted prosthesis.

Osteoarthritis14) is a type of arthritis (damage of the body joints) that is caused bythe breakdown and eventual loss of the cartilage of one or more joints. Cartilage isa proteinaceous substance that serves as a ‘‘lubricant’’ or ‘‘cushion’’ between thebones of the joints. The load transmission between the bones and a joint is relatedto the mechanical properties of all constituents, and thus, changing properties ofthe trabecular may cause damage to the tribological system.

6.4.1Structural Analogies: Cellular Plastics and Metals

For a long time, the development of artificial cellular materials has been aimedat utilizing the outstanding properties of biological materials in technical applica-tions. As an example, the geometry of honeycombs was identically converted intoaluminum structures, which have been used since the 1960s as cores of lightweightsandwich elements in the aviation and space industries [34, 35]. Nowadays, inparticular, foams made of polymeric materials are widely used in all fields oftechnology. For example, Styrofoam� and hard polyurethane foams are widelyused as packaging materials. Other typical application areas are the fields of heat

14) Also known as degenerative arthritis,degenerative joint disease.


10 mm

(a) (b)

Figure 6.13 Open-cell Al foams: (a) Duocel�; (b) M-Pore�.

and sound absorption. During the last few years, techniques for foaming metalsand metal alloys and for manufacturing novel metallic cellular structures have beendeveloped [36]. Figure 6.13 shows two different Al foams that reveal a structurequite similar to that of the bones shown in Figure 6.12. On the basis of thesestructural analogies, it is possible to model the geometry of natural cellular (trabec-ular bone) and artificial cellular materials (plastic or metallic foams) with the samemathematical or engineering approaches despite the fact that the base materialproperties might be quite different.

The approaches found in literature to model the physical behavior of cellularmaterials and to derive respective constitutive equations to predict their behaviorcan be broadly divided into four directions:

• the use of phenomenological models,• the application of equations developed for composite materials [37, 38],• the analysis of model structures representative of the physical structure,• the analysis of the real structure, for example, based on µCT images [39–41].

The last approach, that is, consideration of the real geometry, would require thelowest degree of simplifications since the geometry would be exactly representedfor a certain volume. However, there are still problems connected with this mostadvanced approach:

• It is required to consider a representative volume element (RVE) that comprisesa certain number of cells in order to be representative for the heterogeneousor random cell structure. In addition, a minimum number of cells is requiredto avoid any influence of the edge effects occurring at the ‘‘free’’ surface of thevolume (which can be the boundary of the scanned volume) in order to representmacroscopic values for a larger arrangement of cells [42]. The precise numberdepends, however, on the corresponding cell structure and should be examinedseparately.

• The standard engineering tool to simulate the geometry is the finite elementmethod. The conversion of the geometrical information based on µCT scans [43]is normally automatically done and so-called tetrahedral elements are generated.However, these elements reveal inferior performance in the case of nonlinear


material behavior. Strategies to apply brick (hexaeder) elements by convertingvoxels into finite elements are given, for example, in Keyak et al. [44] and vanRietbergen et al. [45]. However, this approach results in unsmoothed surfaceswith sharp geometric discontinuities, which may give erroneous results [46],especially in the case of nonlinear material behavior

• The numerical approach based on the finite element method with real geometriesimplies high demands with respect to the computational hardware and, forexample, the detailed representation of the size of a mechanical test specimenstill reaches the limits of actual computer memory (RAM). In addition, access toexpensive scanning hardware (CT) is required.

In the following, we will concentrate on the third approach, that is, analysis ofmodel structures. In this case, the real geometry is strongly simplified. On the otherhand, analytical (e.g., Bernoulli–Euler beam theory) or numerical methods (e.g.,finite element method) at a much lower demand may be applied to investigate thephysical properties. The major idea is to create a model structure that representssome characteristics of the real geometry (e.g., porosity) and behaves (e.g., deforms)in a similar way as the real structure (e.g., major deformation behavior representedby beams under bending). However, such a modeling approach cannot aimto determine the precise values of the macroscopic material parameters sincea dependency on the pore geometry and arrangement exits. Nevertheless, thisapproach may aim to determine the principal characteristics of the constitutiveequation, while the exact values of the respective free material parameters needto be determined experimentally for a given real material. In other words, theapproach may be able to derive an equation to describe the material behavior, forexample, as a function of porosity, but there are parameters in the equation thatmust be evaluated through experiments. This approach is, for example, commonfor classical engineering materials, where ductile metals can be described by thevon Mises yield condition (known mathematical equation) whereas the yield stress(free material parameter) must be experimentally determined for each specificductile metal.

Typical expressions for the average mechanical properties such as stiffness ofcellular material are obtained as power laws of density or relative density with thefollowing structure:

mechanicalproperty = C1 · (density)C2 (6.65)

where C1 and C2 are real numbers to be fitted by experimental values. In thefollowing, some classical modeling approaches for the geometry will be described.

The so-called ball-and-strut model by Gent and Thomas [47] was applied torepresent the cellular morphology of open-cell rubber foams, cf. Figure 6.14(a).It consists of circular struts joined at their intersections by polymer spheres.The connecting elements (spheres) were assumed to be undeformable and theorientation distribution of the struts was taken to be random. In addition, it wasassumed that the entire surface of the spherical dead volume is covered by strutelements. This model was later applied by Lederman [49], who assumed that in


(a) (b)

Figure 6.14 Simple idealizations of open-cell structures: (a)the ball-and-strut model [47]; (b) the cubic strut model [48].

general the dead volume does not need to be entirely covered by strut elements. Theimportant geometrical parameter in this model is the ratio of the sphere diameterto the free strut length.

A cubical arrangement of square-section struts (cf. Figure 6.14(b)) was proposedby Gent and Thomas [48], who investigated the elastic behavior and the tensilerupture of open-cell foamed elastic materials. In their analysis, they assumedthat the junctions of the struts are essentially undeformable compared with thestruts themselves, and they referred to this as dead volume. An equivalent modelfor closed-cell systems was proposed by Matonis [50]. Cubic strut models havebeen extensively applied by Gibson and Ashby to different types of materials(bone, polymer, and metal foams [30, 31, 51]) to investigate different macroscopicproperties (e.g., elastic and plastic behavior) by consideration of beam bendingof the struts. Anisotropy can be simply incorporated in this cubical model byuniformly stretching it in one of the principal directions, [52]. It should be notedhere that the cubic strut model shown in Figure 6.14(b) behaves in the linear elasticrange according to Hooke’s law for orthotropic materials with cubic structure, cf.Section 6.3.3. Nevertheless, the derived material properties are in most of the casesassigned to an equivalent isotropic material.

For the cubic plate models shown in Figure 6.15, the primary deformation modeis bending as in the case of the cubic strut model. However, these models weremotivated by the cancellous bone of lower porosity, where the structure transformsinto a more closed network of plates. Some of these plate elements have smallperforations in them resulting in cells that are not entirely closed [53].

Columnar models with a hexagonal cross section are shown in Figure 6.16 for arodlike structure (a) and a platelike structure (b) of cancellous bone. These modelsare motivated by bones where the loading is mainly uniaxial (e.g., as the vertebrae)and the trabeculae often develop a columnar structure with cylindrical symmetry[54, 55]. The elastic behavior of such a model is described by generalized Hooke’slaw for transverse isotropic materials, cf. Section 6.3.4.


(a) (b)

Figure 6.15 (a) Cubic model of plate-like asymmetricstructure [31]; (b) perforated plate model [30].

(a) (b)

Figure 6.16 (a) Hexagonal model of rodlike columnarstructure [31]; (b) prismatic plate model [30].

The cubic block model in Figure 6.17 represents an orthotropic material withcubic structure (cf. Section 6.3.3) and the elastic properties have been investigated byBeaupre and Hayes [56] with the finite element method based on different loadingconditions, that is, uniaxial, compressive, and shear loads. A rather transverseisotropic material is represented by the plate–rod structure shown in Figure 6.17(b).Different deformation mechanisms are activated if a load is applied perpendicularto the plates (bending state) or if loads are applied parallel to the plates [30].A similar model with a regular arrangement of the connecting rods (now anorthotropic structure with two identical principal axes) was applied by Raux et al.[57] to model the trabecular architecture of the human patella.15)

15) The human patella, also known as the kneecap or kneepan, is a thick, triangular bone

which articulates with the femur and coversand protects the knee joint.


(a) (b)

Figure 6.17 (a) Open-celled cubic block [56]; (b) parallel plate model [30].

(a) (b)

Figure 6.18 (a) Tetrakaidecahedron [30]; (b) pentagonal dodecahedron [30].

A different modeling approach is based on so-called polyhedral cells [30]. Typicalrepresentatives are the tetrakaidecahedron, which is composed of six squares andeight hexagons (cf. Figure 6.18(a)) and the pentagonal dodecahedron, which isbuilt up of 12 regular pentagons (cf. Figure 6.18(b)). It should be noted herethat only the tetrakaidecahedron is a true space-filling body and the pentagonaldodecahedron does not pack properly unless distorted or combined with otherstructures.

At the end of this section, let us in addition mention the geometrical model by Ko[58] who considered geometrical shapes of interstices of hexagonal closest packingand FCC closest packing of uniform spheres. His approach can be imaginedas a uniform expansion of each sphere so that a contacting surface becomesflat and tangent to a contact point. Thus, each sphere becomes a polyhedron.As an example, the hexagonal closest packing will deform each sphere intotrapezo-rhombic dodecahedron with six equilateral trapezoids and six congruentrhombics. Most of the interstices are then squeezed to form 24 edges of such adodecahedron.

References 189

6.5Conclusions

Continuum mechanical approaches are an important tool to model the mechanicalbehavior of trabecular bone. The first part of this chapter summarized the commonapproaches based on generalized Hooke’s law in its different forms under sym-metry considerations. The second part of the continuum mechanical approachesintroduced basic ideas for yield and failure surfaces. The major intention of thispart was to collect some kind of basic understanding, which can be extendedto more complex approaches, for example, in the form of anisotropic yield orfailure criteria. The derived and applied constitutive equations can be used in thescope of the finite element method or any other numerical approximation methodto simulate the macroscopic behavior. However, it may turn out that a derivedconstitutive equation is not available in a commercial code. Thus, such laws mustbe implemented based on user-subroutines. Respective approaches for the imple-mentation, that is, integration of the constitutive equations, may be found in thetextbooks on nonlinear finite element analysis by Simo and Hughes [12], Crisfield[59, 60], and Belytschko et al. [61]. In the second part of this chapter, classical modelapproaches were introduced. Despite more and more powerful numerical tools andcomputer hardware, these simple models still remain attractive. Such models allowthe derivation of the basic structure of constitutive equations without given exactpredictions of the involved material constants. However, such material constantsmay be obtained from basic mechanical tests such as the uniaxial tensile or pureshear test.

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Biomechanics of Hard Tissues (Modeling, Testing, and Materials) || Constitutive Modeling of the Mechanical Behavior of Trabecular Bone - Continuum Mechanical Approaches