Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 168

Table 6.6 Formulations of generalized isotropic Hooke's
law based on different elastic constants (elastic compliance
form).

Hooke's law

ε = C − 1 ε ij = C − 1 σ

ijkl σ kl

Formulation based on Lame's constants

λ σ kk δ ij

1
2

λ 1

2

λ

ε =

µ ·

L −

λ 1 ⊗ 1

σ ε ij =

µ ·

σ ij −

2 µ + 3 2 µ + 3

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

· L −

1 + ν 1 ⊗ 1 σ

· σ ij −

1 + ν σ kk δ ij

1 + ν
E

ν 1 + ν

E

ν

ε = ε ij =

Formulation based on bulk modulus and shear modulus

2 G · L −

1 ⊗ 1 σ

2 G · σ ij −

σ kk δ ij

1 3 K − 2 G

9 K

1 3 K − 2 G

9 K

ε = ε ij =

Decomposition in volumetric and deviatoric parts

1
3 K σ m 1 +

1
2 G Ls

1
3 K σ m δ ij +

1
2 G s ij

ε = ε ij =

This inversion can be done using Sherman-Morrison formula [17, 18] given in
general form for a matrix

a and two vectors u and v ( α , β scalars) as

− β A − 1

u ⊗ v · A − 1 T

α A + β ( u ⊗ v ) − 1

T

1
α

A − 1 = (6.40)

α + β v

T

· A − 1 u

as long as α v · A − 1 1

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

T

u =−

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

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