Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 166

Table 6.5 Formulations of generalized isotropic Hooke's law
based on different elastic constants (elastic stiffness form).

Hooke's law

σ = C ε | σ ij = C ijkl ε kl

Formulation based on Lame's constants

σ = ( λ 1 ⊗ 1 + 2 µ L − 1 ) ε | σ ij = λε kk δ ij + 2 µε ij

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

1 + ν · 1 − 2 ν 1 ⊗ 1 + L − 1 ε 1 + ν · ε ij + 1 − 2 ν δ ij ε kk

E ν E ν

σ = σ ij =

Formulation based on bulk modulus and shear modulus

σ = K ) ε · ε ij + δ ij ε kk

−

L − 1

1
3 1 ⊗ 1

3 K 2 G

6 G

1 ⊗ 1 + 2 G ( − σ ij = 2 G

Decomposition in volumetric and deviatoric parts

2 G L − 1 σ = 3 K ε m 1

Volumetric response

+ e

Deviatoric response

σ ij = K ε kk δ ij

Volumetric response

+ 2 Ge ij

Deviatoric response

compression test, from which Young's modulus E and Poisson's ratio ν can be

obtained.
A summary of the different formulations of Hooke's law in matrix form is given
in Table 6.5 and compared with the corresponding tensor form given in common
literature, for example, [2, 10, 15, 16].
The decomposition of the stress vector into spherical and deviatoric components
can also be performed based on projection tensors. The deviatoric stress vector

s

is
obtained by subtracting the spherical state of stress from the actual state of stress.
Thus, we can write

T

1
3 11

1
3 11

1
3 1 ⊗ 1

σ = I σ

T

s = σ − σ m 1 = σ −

σ =

I −

σ =

I − (6.30)

and define the deviatoric projection tensor

1
3 1 ⊗ 1

I = I − (6.31)

to transform the actual stress state in its deviatoric part. Respecting 1

T

1 = 3, we

can indicate the following properties of the deviatoric projection tensor: