Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 157

0.5

0.4

0.3

(b)

(a) 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]).

s m

s yy

s yy − s m

s yx
s xy

s yz
s zy

s yz
s zy

s yx

s m s m

s xy

s xx − s m

s zx s xz s zz − s m

s zx

s xz

s xx

s zz

(a) Total state (b) Hydrostatic part

(change in volume)

(c) Deviatoric part

(change in shape)

Figure 6.2 Decomposition of the stress tensor into its
spherical 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 should
be obtained from well-defined experimental investigations.

6.2
Summary of Elasticity Theory and Continuum Mechanics

6.2.1
Stress Tensor and Decomposition

It is of great importance in the framework of limit or failure surfaces of isotropic
materials to decompose the stress tensor σ ij into a pure volume changing (spherical
or hydrostatic) tensor σ

o
ij and a pure shape changing (deviatoric) stress tensor s ij

(cf. Figure 6.2) 1) :
σ

o
ij

= σ + s ij = σ δ + s ij (6.1) ij m ij

1) It should be noted that in the case of
anisotropic materials, a hydrostatic stress
state may result in a shape change, [6].