Constitutive Modeling of the Mechanical Behavior of Trabecular Bone – Continuum Mechanical Approaches - Biomechanics of Hard Tissues - page 158

1
3 (

σ = σ + σ + σ δ

In Eq. (6.1),

zz ) denotes the mean normal stress 2) and ij the m xx yy

δ = =

Kronecker tensor (

ij is equal to 1 if i

j ,and0if i j ). Furthermore, Einstein's

summation convention was used [7].
Equation (6.1) can be written with components in the following way as

σ σ σ σ m 00

0

s xx s xy s xz

xx xy xz

σ σ σ = σ 0 + s xy s yy s yz (6.2) xy yy yz m

σ σ σ 00 σ s xz s yz s zz xz yz zz m

ij

Stress tensor σ ij Deviatoric tensor s ij

Hydrostatic tensor σ

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

s ij = σ for i = j (6.3) ij

s ij = σ − σ m for i = j (6.4) ij

The hydrostatic part of σ ij has in the case of metallic materials (full dense materials),
for temperatures approximately under 0 . 3 · T kf ( T kf : melting temperature), nearly
no influence on the occurrence of inelastic strains since dislocations slip only
under the influence of shear stresses. On the other hand, the hydrostatic stress has
a considerable influence on the failure or yielding behavior in the case of cellular
materials, in soil or damage mechanics.

6.2.2
Invariants

To ensure the independence of a stress-based description of physical phenomena
from the chosen coordinate system (objectiveness), it is meaningful to use a set
of independent tensor invariants instead of the stress tensor components

ij .
These invariants are independent of the orientation of the coordinate system and
represent the physical content of the stress tensor. The so-called characteristic
equation 3)
det σ

σ

ij =

− λδ 0 (6.5) ij

or in components

σ − λσ σ xx xy xz

σ σ − λσ = 0 (6.6) xy yy yz

σ σ σ − λ xz yz zz

leads to the cubic equation

3 2

λ − I 1 ( σ

ij )

λ + I 2 ( σ

ij )

λ − I 3 ( σ

ij )

= 0 (6.7)

2) Also called the hydrostatic stress. In the
context of soil mechanics, also the pressure
p =− σ m is used. Be aware that some finite
element codes use this definition.

3) det( ) denotes the determinant of a matrix

or tensor.

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