if i ¼ j δ ij ¼ 1,
if i 6 ¼ j δ ik ¼ 0
Hydrostatic stress is the same in all three directions; therefore, in isotropic
materials, hydrostatic stress causes elastic volumetric change only. However,
deviatoric stress is different in all directions; as a result, it can cause shape change
and inelastic (irreversible) deformation. In anisotropic materials, both hydrostatic
and deviatoric stresses can lead to inelastic shape change.
2.6.3 Invariants of the Deviatoric Stress Tensor
Principal deviatoric stresses are in the same planes as the total stress tensor principal
planes. Following the same procedure we used for the total stress tensor, we can
obtain the characteristic equation for the deviatoric stress tensor as follows:
S XX À λ
S YY
S XZ
S YX
S YY À λ
S YZ
S ZX
S ZY
S ZZ À λ
2
6
4
3
7
5 ¼ 0
This equation can be expanded to the following form:
λ
2
À J 2D λ À J 3D ¼ 0
where J 2D and J 3D are the second and third invariants of the deviatoric stress tensor,
respectively. The first invariant of the deviatoric stress tensor is zero,J 1D ¼ 0:
J 2D ¼
1
2
S ij S ij
¼
1
6
σ XX À σ YY
ð
Þ
2 þ σ YY À σ ZZ
ð
Þ
2 þ σ XX À σ ZZ Þ
2
þ σ
2
XY þ σ
2
XZ þ σ
2
YZ
À
Ã
h
J 2D ¼
1
2
S
2
XX þ S
2
YY þ S
2
ZZ
À
Á þ σ
2
XY þ σ
2
XZ þ σ
2
YZ
J 2D ¼
1
2
S ij S jk S ki
2.6 Stress Tensor Invariants
27