2.6.2 Representation of Stress Tensor in Spherical
and Deviatoric Components
Hydrostatic stress component for any stress tensor is given by
p ¼
1
3
σ 11 þ σ 22 þ σ 33
ð
Þ
ð 3:11Þ
Using this definition, we can represent the stress tensor as a summation of
hydrostatic stress tensor and the remainder, which is called deviatoric stress tensor:
σ
½ Š ¼
p 0 0
0 p 0
0 0 p
2
6
4
3
7
5 þ
σ xx À p
σ xy
σ xz
σ yx
σ yy À p
σ yz
σ zx
σ zy
σ zz À p
2
6
4
3
7
5
ð3:12Þ
In the Eq. (3.12), the first tensor is called hydrostatic stress tensor and the second
tensor is called deviatoric stress tensor. Hydrostatic tensor is also called spherical
stress tensor. We will use S ij to represent deviatoric stress tensor:
S ij
 à ¼
S xx S xy S xz
S yx S yy S yz
S zx S zy S zz
2
6
4
3
7
5
where
S XX ¼ σ XX À
1
3
σ xx þ σ YY þ σ ZZ
ð
Þ
S XY ¼ σ XY , S XZ ¼ σ XZ , S YZ ¼ σ YZ
In the same fashion S YY and S ZZ can be written in Cartesian form, deviatoric
stress tensor is given by
S ij ¼ σ ij À
1
3
σ kk δ ij
where δ ij is the Kronecker delta, when
A
B
r = 0
r = l
y
x
z
O
Fig. 2.11 Definition of
local coordinates
26
2 Stress and Strain in Continuum