Page 73 - Physical Principles of Sedimentary Basin Analysis

3.10 Deviatoric stress 55

T
k n = n Kn, where n is a unit vector, while the stress ellipsoid expresses all possible stress
vectors S = σn.
Exercise 3.10 Show how the strain tensor can be written as an ellipsoid in an analogous
way to the stress tensor.

3.10 Deviatoric stress
The ductile flow of rocks is often related to how far away the stress is from an isotropic
state. A measure for this difference is the deviatoric stress defined as the stress tensor minus
the mean stress
⎛ ⎞
σ xx − σ m σ xy σ xz

σ = ⎝ σ yx σ yy − σ m σ yz ⎠ (3.78)
σ zx σ zy σ zz − σ m
where the mean stress is
1 1
σ m = σ ii = σ xx + σ yy + σ zz . (3.79)
3 3
Using the more compact notation we write that

σ = σ ij − σ m δ ij . (3.80)
ij
In the principal system we have
⎛ ⎞
σ 1 − σ m 0 0

σ = ⎝ 0 σ 2 − σ m 0 ⎠ (3.81)
0 0 σ 3 − σ m
where σ 1 , σ 2 and σ 3 are the principal stress components. We notice that the component
σ = σ 3 − σ m is always less than (or equal to) zero. The deviatoric stress tensor therefore

3
contains negative components, regardless of how compressive the stress state might be.
The mean stress is one third of the stress invariant I 1 , and we therefore have
1
1 1
σ m = σ xx + σ yy + σ zz = σ 1 + σ 2 + σ 3 = I 1 . (3.82)
3 3 3
The deviatoric stress measures how far away a stress state is from the isotropic mean stress.
An often-used alternative to the deviatoric stress is the differential stress, defined as the
difference between the largest and the least principal stress components,
σ d = σ 1 − σ 3 . (3.83)

In the quite common situation with σ 2 = σ 3 the deviatoric stress is proportional to the
differential stress
⎛ 2 ⎞
3 0 0
1
σ = ⎝ 0 − 0 ⎠ (σ 1 − σ 3 ), (3.84)
3
0 0 − 1
3