Page 74 - Physical Principles of Sedimentary Basin Analysis

56 Linear elasticity and continuum mechanics

a property shown in Exercise 3.11. Nevertheless, deviatoric stress should not be confused
with differential stress, because differential stress is just one value, while the deviatoric
stress is a tensor with up to six different values.

Exercise 3.11 Show that equation (3.84) gives the deviatoric stress in the principal sys-
tem when σ 2 = σ 3 . Such a stress state is common in triaxial testing where a cylindrical
rock specimen is subjected to axial compression with the largest principal stress, and
confinement with σ 2 = σ 3 . See Figure 8.9.
1
Solution: Since σ m = (σ 1 + 2σ 3 ) we get
3
1 2

σ = σ 1 − (σ 1 + 2σ 3 ) = (σ 1 − σ 3 )
1
3 3
1 1

σ = σ 3 − (σ 1 + 2σ 3 ) =− (σ 1 − σ 3 ).
3
3 3

Exercise 3.12 Show that the first invariant I for a deviatoric stress tensor is 0.
1
3.11 Linear stress–strain relations
We have seen that stress and strain are related for linear elastic materials, and we have
defined three moduli for such materials. It is now time to generalize these relations in
terms of the stress and strain tensors. We start by adding the strain contributions in the
x-direction, where we have

1
(1)
ε = (3.85)
xx σ xx
E
ν
(2)
ε xx =−νε yy =− σ yy (3.86)
E
ν
(3)
ε =−νε zz =− σ zz . (3.87)
xx
E
The first part is Hooke’s law (3.1), and the next two parts are the ratio of transverse and lon-
gitudinal strain (3.2)inthe y- and z-directions. The transverse strain (ε yy and ε zz ) is related
to the transverse stress (σ yy and σ zz ) by Hooke’s law. The total strain in the x-direction is
the sum of all three contributions, because we are dealing with a linear elastic material:
1 ν ν
ε xx = σ xx − σ yy − σ zz . (3.88)
E E E
The total strain in the y- and z-directions is obtained in a similar way, and these are
respectively

ν 1 ν
ε yy =− σ xx + σ yy − σ zz (3.89)
E E E
ν ν 1
ε zz =− σ xx − σ yy + σ zz . (3.90)
E E E