Page 66 - Physical Principles of Sedimentary Basin Analysis

48 Linear elasticity and continuum mechanics

σ ij n j = σn i . (3.41)

The unit normal vector n and the normal stress σ are seen to be the eigenvector and
the eigenvalue, respectively, of the stress tensor. It is shown in Notes 3.2 and 3.3,using
linear algebra, that there are three real eigenvalues (σ 1 , σ 2 and σ 3 ) and that the three cor-
responding eigenvectors (n 1 , n 2 and n 3 ) are orthogonal, because the stress tensor is real
and symmetric. Consequently, the eigenvectors give three orthogonal directions with only
normal stress. A plane oriented with one of the eigenvectors as the normal vector will have
the corresponding eigenvalue as the normal stress. The orthogonal system is the principal
system, and the corresponding stress is the principal stress. By convention, the largest prin-
cipal stress is numbered σ 1 and the least principal stress is numbered σ 3 . The eigenvalues
are a solution of

σ 11 − σ σ 12 σ 13

σ 21 σ 23
σ 22 − σ = 0 (3.42)

σ 33 − σ
σ 31 σ 32
which becomes the polynomial
2
3
σ − I 1 σ + I 2 σ − I 3 = 0 (3.43)
that has three solutions σ i , i = 1, 2, 3. The coefficients of the polynomial are

I 1 = σ 11 + σ 22 + σ 33
I 2 = σ 22 σ 33 − σ 23 σ 32 + σ 33 σ 11
− σ 31 σ 13 + σ 11 σ 22 − σ 12 σ 21 (3.44)
I 3 = σ 11 σ 22 σ 33 − σ 11 σ 23 σ 32 + σ 12 σ 23 σ 31
− σ 12 σ 21 σ 33 + σ 13 σ 21 σ 32 − σ 13 σ 22 σ 31
and they are called invariants because the solutions for the normal stress σ i have to be
independent of the coordinate system used to represent the stress tensor. The eigenvalues
σ i are the same regardless of how the stress tensor σ is rotated. It is shown rigorously
in Exercise 3.8 that I 1 , I 2 and I 3 really are invariant under rotation. We also have that
any combination of the invariants is also an invariant. An important use of the invari-
ants is the formulation of yield criteria. We will encounter the first invariant later because
1
it is proportional to the mean stress σ m = I 1 . The stress tensor in the principal system
3
is simply
⎛ ⎞
σ 1 0 0
σ = ⎝ 0 σ 2 0 ⎠ (3.45)
0 0 σ 3
and the normalized eigenvectors become the unit vectors of the orthonormal system

T T T
e 1 = (1, 0, 0) , e 2 = (0, 1, 0) , e 3 = (0, 0, 1) . (3.46)