Page 64 - Physical Principles of Sedimentary Basin Analysis

46 Linear elasticity and continuum mechanics

3.6 Rotation of stress and strain
The stress and strain are often wanted in a rotated coordinate system. The rotation of the
stress and strain tensor is done in the same way as rotation of the permeability tensor in
Section 2.9. Force is a vector, and the force vector that acts on a plane with area A and out-
ward unit normal vector n j is f i = σ ij n j A. The same expression in vector–matrix notation
is f = σnA, where σ = (σ ij ) is the stress tensor. The force vector f becomes rotated by
multiplication by the rotation matrix R, and the components in the rotated coordinate sys-
tem are denoted by a prime. (The rotation matrix is derived in Section 2.8.) The rotated
force becomes

f = Rf (3.27)
= RσnA (3.28)
= RσR −1 RnA (3.29)

= σ n A (3.30)
where we had R −1 R = 1 (the identity matrix). Recall the nice property of the rotation
T
matrix that the inverse matrix is the transpose, R −1 = R (see Exercise 2.13). The stress

tensor in the rotated coordinate system is therefore σ = RσR −1 ,or

σ = R ik σ kl R −1 (3.31)
ij lj
when it is written out with indices. The strain tensor is rotated the same way as the stress
tensor. A small line segment represented by the vector ds = dx is rotated into

ds = Rds
= R ds
= R R −1 Rds

= ds (3.32)
which gives that the rotated strain tensor is = R R −1 .

Note 3.1 The strain tensor in the rotated coordinate system is

1 ∂u ∂u

j
i
= + (3.33)
ij
2 ∂x ∂x
j i

where u and x are the displacement field and coordinates of the rotated system, respec-
i j
tively. They are both related to the unrotated displacement field and coordinates by use

of the rotation matrix, u = R ik u k and x = R lm x m , where the inverse relationship is
i l
−1
x l = R lm x . The rotated strain tensor becomes
m