Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 37

The stress tensor has only six independent Cartesian components: three normal
stresses, τ 11 , τ 22 , τ 33 and three shear stresses, τ 12 , τ 23 , τ 13 .

1.4.3 Analysis of deformation
The deformation of a body is described by the vector displacement field
u i ( x 1 , x 2 , x 3 ), which is the vector displacement the material particle originally at
( x 1 , x 2 , x 3 ) experienced during deformation. The displacement of a material particle
P , originally at x k + dx k , relative to another material particle P , originally at x k ,

is then

∂ u i
∂ x j dx j .

du i = (1.222)

The vector gradient ∂ u i /∂ x j is a second-order tensor. Since du i and dx j are vec-
tors, in a new co-ordinate system they become

du i = c ij du j , dx j = c jk dx k , (1.223)

and in reverse transformation,

c lk dx l . dx k = (1.224)

Then,

∂ u i
∂ x j

c ij ∂ u j c ij c lk ∂ u j du i = dx j =

∂ x k dx l ,

c ij du j =

∂ x k dx k

= (1.225)

or

∂ u i dx l =

∂ x l − c ij c lk ∂ u j

0. (1.226)

∂ x k

Thus,

∂ u i

∂ x l = c ij c lk ∂ u j ∂ x k ,

(1.227)

in accordance with the transformation law for second-order tensors.
The vector gradient can be separated into symmetric and antisymmetric parts by
writing

∂ u i
∂ x j +

∂ u i
∂ x j −

∂ u i
∂ x j =

1
2

∂ u j
∂ x i

1
2

∂ u j
∂ x i

+ = e ji + ω ji , (1.228)

where

∂ u j ∂ u j

1
2 ∂ x i + ∂ u i

1
2 ∂ x i − ∂ u i

e ij = , ω ij = (1.229)

∂ x j ∂ x j

are both second-order tensors.