Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 35

X 3

33

32

23

31

22

13

X 2

21 21

12

11

X 1

Figure 1.4 A rectangular parallelepiped, with faces parallel to the co-ordinate
axes, in the stressed medium.

forces vanish more strongly than the surface forces, and that, in the limit, force
equilibrium demands that

F ν = F 1 cos(ν, x 1 ) + F 2 cos(ν, x 2 ) + F 3 cos(ν, x 3 ). (1.213)

The state of stress in a medium can therefore be described by a quantity, which
associates a vector with each spatial direction, by means of an expression that is
linear and homogeneous in the direction cosines. In other words, stress is a second-
order tensor .
If we write the components of F 1 as (τ 11 ,τ 12 ,τ 13 ), those of F 2 as (τ 21 ,τ 22 ,τ 23 )
and those of F 3 as (τ 31 ,τ 32 ,τ 33 ), we have

F ν i = τ ji ν j , (1.214)

where the range and summation conventions are taken to apply. The τ ji are the
Cartesian components of the second-order stress tensor . They represent the com-
ponents of the forces per unit area on the faces of the rectangular parallelepiped
shown cut from the medium in Figure 1.4. The components τ 11 , τ 22 , τ 33 are
referred to as normal stresses , while τ 12 , τ 13 , τ 21 , τ 23 , τ 31 , τ 32 are referred to as
shear stresses .