Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 34

X 3

n

F n

X 2

Δ S

X 1

Figure 1.3 A small tetrahedron, in a stressed medium, cut out by the Cartesian
co-ordinate planes.

Aforce F acts on the outside of the small surface element, while an equal and
opposite force F acts on the inside. The orientation of the surface element is

specified by its unit outward normal vector ν . To specify the state of stress in
the medium we must, in general, specify both F and ν . To do this, consider the
tetrahedron cut out by the Cartesian co-ordinate planes shown in Figure 1.3.
The faces of the tetrahedron orthogonal to the three co-ordinate directions have
areas

−

Δ S 1 , Δ S 2 and Δ S 3 , respectively, while the fourth face has area Δ S . Then,

Δ S 1 =− cos(ν, x 1 ) Δ S ,

Δ S 2 =− cos(ν, x 2 ) Δ S , (1.211)

Δ S 3 =− cos(ν, x 3 ) Δ S ,

where an area is counted positive if it is on the side of the normal vector or co-
ordinate direction, negative otherwise, and where cos (ν, x i ) is the cosine of the
angle between ν and e i , the unit vector in the x i -direction. Let the force per unit
area acting on S be F ν , that acting on Δ Δ S i be F i . The sum of the surface forces

acting on the tetrahedron is therefore
F ν

F 3 cos(ν, x 3 ) − F 1 cos(ν, x 1 ) − F 2 cos(ν, x 2 ) − Δ S . (1.212)

Body forces could also be acting on the medium enclosed by the tetrahedron.
Body forces are proportional to volume. Suppose L is a characteristic length of
the tetrahedron. Then, the surface forces are proportional to L 2 , the body forces
are proportional to L 3 .Ifwelet Δ S → 0 with ν fixed, it is evident that the body