Page 61 - Physical Principles of Sedimentary Basin Analysis

3.5 Stress 43

The displacement vector between the two points in the deformed state is a rotation by an
angle a of the displacement vector of the initial state. This rotation matrix is identical to
therotationmatrixin(2.54), which applies for small rotation angles.

3.5 Stress
Stress is a generalization of pressure, and similar to pressure it is defined as force per unit
area of a surface. The pressure in a fluid is a special case of stress where the stress is always
normal to the surface. Let the force F be acting on a (small) plane with surface area A,as
shown in Figure 3.6. It is assumed that the plane is so small that the force can be considered
constant over the surface. The stress vector acting on the plane is then the force divided
by the surface area, S = F/A. The vector S can be decomposed into a part normal to the
plane, S n , and a part tangential to the plane, S t , and we have that S = S n + S t . Notice that
the stress on the opposite side of the plane is −S because the force on that side of the plane
is −F. This follows from Newton’s law which says that the sum of all forces is zero on a
body at rest.
A stress state can be related to a coordinate system by using the forces acting on the sur-
face planes of a cube oriented parallel to the coordinate axes, (see Figure 3.6). The force
acting on the plane that is normal to the x-direction gives the stress vector σ x , when it
is divided by the surface area. The vector σ x can be written in terms of its vector com-
ponents as σ x = (σ xx ,σ xy ,σ xz ), where σ xx is the normal stress and where σ xy and σ xz
are the tangential (or shear) stress decomposed in the y- and z-directions respectively. The
stresses on the sides with outward unit vector pointing in the positive y- and z-directions
are found in the same way. We then get three stress vectors with nine components:

σ x = (σ xx ,σ xy ,σ xz ) (3.16)
σ y = (σ yx ,σ yy ,σ yz ) (3.17)
σ z = (σ zx ,σ zy ,σ zz ). (3.18)

z

n
S
S n
dz
σ
A xz y
S t σ
σ xx xy dx
dy
x
(a) (b)
Figure 3.6. (a) The stress acting on a plane can be decomposed into a normal stress component (S n )
and a tangential (or shear) component (S t ). (b) The stress components on the sides of a cube.