Page 12 - Applied Petroleum Geomechanics

2 Applied Petroleum Geomechanics

Keywords: Anisotropy of rock; Constitutive equation; Effective stress; Mohr’s circle;
Stress and strain.

1.1 Stresses
1.1.1 Normal and shear stresses
The stress is equal to the force divided by the area. On a real or imaginary
plane through a rock, there can be normal force (DN ) and shear force (DS),
as shown in Fig. 1.1. The forces induce normal and shear stresses in the
rock. It should be noted that a solid can sustain a shear force and shear stress,
whereas a liquid or gas cannot (Hudson and Harrison, 1997). A liquid or gas
contains a pressure, i.e., a force per unit area, which acts equally in all di-
rections and hence is a scalar quantity. However, stresses in rocks normally
are not equal in different directions, and they are vectors.
The normal (shear) stress is the normal (shear) force per unit area as
shown in Fig. 1.1A. The normal and shear forces and normal and shear stress
components are shown in Fig. 1.1B. The normal stress is perpendicular to
each of the planes, but the shear stress is parallel to each of the planes as
shown in Fig. 1.1B. The normal and shear stresses can be mathematically
defined as follows when the size of the small area is reduced to zero:
DN
normal stress; s n ¼ lim ðDA/0Þ (1.1)
DA
DS
shear stress; s ¼ lim ðDA/0Þ (1.2)
DA

Figure 1.1 (A) Normal force (DN) and shear force (DS) and their acting area (DA). (B)
Normal stress (s) and shear stress (s) induced by normal and shear forces plotted in a
two-dimensional small element.