Page 13 - Applied Petroleum Geomechanics

Stresses and strains 3

1.1.2 Stress components
If an infinitesimal cube is cut within the rock, it will have normal and shear
stresses acting on each plane of the cube. The compressive normal stress is
positive, and the tensile normal stress is treated as negative in rock
mechanics sign convention. Each normal stress is perpendicular to each of
the planes, as shown in Fig. 1.2. However, the case of the shear stresses is
not so direct because the resulting shear stresses on any face will not
generally be aligned with these axes. The shear stress on any face in Fig. 1.2
has two perpendicular components that are aligned with the two axes
parallel to the edges of the face. Therefore, there are nine stress components
comprising three normal components and six shear components acting on a
cubic element. The stress tensor can be expressed as follows:
2 3
s x s xy s xz
6 7
s ¼ 4 s yx s y s yz 5 (1.3)
s zx s zy s z
By considering equilibrium of moments around the x, y, and z axes, the
shear stresses have the following relations:

s xy ¼ s yx ; s yz ¼ s zy ; s xz ¼ s zx (1.4)

Therefore, the state of stress at a point is defined completely by six
independent components. These are three normal stress components
(s x , s y , s z ) and three shear stress components (s xy , s yz , s zx ).

1.1.3 Stresses in an inclined plane
The principal stresses in two dimensions are very useful because many
engineering problems of practical interest are effectively two-dimensional,
such as the borehole problem during drilling operations, which can be

σ z
τ
τ zy zx
τ
τ yz xz τ
τ yx xy σ x x x

σ y
y
z
Figure 1.2 Normal and shear stress components on an infinitesimal cube in the rock.