Page 14 - Applied Petroleum Geomechanics

4 Applied Petroleum Geomechanics

y

θ
σ

σ x τ
τ xy
θ
x
τ yx
σ y
Figure 1.3 Force equilibrium on a small triangle element, assuming that all the stress
components are positive.

simplified as the state of plane strain. Consider a two-dimensional small
triangular element of the rock in which the normal stresses s x and s y and
shear stress s xy act in the xy-plane. The normal (s) and shear (s) stresses at a
surface oriented normal to a general direction q in the xy-plane (Fig. 1.3)
can be calculated as follows:
s x þ s y s x s y
s ¼ þ cos 2q þ s xy sin 2q
2 2
(1.5)
s y s x
s ¼ sin 2q þ s xy cos 2q
2
By proper choice of q, it is possible to obtain s ¼ 0. From Eq. (1.5) this
happens when:
2s xy
tan 2q ¼ (1.6)
s x s y

Eq. (1.6) has two solutions, q 1 and q 2 . The two solutions correspond to
two directions for which the shear stress s vanishes. These two directions
are named the principal axes of stress. The corresponding normal stresses,
s 1 and s 3 , are the principal stresses, and they are found by introducing
Eq. (1.6) into the first equation of Eq. (1.5):

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
s x þ s y 2 ðs x s y Þ
s 1 ¼ þ s þ
xy
2 4
(1.7)
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
s x þ s y 2 ðs x s y Þ
s 3 ¼ s þ
xy
2 4