Page 127 - Applied Petroleum Geomechanics

Rock strengths and rock failure criteria 119

(A) (B)

Figure 3.28 Shear failure plane (A) and the Mohr circle (B) in the rock with a set of
parallel planes of weakness.

Fig. 3.28 plots the MohreCoulomb failure envelope and the Mohr
circle representation of the rock containing a set of parallel planes of
weakness. The shear failure angle (b w ) in the shear failure plane of the rock

can be obtained from the Mohr circle, i.e., b w ¼ 45 þ 4 w /2. Compared to
Eq. (3.51), the minimum strength or the shear failure that the anisotropic
rock is most likely to occur is when the bedding angle, b, is equal to the
shear failure angle (b w ), i.e., b ¼ b w ¼ b min .

3.4.4 DruckerePrager failure criterion

The MohreCoulomb failure criterion does not consider the effect of the
intermediate principal stress. Laboratory data have shown that the inter-
mediate principal stress plays an important role in the failure of rocks. The
DruckerePrager failure criterion can describe rock failures in the domain of
three principle stresses (Drucker and Prager, 1952). The DruckerePrager
failure criterion can be expressed in the effective stress form, i.e.,
p ffiffiffiffi
J 2 ¼ a 0 I þ k (3.52)
0
1
0
where a 0 and k are material constants; I 1 and J 2 are the stress invariants,
which can be expressed as:
0
0
0
I ¼ s þ s þ s 0 (3.53)
1 1 2 3
1 2 2 2
J 2 ¼ ðs 1 s 2 Þ þðs 2 s 3 Þ þðs 3 s 1 Þ (3.54)
6