Page 154 - Applied Petroleum Geomechanics

Basic rock fracture mechanics 147

The cohesive crack model can be used to describe the nonlinear
deformation ahead of a crack tip in rock. It is actually a modification to
Dugdale’s crack model originally developed for metals (Dugdale, 1960). In
this model a notional crack with an effective crack length is assumed. This
effective crack length consists of a traction-free portion (true crack length)
and a length of the FPZ over which a cohesive stress, tending to close the
crack, is distributed (refer to Fig. 4.7). Such a hypothesized crack is also
referred to as a fictitious crack (Whittaker et al., 1992). The cohesive
crack model has the following assumptions (Labuz et al., 1985; Whittaker
et al., 1992):
(a) The rock in the FPZ is partially damaged but still able to carry closing
stress s (x), which is transferred from one surface to the other of the
crack. The rock outside the FPZ is assumed to be linear elastic.
(b) The FPZ starts to develop when the maximum principal stress reaches
the tensile strength T 0 and the corresponding true crack tip opening
displacement d t is zero. With increasing d t , the stress is decreased until
zero and the corresponding d t reaches a critical value d c .
(c) The closing cohesive stress is a function of true crack tip opening
displacement d t .
(d) Overall stress intensity factor at the notional crack tip no longer exists,
i.e., stress singularity at the notional crack tip disappears.
In Dugdale’s crack model a constant closing stress distribution was used.
For rocks, the closing stress, s (x), should probably be a function of the

y

Traction Elastic stress
free zone distribution zone
T
δ c σ(x)

O x

True crack FPZ
length: a length: l
Effective crack
half length: a + l
Figure 4.7 Representation of the fracture process zone of rock in the cohesive
crack model.