Page 133 - Rock Mechanics For Underground Mining

STRENGTH CRITERIA FOR ISOTROPIC ROCK MATERIAL

An instructive and practically useful interpretation of the Hoek-Brown criterion
for brittle intact rock has been provided by Martin (1997) and others (e.g. Martin and
Chandler, 1994, Hajiabdolmajid et al., 2002, Martin et al., 1999), who studied the
laboratory and field behaviour of Lac du Bonnet granite. Martin (1997) found that,
in a manner consistent with that described in Section 4.3.7, the start of the fracture or
failure process began with the initiation of damage caused by small cracks growing in
the direction of the maximum applied load. For unconfined Lac du Bonnet granite, this
occurred at an applied stress of 0.3to0.4 c . As the load increased, these stable cracks
continuedtoaccumulate.Eventually,whenthesamplecontainedasufficientdensityof
these cracks, they started to interact and an unstable cracking process involving sliding
was initiated. The stress level at which this unstable cracking process is initiated is
referred to as the long term strength of the rock, cd . Martin (1997) argued that,
in terms of the Coulombic concepts of cohesion and friction, the mobilised strength
to this stage is cohesive. After the stress cd has been reached, cohesion is lost and
frictional strength is mobilised.
As illustrated in Figure 4.31, Martin (1997) determined the laboratory peak, long
term and crack initiation strengths for the Lac du Bonnet granite. He was able to fit
Hoek-Brown failure envelopes to these curves, although the laboratory crack initia-
tion curve was found to be a straight line on 1 versus 3 axes. Subsequently, in a
field experiment carried out at the URL site, the initiation of cracks around a tunnel
excavated in the Lac du Bonnet granite was recorded using microseismic emissions
(see section 18.2.7). As shown in Figure 4.31, these data correspond well with the
laboratory crack initiation data. It was found that crack initiation at approximately
constant deviatoric stress, ( 1 − 3 ), could be well represented by the Hoek-Brown
criterion with m b = 0 and s = 0.11 (Martin et al., 1999). This important result will
be used in later chapters of this book.

4.5.6 Yield criteria based on plasticity theory
The incremental theory of plasticity (Hill, 1950) is a branch of continuum mechanics
that was developed in an attempt to model analytically the plastic deformation or
flow of metals. Plastic deformation is permanent or irrecoverable; its onset marks the
yield point. Perfectly plastic deformation occurs at constant volume under constant
stress. If an increase in stress is required to produce further post-yield deformation,
the material is said to be work- or strain-hardening.
As noted in section 4.4.3, plastic or dissipative mechanisms of deformation may
occur in rocks under suitable environmental conditions. It would seem reasonable,
therefore, to attempt to use plasticity theory to develop yield criteria for rocks. The
relevant theory is beyond the scope of this introductory text and only the elements of
it will be introduced here.
Because plastic deformation is accompanied by permanent changes in atomic posi-
tions, plastic strains cannot be defined uniquely in terms of the current state of stress.
Plastic strains depend on loading history, and so plasticity theory must use an incre-
mental loading approach in which incremental deformations are summed to obtain the
total plastic deformation. In some engineering problems, the plastic strains are much
larger than the elastic strains, which may be neglected. This is not always the case
for rock deformation (for example, Elliott and Brown, 1985), and so an elastoplastic
analysis may be required.

115