Civil Engineering Reference
In-Depth Information
A more complete equivalent-continuum model
that includes the effects of joint orientation
and spacing is a micropolar (Cosserat) plasticity
model. The Cosserat theory incorporates a local
rotation of material points as an independent
parameter, in addition to the translation assumed
in the classical continuum, and couple stresses
(moments per unit area) in addition to the clas-
sical stresses (forces per unit area). This model, as
implemented in FLAC, is described in the context
of slope stability by Dawson and Cundall (1996).
The approach has the advantage of using a con-
tinuum model while still preserving the ability
to consider realistic joint spacing explicitly. The
model has not yet (as of 2003) been incorporated
into any publicly available code.
The most common failure criterion for rock
masses is the Hoek-Brown failure criterion (see
Section 4.5). The Hoek-Brown failure criterion
is an empirical relation that characterizes the
stress conditions that lead to failure in intact rock
and rock masses. It has been used successfully
in design approaches that use limit equilibrium
solutions. It also has been used indirectly in
numerical models by finding equivalent Mohr-
Coulomb shear strength parameters that provide
a failure surface tangent to the Hoek-Brown fail-
ure criterion for specific confining stresses, or
ranges of confining stresses. The tangent Mohr-
Coulomb parameters are then used in traditional
Mohr-Coulomb type constitutive relations and
the parameters may or may not be updated during
analyses. The procedure is awkward and time-
consuming, and consequently there has been little
direct use of the Hoek-Brown failure criterion
in numerical solution schemes that require full
constitutive models. Such models solve for dis-
placements, as well as stresses, and can continue
the solution after failure has occurred in some loc-
ations. In particular, it is necessary to develop a
“flow rule,” which supplies a relation between
the components of strain rate at failure. There
have been several attempts to develop a full con-
stitutive model from the Hoek-Brown criterion:
for example, Pan and Hudson (1988), Carter
et al
. (1993) and Shah (1992). These formulations
assume that the flow rule has some fixed relation
to the failure criterion and that the flow rule is
isotropic, whereas the Hoek-Brown criterion is
not. Recently, Cundall
et al
. (2003) has proposed
a scheme that does not use a fixed form of the flow
rule, but rather one that depends on the stress
level, and possibly some measure of damage.
Real rock masses often appear to exhibit pro-
gressive failure—that is, the failure appears to
progress over time. Progressive failure is a com-
plex process that is understood poorly and diffi-
cult to model. It may involve one or more of the
following component mechanisms:
•
Gradual accumulation of strain on principal
structures and/or within the rock mass;
•
Increases in pore pressure with time; and
•
Creep, which is time-dependent deformation
of material under constant load.
Each of these components is discussed briefly later
in the context of slope behavior.
Gradual accumulation of strain on principal
structures within the rock mass usually results
from excavation, and “time” is related to the
excavation sequence. In order to study the pro-
gressive failure effects due to excavation, one
must either introduce characteristics of the post-
peak or post-failure behavior of the rock mass
into a strain-softening model or introduce similar
characteristics into the explicit discontinuities. In
practice, there are at least two difficulties asso-
ciated with strain-softening rock mass models.
The first is estimating the post-peak strength and
the strain over which the strength reduces. There
appear to be no empirical guidelines for estimat-
ing the required parameters. This means that the
properties must be estimated through calibration.
The second difficulty is that, for a simulation in
which the response depends on shear localization
and in which material softening is used, the res-
ults will depend on the zone sizes. However, it
is quite straightforward to compensate for this
form of mesh-dependence. In order to do this,
consider a displacement applied to the boundary
of a body. If the strain localizes inside the body,
the applied displacement appears as a jump across
the localized band. The thickness of the band
