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安徽理工大学毕业论文
Failure Properties of
Fractured Rock Masses as Anisotropic
Homogenized Media
Introduction
It is commonly
acknowledged that rock masses always display
discontinuous surfaces of
various sizes
and orientations, usually referred to as fractures
or joints. Since the latter have
much
poorer mechanical characteristics than the rock
material, they play a decisive role in the
overall behavior of rock
structures,whose deformation as well as failure
patterns are mainly
governed by those
of the joints. It follows that, from a
geomechanical engineering standpoint,
design methods of structures involving
jointed rock masses, must absolutely account for
such
‘‘weakness’’ surfaces in their
analysis.
The most
straightforward way of dealing with this situation
is to treat the jointed rock
mass
as
an
assemblage
of
pieces
of
intact
rock
material
in
mutual
interaction
through
the
separating
joint
interfaces.
Many
design-oriented
methods
relating
to
this
kind
of
approach
have been developed in the past
decades, among them,the well-
known
‘‘block theory,’’ which
attempts to
identify poten-
tially unstable lumps
of rock from geometrical and kinematical
considerations (Goodman
and
Shi
1985;
Warburton
1987;
Goodman
1995).
One
should
also
quote
the
widely
used
distinct element
method, originating from the works of Cundall and
coauthors (Cundall and
Strack
1979;
Cundall
1988),
which
makes
use
of
an
explicit
?nite
-difference
numerical
scheme
for
computing
the
displacements
of
the
blocks
considered
as
rigid
or
deformable
bodies. In this
context, attention is primarily focused on the
formulation of realistic models
for
describing the joint behavior.
Since
the previously mentioned direct approach is
becoming highly complex, and then
numerically
untractable,
as
soon
as
a
very
large
number
of
blocks
is
involved,
it
seems
advisable
to
look
for
alternative
methods
such
as
those
derived
from
the
concept
of
homogenization.
Actually,
such
a
concept
is
already
partially
conveyed
in
an
empirical
fashion by the famous Hoek and Brown’s
criterion (Hoek and
Brown 1980; Hoek
1983). It
stems from the intuitive idea
that from a macroscopic point of view, a rock mass
intersected
by
a
regular
network
of
joint
surfaces,
may
be
perceived
as
a
homogeneous
continuum.
Furthermore, owing to the existence of
joint preferential orientations, one should expect
such
a homogenized material to exhibit
anisotropic properties.
The
objective
of
the
present
paper
is
to
derive
a
rigorous
formulation
for
the
failure
criterion of a jointed rock mass as a
homogenized medium, from the knowledge of the
joints
and rock material respective
criteria. In the particular situation where
twomutually orthogonal
joint sets are
considered, a closed-form expression is obtained,
giving clear evidence of the
related
strength anisotropy. A comparison is performed on
an illustrative example between the
results
produced
by
the
homogenization
method,making
use
of
the
previously
determined
criterion,
and
those
obtained
by
means
of
a
computer
code
based
on
the
distinct
element
method.
It
is
shown
that,
while
both
methods
lead
to
almost
identical
results
for
a
densely
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安徽理工大学毕业论文
fractured rock mass, a ‘‘size’’ or
‘‘scale effect’’ is observed in the case of a
limited number of
joints. The second
part of the paper is then devoted to proposing a
method which attempts to
capture such a
scale effect, while still taking advantage of a
homogenization technique. This is
achieved by resorting to a micropolar
or Cosserat continuum description of the fractured
rock
mass,
through
the
derivation
of
a
generalized
macroscopic
failure
condition
expressed
in
terms of stresses and
couple stresses. The implementation of this model
is ?nally illustrated
on a simple
example, showing how it may actually account for
such a scale effect.
Problem Statement
and Principle of Homogenization Approach
The problem under consideration is that
of a foundation (bridge pier or abutment) resting
upon a fractured bedrock (Fig. 1),
whose bearing
capacity
needs
to
be
evaluated
from
the
knowledge
of
the
strength
capacities
of
the
rock
matrix and the joint
interfaces. The failure condition of the former
will be expressed through
the
classical
Mohr-Coulomb
condition
expressed
by
means
of
the
cohesion
C
m
and
the
friction angle
?
m
.
Note that tensile stresses will be counted
positive throughout the paper.
Likewise,
the
joints
will
be
modeled
as
plane
interfaces
(represented
by
lines
in
the
?gure’s plane). Their
strength properties are described by means of a
condition involving the
stress vector
of components (σ, τ) acting
at any
point of those interfaces
According
to
the
yield
design
(or
limit
analysis)
reasoning,
the
above
structure
will
remain safe under a
given vertical load Q(force per unit length along
the Oz axis), if one can
exhibit
throughout the rock mass a stres
s
distribution which satis?es the equilibrium
equations
along
with
the
stress
boundary
conditions,while
complying
with
the
strength
requirement
expressed at any
point of the structure.
This
problem
amounts
to
evaluating
the
ultimate
load
Q
﹢
beyond
which
failure
will
occur, or equivalently within which its
stability is ensured. Due to the strong
heterogeneity of
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安徽理工大学毕业论文
the
jointed rock mass, insurmountable dif?culties are
likely to arise when trying to implement
the above reasoning directly. As
regards, for instance, the case where the strength
properties
of the joints
are
considerably
lower than
those of the
rock matrix, the
implementation
of a
kinematic
approach
would
require
the
use
of
failure
mechanisms
involving
velocity
jumps
across the joints, since the latter
would constitute preferential zones for the
occurrence of
failure. Indeed, such a
direct approach which is applied in most classical
design methods,
is
becoming
rapidly
complex
as
the
density
of
joints
increases,
that
is
as
the
typical
joint
spacing l is becoming small in
comparison with a characteristic length of the
structure such as
the foundation width
B.
In
such
a
situation,
the
use
of
an
alternative
approach
based
on
the
idea
of
homogenization and related concept of
macroscopic equivalent continuum for the jointed
rock
mass,
may
be
appropriate
for
dealing
with
such
a
problem.
More
details
about
this
theory,
applied in the context of reinforced
soil and rock mechanics, will be found in (de
Buhan et al.
1989; de Buhan and Salenc
,on 1990; Bernaud et al. 1995).
Macroscopic Failure Condition for
Jointed Rock Mass
The
formulation
of
the
macroscopic
failure
condition
of
a
jointed
rock
mass
may
be
obtained from the
solution of an auxiliary yield design boundary-
value problem attached to a
unit
representative cell of jointed rock (Bekaert and
Maghous 1996; Maghous et al.1998). It
will now be explicitly formulated in
the particular situation of two mutually
orthogonal sets of
joints under plane
strain conditions. Referring to an orthonormal
frame O
?
1
?
2
whose axes are
placed
along the joints directions, and introducing the
following change of stress variables:
such a macroscopic failure condition
simply becomes
where it
will be assumed that
A convenient representation of the
macroscopic criterion is to draw the strength
envelope
relating to an oriented facet
of the homogenized material, whose unit normal n I
is inclined by
an angle a with respect
to the joint direction. Denoting by
?
n
and
?
n
the normal and
shear
components of the stress vector
acting upon such a facet, it is possible to
determine for any
value of a the set of
admissible stresses
(
?
n
,
?
n
) deduced from
conditions (3) expressed in
terms
of
(
?
1
1
,
?
22
,
?
12
).
The
corresponding
domain
has
been
drawn
in
Fig.
2
in
the
particular case where
?
?
?
m
.
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安徽理工大学毕业论文
Two comments are worth being made:
1.
The
decrease
in
strength
of
a
rock
material
due
to
the
presence
of
joints
is
clearly
illustrated
by
Fig.
2.
The
usual
strength
envelope
corresponding
to
the
rock
matrix
failure
condition
is
‘‘truncated’’
by
two
orthogonal
semilines
as
soon
as
condition
H
j
?
H
m
is
ful?lled.
2.
The
macroscopic
anisotropy
is
also
quite
apparent,
since
for
instance
the
strength
envelope drawn in
Fig. 2 is dependent on the facet orientation a.
The usual notion of intrinsic
curve
should therefore be discarded, but also the
concepts of anisotropic cohesion and friction
angle as tentatively introduced by
Jaeger (1960), or Mc Lamore and Gray (1967).
Nor
can
such
an
anisotropy
be
properly
described
by
means
of
criteria
based
on
an
extension
of
the
classical
Mohr-Coulomb
condition
using
the
concept
of
anisotropy
tensor(Boehler and Sawczuk 1977; Nova
1980; Allirot and Bochler1981).
Application to Stability of Jointed
Rock Excavation
The closed-form
expression (3) obtained for the macroscopic
failure condition, makes it
then
possible to perform the failure design of any
structure built in such a material, such as the
excavation shown in Fig. 3,
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安徽理工大学毕业论文
where
h
and
β
denote
the
excavation
height
and
the
slope
angle,
respectively.
Since
no
surcharge
is
applied
to
the
structure,
the
speci?c
weight
γ
of
the
constituent
ma
terial
will
obviously constitute the sole loading
parameter of the ing the stability of this
structure will amount to evaluating the
maximum possible height h
+
beyond which failure will
occur. A standard dimensional analysis
of this problem shows that this critical height
may be
put in the form
where
θ=joint
orientation
and
K
+
=nondimensional
factor
governing
the
stability
of
the
excavation.
Upper-bound
estimates
of
this
factor
will
now
be
determined
by
means
of
the
yield design kinematic approach, using
two kinds of failure mechanisms shown in Fig. 4.
Rotational Failure
Mechanism [Fig. 4(a)]
The ?rst class of
failure mechanisms considered in the analysis is a
direct transposition
of
those
usually
employed
for
homogeneous
and
isotropic
soil
or
rock
slopes.
In
such
a
mechanism
a volume of homogenized jointed rock mass is
rotating about a point Ω with an
angular velocity ω. The curve
separating this volume from the rest of the
structure which is
kept motionless is a
velocity jump line. Since it
is an arc
of the log spiral of angle
?
m
and
focus Ω the velocity discontinuity at
any point of this line is inclined at angle wm
with respect
to the tangent at the same
point.
The
work
done
by
the
external
forces
and
the
maximum
resisting
work
developed
in
such
a mechanism may be written as (see Chen and Liu
1990; Maghous et al. 1998)
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安徽理工大学毕业论文
where
w
e
and
w
me
=dimensionless
functions,
and
μ
1
and
μ
2
=angles
specifying
the
position of the center of rotation
Ω.Since the kinematic approach of yield
design states that a
necessary
condition for the structure to be stable writes
it follows from Eqs. (5)
and (6) that the best upper-
bound
estimate derived from this ?rst
class
of mechanism is obtained by
minimizati
on with respect to
μ
1
and
μ
2
which may be determined numerically.
Piecewise Rigid-Block Failure Mechanism
[Fig. 4(b)]
The second class of failure
mechanisms involves two translating blocks of
homogenized
material. It is de?ned by
?ve angular parameters.
In order to
avoid any misinterpretation, it
should
be
speci?ed
that
the
terminology
of
block
does
not
refer
here
to
the
lumps
of
rock
matrix
in
the
initial
structure,
but
merely
means
that,
in
the
framework
of
the
yield
design
kinematic approach, a wedge of
homogenized jointed rock mass is given a (virtual)
rigid-body
motion.
The
implementation
of
the
upper-bound
kinematic
approach,making
use
of
of
this
second class of failure
mechanism, leads to the following results.
where U
represents
the
norm of the velocity
of
the lower block. Hence, the following
upper-bound estimate for
K
+
:
Results and Comparison with Direct
Calculation
The optimal bound has been
computed numerically for the following set of
parameters:
The
result obtained from the homogenization approach
can then be compared with that
derived
from a direct calculation, using the UDEC computer
software (Hart et al. 1988). Since
the
latter can handle situations where the position of
each individual joint is speci?ed, a series
of calculations has been performed
varying the number n of regularly spaced joints,
inclined
at the same angleθ=10° with
the horizontal, and intersecting the facing of the
excavation, as
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安徽理工大学毕业论文
sketched in Fig. 5. The
corresponding estimates of
the stability factor have been plo
tted
against n in the same ?gure.
It
can
be
observed
that
these
numerical
estimates
decrease
with
the
number
of
intersecting
joints
down
to
the
estimate
produced
by
the
homogenization
approach.
The
observed
discrepancy between homogenization and
direct app
roaches, could be regarded as
a ‘‘size’’ or
‘‘scale effect’’ which is
not included in the classical
homogenization model. A possible way to
overcome such a limitation of the latter, while
still
taking
advantage
of
the
homogenization
concept
as
a
computational
time-saving
alternative
for
design
purposes,
could
be
to
resort
to
a
description
of
the
fractured
rock
medium
as
a
Cosserat
or
micropolar
continuum,
as
advocated
for
instance
by
Biot
(1967);
Besdo(1985);
Adhikary and Dyskin (1997); and Sulem and
Mulhau
s (1997) for strati?ed or
block
structures.
The
second
part
of
this
paper
is
devoted
to
applying
such
a
model
to
describing the failure properties of
jointed rock media.
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