Tectonophysics, 226 (1993) 187-198
Elsevier Science Publishers B.V., Amsterdam
A rheological model of a fractured solid
Vladimir Lyakhovsky
a,
Yury Podladchikov and Alexei Poliakov
Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,
69978 Tel Auiv, Israel
Institute of Experimental Mineralogy, Chernogolovka, Moscow District 142432, Russia
Hans Ramberg Tectonic Laboratory, Institute of Geology, Uppsala University, Box 555, 751 22 Uppsala, Sweden
(Received April 23,1993; revised version accepted May 26,1993)
ABSTRACT
Experiments to study the behavior of various materials point to the relation that exists between elastic properties and the
type of stress. The influence of the state of stress on the elasticity of a fractured material will be discussed for a physically
non-linear model of an elastic solid.
The strain-dependent moduli model of material, presented in this paper, makes it possible to describe this feature of a
solid. It also permits to simulate a dilatancy of rocks.
A damage parameter, introduced into the model using a thermodynamical approach, allows to describe a rheological
transition from the ductile regime to the brittle one, and to simulate the rock's memory, narrow fracture zone creation and
strain rate localization. Additionally, the model enables the investigation of the final geometry of fracture zones, and also to
simulate their creation process, taking into account pre-existing fracture zones.
The process of narrow fracture zone creation and strain rate localization was simulated numerically for single axis
compression and shear flow.
Introduction
The early experiments of Adams and his colleagues Von Karman (1911) and Griggs (1936)
demonstrated that rheological properties of rocks
depend on the applied load and temperature.
Two basic regimes of rock behavior have been
revealed in these and later experiments (Kirby,
1980):
(1) The brittle regime, in which specimen deformation results from localized displacements or
shear fractures or faults.
(2) The ductile regime, which is characterized
by a very small effect of pressure on strength, no
significant dilatation and negligable evidence of
intercrystalline glide or other mechanisms of
plastic deformation.
The transition between brittle and ductile behavior involves a broad region of semibrittle behavior in which both stable microfracturing and
ductile processes occur.
Empirical relationships between the strength
and the critical least principal stress of the brit-
tle-ductile transition were investigated by Byerlee (1968), Kirby (1980) and many others. Goetz
and Evans (1979) paid attention to the fact that
the application of a number of well-known constitutive relationships, such as perfectly elastic,
visco-elastic, elasto-plastic to plastic behavior of
the lithosphere, fails to describe a number of
basic phenomena. Among them the dependence
of the deformational parameters of the material
on the type of loading, the ability of the material
to remember the history of a fracture process,
etc. The main difficulty lies in describing the
brittle regime. There are different ways of expressing the constitutive relationships for non-homogeneous or fractured media. One of them is
based on the assumption that a body consists of a
matrix and a set of cracks or inclusions. All
physical characteristics of the matrix and the inclusions are supposed to be known. This approach was put forward by Eshelby (1957). and
has been successfully applied to composite materials (Christensen, 1979). For higher crack densities when crack interactions cannot be neglected,
0040-1951/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
V.LYAKHOVSKY ET AL.
O'Connel and Budiansky (1974) and Budiansky
and O'Connel (1976) proposed a self-consistent
method for the calculation the elastic moduli for
random crack orientation statistics. In this method
the effect of crack interactions is included by
assuming that each crack is embedded in a
medium with the effective stiffness of the cracked
body. Bruner (1976) and Henyey and Pomphrey
(1982) have pointed out that this scheme may
overestimate the crack interactions and have proposed an alternative, differential scheme. Another method has been proposed by Kachanov
(1987), Sayers and Kachanov (1991) for finding
the effective properties for solids having interacting cracks with arbitrary crack interactions. However, these results are obtained for each given
arrangement of cracks rather than in statistical
terms. Thus, the only schemes available at pxesent for calculating the effective elastic constants
at finite crack densities are the self-consistent
and differential schemes. Both of them are not
applicable for the description of a fracture distribution and fragmentation of rocks which have
been identified as power law phenomena and
indicates a fractal relation (e.g., Turcotte, 1986;
Herrmann and Roux, 1990; Velde et al., 1991).
Fractal analysis has been used to characterize the
nature of the pattern of fractures in natural rocks
and particularly the pattern in major fault systems (e.g., Okubo and &, 1987; Aviles et al.,
1987). In this case, there is another method to
simulate effective rheological parameters based
on a phenomenological approach.
Experimental studies of the behavior of various granular materials show noticeable dependence of elastic properties on the type of loading.
This is particularly manifest in the difference
between strain characteristics of the material under tension and under compression. Opening and
closing of cracks under tension and under compression produce variations of elastic moduli and
including abrupt changes when the strain reverses. The dependence of elastic properties on
the type of loading has been observed in experiments with a variety of design materials (Collins,
1981) and rocks (e.g., Zoback and Byerlee, 1975;
Stavrogin and Protosenya, 1979; Alm et al., 1985).
Several mechanical models with strain-depen-
dent moduli have been suggested; the one of
Arnbartsumyan-Khachatryan (Ambartsumyan,
1982) assumes that in the stress-strain law each
of the compliances (Poisson's ratio divided by
Young's modulus) changes whenever the stress
for which this compliance acts as a factor reverses. Jones (1977) suggested a model of a material with a matrix of weighted compliances; the
weights depend on absolute values of the principal stresses. For materials with a weakly non-linear response it is possible to assume that the
elastic moduli depend only on the type of stress.
This assumption was the basis for the model of
Lomakin and Rabotnov (1978).
A long-term action of stress and temperature
will cause growth or healing of microcracks resulting in a variation of the rheological properties
of a solid. The recent investigations by Yukutake
(1989) and Lockner et al. (1991) of a fracturing
process in granite show that this process may
hardly be described in terms of the classical Griffits model of single crack propagation. hckner
and Madden (1991) put forward a multiple-crack
model of brittle fracture. Many attempts have
been made to define and estimate suitable variables to describe the extent of fracturing. Such
variables should be able to characterize the processes of fracturing quantitatively and should be
suitable for modeling from the point of view of
continuum mechanics. Among such approaches
are Robinson's (1952) linear cumulative creep
damage law, Hoff's (1953) ductile creep rupture
theory, Kachanov's (1958, 1986) brittle rupture
theory, Rabotnov's (1969) coupled damage creep
theory and many modifications of these theories.
Following this approach and using the strain-dependent moduli model, a description of the damage evolution of a fractured medium under longterm stress has been put forward (Lyakhovsky
and Myasnikov, 1985; Myasnikov et al., 1990;
Lyakhovsky et al., 1991). This model has made it
possible to simulate the evolution of fracturing
under tectonic stress, resulting in the appearance
of seismic boundaries of rheological nature (Mints
et al., 1987; Lyakhovsky and Myasnikov, 1987,
1988) and investigate a faulting process in the
northern Dead Sea rift (Ben-Avraham and
Lyakhovsky, 1992).
RHEOLOGICAL MODEL OF A FRACTURED SOLID
Deformational properties of rocks
Following Lyakhovsky and Myasnikov (1984),
the dependence of the elastic potential U e on the
invariants Il = eii and I2= eijeij of the strain
tensor eij may be written as:
where i, j = 1, 2, 3; p is the density, A, p are
Lame constants; the coefficient v is an additional
elastic modulus; here and later, repeating indexes
imply summation.
Apart from the quadratic terms containing I:
and I, common to elastic solids, the potential (1)
includes the non-analytical second-order term
VI,&
which accounts for the effects of fracturing in the material. Such an addition to the
elastic potential is the simplest one for which the
requirement of being analytical is rejected. It
allows for differing properties of a solid under
tension and under compression. For models in
which non-linearity is taken into account by means
of higher-order terms, non-linear effects vanish at
small strains. The suggested model preserves its
properties under arbitrarily small deformations
since the principal terms and the additional term
are of the same order.
In a solid described by potential (I), the relation between the stress tensor uij and the strain
tensor is:
where 6,, = 1, if i =j, and 0, if i # j.
Using the function 6 = I l / & to describe the
form of stress state, effective elastic moduli can
be introduced by:
To find the value 6 in terms of uij, the following
equation should be solved:
where:
Volume deformation [10**4]
Fig. 1. Average pressure versus volume strain of Westerly
granite under different type u3/a, of 3-D load (dash lines
with markers). Simulated average pressure versus volume
strain (heavy lines).
Constructing a numerical method to find the
strain tensor in terms of a known stress tensor to
solve eqn. (3), we have to search for roots of a
polynomial with real coefficients. In the case of
pure shear stress (uii= O), eqn. (3) has the form:
~ t ~ - ( 3 ~ + 2 ~ ) t + 3 ~ = 0
and has the following solution:
The sign in front of the radical is chosen to give
regularity of the solution as v -+ 0. For small v
we get the solution by neglecting higher terms:
3v
3 ~ + 2 , ~
'=
The obtained expression for 5 is strictly positive.
This means that a shear stress in the solid causes
extensional strain (dilatation). The effect of dilatancy in rocks was discovered by Bridgeman in
1949 and has been studied by Reynolds (1885)
and has been studied by many researchers. Figure
1 shows the average pressure p = (2c1 + u3)/3of
Westerly granite under 3-D load versus volume
strain of sample (Schock, 1977); u3 is the maximal tension stress; a, = u2 are compressive
stresses. The results of analogous simulations using the above described scheme are also shown
V. LYAKHOVSKY ET AL.
cm2. This means a necessity to take into account
the dependence of elastic moduli on the type of
loading and makes it possible to estimate the
value of additional modulus incorporated in eqn.
(1). Using the results of experiment shown in
Figure 2, the estimation of v / ( A 2 ~ for) marble is about 10% and for diabase is about 2.5%.
This difference is very natural because the
strength of diabase is much higher than that of
marble.
Figure 3 shows results of estimations of effective elastic moduli A, p for different types of
loading 6 and the approximation of a strain-dependent moduli model. Experimental points were
calculated using results of granite sample testing
collected by Volarovich (1988). Each point represents an independent measurement of the load
and the resulting strain tensor that allows an easy
calculation of the value of elastic moduli. The
approximation represented by Figure 3 is much
better then the one by Hook which provide constant (independent on type of loading) estimation
of elastic moduli. Evaluation of the three elastic
moduli as a best fit to the experimental data
gives: A = 3.95 x l o 4 MPa; p = 0.11 X lo4 MPa;
Y = 0.12 x lo4 MPa.
+
Fig. 2. Stress-strain relation for marble ( 0 )and diabase ( A 1.
3-D compression - heavy lines; 1-D compression, a = ul,u2
= r3= 0,
- compression (dash lines), E , - tension (dotted
lines).
on Figure 1. The elastic moduli were assumed to
be h = p , v = 0.2A. Comparison of the slopes of
the average pressure shows a good agreement
between the simulated description of the dilatancy and the experiment.
Figure 2 shows the results of sample testing
with marble and diabase under 1-D and 3-D
compression (Stavrogin and Protosenia, 1979).
The connection between stress and strain for
small deformation is linear, but the value of elastic moduli depends on the type of loading. According to this experiment bulk modulus of marble under 3-D compression is 3.3 x l o 5 kg/cm2,
but under 1-D compression it is 2.5 X lo5 kg/
4 :
0
1
0.2
I
0.4
I
0.8
I
0.8
I
1
I
IS
5 p e of deformation
I
1.4
1
1.8
1
1.B
Model of medium with damage
Many writers on continuum thermodynamics
have postulated it as a state function of all the
0.10 1
A
d.2
d.4
d.6
d.s
i
i.2
vpe of deformation
Fig. 3. Elastic moduli h (a) and p (b) VS. type of deformation of granite from Kola Peninsula.
4
i.8
I
1.8
RHEOLOGICAL MODEL O F A FRACTURED SOLID
state variables, including "hidden variables" (e.g.,
Malvern, 1969) not available for macroscopic observation. Colernan (1964) and Coleman and
Mizel (1964) considered the class of materials in
which thermodynamical state functions depend
not only on the instantaneous values of the variables, but also on their gradients. Coleman's approach promises to be useful since it deals with a
limited numbers of explicitly enumerated macroscopic state variables and not a vague collection
of unspecified substate variables. Edelen and
Laws (1971) and Edelen et al. (1971) demonstrated the dependence of thermodynamical state
functions on gradient of "hidden variables" results in nonlocal properties of continuum. This
approach was used by Truskinovsky (1991) to
describe the process of phase transition.
In order to simulate a fracturing process in
terms of fractal analysis an non-dimensional damage parameter a! was incorporated. This parameter lies in the interval from 0 to 1 to describe the
evolution of the medium from the ideal undestroyed elastic solid to the absolutely destroyed
material.
Following this approach we represent the free
energy of a solid F as:
where Fo = F, (T, a); U e is the elastic potential
(1)with the elastic moduli A, p , v to be functions
of a; p is the coefficient of temperature expansivity, and T the temperature.
We express the elastic strain tensor eij in
terms of a metric tensor describing the current
state of medium gjj and a metric tensor g:
describing the irreversible viscous deformation:
entropy can be written (e.g., Malvern, 1969). In
the case of lack of damage variation and viscous
flow, the process of deformation must be reversible.
Using this condition we arrive at the ordinary
equation for the stress tensor (Myasnikov et al.,
1990):
It is usually assumed in irreversible thermodynamics (e.g., Fitts, 1962; De Croot and Mazur,
1962) according to a principle of maximum rate
of entropy production or maximum dissipation
power, that the constitutive equations give the
fluxes as functions of the forces, and that at least
"in the neighborhood of equilibrium" the constitutive equations (called phenomenological equations) are equations of the form:
da
aF
- = -cdt
aa
where C is a positive constant describing the
temporal scale of a damage process.
This equation is well-known as the GinzburgLandau equation, widely used in the theory of
phase transitions (e.g., Honenberg and Halperin,
1977; Urnantsev and Roitburd, 1988; Bronsard
and Kohn, 1991; Truskinovsky, 1991). Analogous
to the J-integral in fracture mechanics, the thermodynamic force aF/aa can be interpreted
(Lemaitre, 1985) as the energy release rate resulting from the increase of damage.
To describe a viscous flow we use a common
rheological law for a incompressible Maxwell
body:
The expression for the whole strain rate tensor
e j j = f(dgij/dt) is similar to the common used
form, usually formulated in terms of velocities
To close the set of eqns. (5-7), the equation of
motion have to be added ( f i forces):
(vi):
Since the free energy is a function of T, eij, a ,
taking into account the irreversible changes of T ,
a and gt, the balance equations of energy and
The final system of eqns. (5-8) describes elastic
deformation, viscous flow, and damage evolution
of the media. Detals of the application of the
thermodynamics equations to the damage process
V. LYAKWOVSKY ET AL.
were discussed in Myasnikov et al. (1990) and
Lyakhovsky et al. (1991).
Parameters and main features of the model
The simplest formulation of the model is to
assume a linear dependence of elastic moduli A,
p, v on the damage parameter a. For a = 0 the
material is expected to be an ideal undestroyed
solid, i.e., a linear elastic Hook's model. So, for
a = 0 the third elastic modulus u in eqn. (1) have
to be equal to zero. For a! = 1 the material is
absolutely destroyed, i.e., it is impossible to create stresses resulting in expansion. Mathematically this means that elastic potential U e (eqn. 1)
is not convex as a function of strain tensor eij for
I, > 0. This condition gives us the maximum value
for v = V, as a function of A = A,, ,u = p, for
a=l:
Using this assumption the dependence of elastic
moduli on a may be written as:
The next step is to describe the existence of
two different stable regimes which may be interpreted as ductile and brittle, and the process of
rheological transition (bifurcation). From the theory of catastrophe it is known that to describe the
bifurcation process we have to assume the energy
to be a function of the fourth order of its parameter (e.g., Hale and Kocak, 1991). Following this
recommendation, we represent 1F, as:
where K is a constant and a,, a, are levels of
damage corresponding to the ductile and brittle
regimes, respectively. Using eqns. (6) and (9) the
damage evolution is:
Fig. 4. Free energy as a function of damage parameter a for
different stresses.
where:
Equation (10) has a number of stationary solutions. The dependence of the free energy F on a
in the case of an unloaded sample (uij = 0) is
shown in Figure 4a. There are two equivalent
stable solutions corresponding to the ductile ( a =
a,) and brittle ( a = a 2 ) regimes and one unstable
[a = (a, a2)/2] solution. Thus, the damage of
unloaded homogeneous solid comes to one of
these two solutions. This process may be very
slow and the solid can "remember" its initial
+
RHEOLOGICAL MODEL OF A FRACTURED SOLID
to describe the process of creation of narrow high
fractured zones with the strain rate localization.
Since the modeled solid has "memory" it is possible to introduce some pre-existing high- and lowdamage zones and to simulate their temporal
evolution. After the unloading, in contrast to the
models of plasticity, the simulated material "remembers" information about the created damage
zones. The next stage of loading may reactivate
these old zones or provoke some new fracturing.
Numerical simulation of damage evolution under
single axis compression
o!
0
I
0.5
I
1
I
1.5
I
2
I
2.5
I
I
I
I
I
3
3.5
4
4.5
5
Confining pressure &bad
Fig. 5. Differentia1 stress versus confining pressure at the
transition to brittle behavior; I = granite Kola Peninsula
(Stavrogin and Protosenia, 1985); 2 = Spruce Pine Dunite;
3 = Cabramurra Serpentinite; 4 = Nahant Gabbro (Raleigh
and Paterson, 1965). Solid line is the boundary between the
brittle and ductile regions determined from the model.
level of damage for a very long time. Loading of a
sample results in a variation of its elastic energy
causing an increase or decrease of damage. A
relatively high shear stress promotes the increase
of damage. The energy (Fig. 4b) corresponding to
the ductile regime becomes much higher than the
energy of the brittle one. The subsequent increase of the stress leads to a disappearance of
the first solution (Fig. 4c).
It is impossible for the sample to keep its
initial ductile regime and it yields an abrupt
change of the damage. At this moment a rheological transition from the ductile to the brittle
regime occurs. Under relatively high pressures
this process may reverse and will result in a
healing of damage.
The level of stresses at the ductile-brittle transition depends on the type of loading. Figure 5
represents the comparison of the simulated differential stress versus confining pressure at the
ductile-brittle transition with results of experiments for different crystalline rocks.
The problem of space-temporal damage evolution is of special interest for various geological
problems. One of the main features of the model,
as it will be shown numerically, is the possibility
First, we start the simulation with the most
simple example: single axis compression of the
rectangle sample neglecting the process of viscous
relaxation of stresses. The process of damage
evolution is assumed to be quasi-static, i.e., the
stress distribution satisfies the equilibrium equation of an elastic solid for each time step. The
new pattern of damage is calculated using eqn.
(10) for the known stress distribution and the new
distribution of a or new elastic moduli results in
a new stress field at the next step. To simulate
elastic stresses at each time step of the simulation, the finite-element method based on an algorithm by Zienkiewicz (1971) was used. Application of the finite-element method to the solution
of different variations of the elastic problem was
widely discussed in literature. In this case,
quadratic interpolation of displacements within
two-dimensional triangular elements was used.
The area integrals were evaluated by GaussLegendre quadrature. To take into account the
dependence of effective elastic moduli on the
type of loading simple iteration method was used.
Figure 6 shows different time steps of the
damage evolution in a rectangle area under one
axis compression. To initiate the beginning of the
fracture, a small zone of high damage value was
located at the boundary. Stress distribution and
evolution of damage at the tip of the fracture
zone is similar to the problem of crack growth in
elastic media. In contrast to the problem of crack
evolution in elastic media, due to the finite size
of a gradient zone between destroyed and undestroyed area, there is no singularity. Stress distri-
V. LYAKHOVSKY ET AL.
194
possible. The initially continuous area breaks into
two pieces.
Shear flow simulation
To simulate any flow it is necessary to define
an effective viscosity as a function of the parameters of a solid. Power law viscosity is usually
assumed for rocks in the ductile regime. For the
brittle regime hardly anything is known about the
viscosity, except its very strong dependence on
the level of fracturing. To take into account
stronger viscosity variations in comparison with
elastic moduli, we assume the viscosity 77 to be an
exponential function of the damage parameter a:
Fig. 6. Distribution of damage parameter a for the simulated
evolution of the narrow fracture zone in rectangular area
under l - compression.
~
(a) Geometry of the pre-existing
damage zone to initiate the fracture process. (b) Intermediate
stage of the damage zone propagation. (c) Final stage of
simulation.
bution has a regular solution at every point and
eqn. (10) provides the fracture zone with a finite
rate of growth. At the intermediate stage (Fig.
6b), similar to the process of crack propagation in
elastic media, there are two different possible
directions for the future damage increase. The
selection of the direction is very sensitive to the
stress distribution and the accuracy of the numerical scheme strongly influences this process. During the final stage of the simulation (Fig. 6c), the
damage 'One reaches the opposite boundary of
the area and the continuation of the simulation in
the frame of elastic displacements becomes im-
where A, B are constants.
Figure 7 shows a comparison of a stress-strain
rate relation obtained by this model and by power
law for the order n = 3 and n = 10.
Numerical simulation has been done using
"FLAC" algorithm (Cundall and Board, 1988;
Cundall, 1989). The formulation is explicit-intime, using an updated Lagrangian scheme to
provide the capability for large strains. FLAC is
believed to offer advantages over conventional
finite-element schemes in cases where flow and
material instability occur. Physical instability can
be modeled without numerical instability if internal terms are included in the equilibrium equa-
________----------------............
...................
.....................
4
Strain rate
Fig. '7. Stress versus strain rate for ductile regime. Heavy line
- ,,,ested
model: dotted and dash lines - mwer law
-viscosity for n = 3 and n = 10,respectively.
RHEOLOGICAL MODEL OF A FRACTURED SOLID
tions. An application of this method to the simulation of a visco-elastic flow is described in Bolshoi et al. (1992).
The computational mesh consists of quadrilateral elements, which are subdivided into pairs of
constant-strain triangles, with different diagonals.
This overlay scheme ensures symmetry of solution
by averaging the results obtained on two meshes
(Cundall and Board, 1988). Linear, triangularelement shape functions L, ( k = 1,3) are defined
as:
,
where a,, b,, are constants and (x,y) are grid
coordinates. These shape functions are used to
interpolate the nodal velocities u/kj within each
triangular element (e):
This formula enables the calculation of the strain
increments in each triangle and the element
stresses using eqn. (5). When the stresses in each
triangle are known, the forces at the node number n ( F Y I ) are calculated by projecting the
stresses from all elements surrounding the node:
where n j is the jth component of the unit vector
normal to each side of the two element adjacent
to the node n. The length of each side is denoted
by dl. The minus sign is a consequence of Newton's Third Law. New velocities are computed by
integrating eqn. (8) over a given time step A t :
+
~ ( " ) ( t A t ) = vIn)(t )
+
At
F,(")--
P
If a body is at mechanical equilibrium, the net
forces on each node are zero; otherwise, the
nodes are accelerated. New coordinates of the
grid nodes are computed by:
xln'(t + A t )
= xin)(t )
Fig. 8. Computer simulation of the shear deformation of
initially homogeneous rectangle area. (a) Initial shape of the
area. (b) The beginning of the deformational process. (c) The
final stage demonstrates the three rigid blocks which moves
separately.
+ uln)At
and then calculations are repeated for the new
configuration. Together with the deformation, a
new damage distribution is calculated using
known stresses. Thus, visco-elastic deformation of
a damage material is simulated.
The FLAC method has advantage over implicit
methods because it is computationally inexpensive for each time step and it is memory-efficient
because matrices storing the system of equations
are not required.
A computer simulation of the shear deformation of an initially homogeneous rectangular body
and the damage distribution is shown in Figures 8
and 9. The boundary conditions of this example
were as follow: shear movement at the top and
the bottom and free surface at the left and right.
The first step of deformation (Fig. 8b) results in
V. LYAKHOVSKY ET AL.
Fig. 9. Same as in Fig. 8 showing the damage increase (isolines correspond to a level of the damage).
homogeneous deformation similar to a shear flow
in a linear viscous body. The deformation causes
a change of the shape of the area and creates
conditions for stress concentration at the two
corners (Fig. 9b). The damage propagates from
the corners into the area (Fig. 9c, d) and finally
yields two parallel zones separating the sample
into three rigid blocks (Fig. 9e). This result of
three rigid blocks separated by fractured zones
can be also seen in Figure 8c.
Conclusions
Experiments for studying the behavior of various materials point to the relation that exists
between elastic properties and the type of stress.
The influence of the state of stress on the elastic-
ity in a fractured material can be taken into
account in a physically non-linear model of an
elastic solid. The additional non-analytical second-order term included in the elastic potential
makes it possible to simulate the dependence of
effective elastic moduli on the type of loading,
particularly under tension and compression. Such
an addition to the elastic potential is the simplest
one for which the requirement of being analytical
is rejected. For models in which non-linearity is
taken into account by means of higher-order
terms, non-linear effects become negligable at
small strain. The suggested model preserves its
properties under arbitrarily small deformations
since the principal terms and the additional term
are of the same order. It is also shown that a
shear stress in the simulated solid causes extensional strain (dilatation). A comparison of the
slopes of the average pressure for the sample
testing (Fig. la) and numerical experiment (Fig.
lb) proves a good agreement between the model
and experiment.
A long-term action of stress and temperature
will cause growth or healing of microcracks resulting in a change in rheological properties of a
solid and rheological transition between brittle
and ductile regimes. The present study of the
rheological model of rocks is based on the thermodynamical approach in continuum mechanics.
The crucial point of the model is the description
of the existence of two different steady regimes
which may be interpreted as ductile and brittle,
and the process of rheological transition (bifurcation). The simulated state of stress at the
ductile-brittle transition (Fig. 5) is close to the
empirical Coulomb-Mohr failure criterion and is
in a good agreement with laboratory observations. The description of slow flow of material in
a ductile regime (Fig. 7) is analogous to the
power law viscous model.
The results of some numerical experiments are
presented in order to study mechanisms involved
in strain localization and fracturing processes.
These numerical examples show one of the main
features of the model, i.e., the possibility to simulate the process of fracture zone propagation. In
contrast to the model of single crack propagation
in elastic media, there is no singularity at the tip
I
i
i
RHEOLOGICAL MODEL OF A FRACTURED SOLID
of a propagating fracture zone and a finite rate of
its growth. The final stage of the evolution of the
fracture zones is similar to the same numerical
experiments for visco-plastic models, but the present model makes it possible to investigate all
intermediate stages of the process too.
This approach makes it also possible to take
into account the effect of the rock's memory, or
the ability of the material to remember a history
of fracture process. Thus, some pre-existing fracture zone might be introduced into the simulation. This feature of the model has been used for
the investigation of a faulting process in the
northern Dead Sea rift (Ben-Avraham and
Lyakhovsky, 1992).
References
Am, O., Jaktlund, L.L. and Kou, S., 1985. The influence of
microcrack density on the elastic and fracture mechanical
properties of Stripa granite. Phys. Earth Planet. Inter., 40:
161-179.
Ambartsumyan, S.A., 1982. Raznomodulnaya Teoria Uprugosti (Strain-Dependence Elastic Theory). Nauka, Moscow, 320 pp. (in Russian).
Aviles, C.A., Scholz, C.H. and Boatwright, J., 1987. Fractal
analysis applied to characteristic segments of the SanAndreas fault. J. Geophys. Res., 92: 331-344.
Ben-Avraham, 2.and Lyakhovsky, V., 1992. Faulting process
along the northern Dead Sea transform and the Levant
margin. Geology, 20: 1143-1 146.
Bolshoi, A.N., Cundall, P.A., Podladchikov, Y.Y. and
Lyakhovsky, V.A., 1992. An explicit inertial method for
the simulation of viscoelastic flow: an evaluation of elastic
effects on diapiric flow in two and three layers models.
UMSI Rep. 92/110, 20 pp.
Bronsard, L. and Kohn, R.V., 1991. Motion by mean curvature as a singular limit of Ginzburg-Landau Dynamics. J.
Diff. Eqn., 90: 211-237.
Bruner, W.M., 1976. Comments on "Seismic velocities in dry
and saturated cracked solids" by O'Connel and Budiansky.
J. Geophys. Res., 81: 2573-2576.
Budiansky, B. and O'Connel, R.J., 1976. Elastic moduli of a
cracked solid. Int. J. Solids Struct., 12: 81-97.
Byerlee, J.D., 1968. Brittle-ductile transition in rocks. J.
Geophys. Res., 73: 4741-4750.
Christensen, R.M., 1979. Mechanics of Composite Materials.
Wiley, New York, NY, 330 pp.
Coleman, B.D., 1964. Thermodynamics of materials with
memory. Arch. Ration. Mech. Anal., 17: 1-46.
CoIeman, B.D. and Mizel, V.J., 1964. Existence of caloric
equations of state in thermodynamics. J. Chem. Phys., 40:
1116-1125.
Collins, J.A., 1981. Failure of Materials in Mechanical Design.
Wiley, New York, NY, 620 pp.
Cundall, P.A., 1989. Numerical experiments on localization in
frictional materials. Ing. Arch., 59: 148-159.
Cundall, P.A. and Board M., 1988. A microcomputer program
for modeling large-strain plasticity problems. In: C. Swoboda (Editor), Numerical Methods in Geomechanics. Proc.
6th Int. Conf. on Numerical Methods in Geomechanics,
Innsbruck. Balkema, Rotterdam, pp. 2101-2108.
De Groot, S.R. and Mazur, P., 1962. Nonequilibrium Thermodynamics. North-Holland Publishing Co., Amsterdam.
Edelen, D.G.B. and Laws, N., 1971. On the thermodynamics
of systems with nonlocality. Arch. Ration. Mech. Anal., 43:
24-35.
Edelen, D.G.B., Green, A.E. and Laws, N., 1971. Nonlocal
continuum mechanics. Arch. Ration. Mech. Anal., 43:
36-44.
Eshelby, J.D., 1957. The determination of the elastic field of
an ellipsoidal inclusion and related problems. Proc. R.
Soc. London, Ser. A, 241: 376-396.
Fitts, D.D., 1962. Nonequilibrium Thermodynamics. McGrawHill, New York, NY,
Griggs, D.T., 1936. Deformation of rocks under high confining pressure, J. Geol., 44: 541-577.
Goetz, C. and Evans, B., 1979. Stress and temperature in the
bending lithosphere as constrained by experimental rock
mechanics. Geophys. J.R. Astron. Soc., 59: 436-478.
Hale, J. and Kocak, H., 1991. Dynamics and Bifurcations.
Springer-Verlag, New York, NY, 568 pp.
Henyey, F.S. and Pomphrey, N., 1982. Self-consistent moduli
of a cracked solid. Geophys. Res. Lett., 9: 903-906.
Herrmann, H.J. and Roux, S. (Editors), 1990. Statistical Models for the Fracture of Disordered Media. Elsevier, Amsterdam, 353 pp.
Hoff, N.J., 1953. The necking and rupture of rods subjected to
constant tensile loads. J. Appl. Mech., 20: 105-108.
Hohenberg, P.C. and Halperin, B.I., 1977. Theory of dynamic
critical phenomena. Rev. Mod. Phys., 49: 435-480.
Jones, R.M., 1977. Stress-strain relation for materials with
different moduli in tension and compression. AIAA J., 15:
62-73.
Kachanov, L.M., 1958. Time of the rupture process under
creep conditions. Izv. Acad. Nauk USSR, Otd. Tech., 8:
26-31 (in Russian).
Kachanov. L.M., 1986. Introduction to Continuum Damage
Mechanics. Martinus Nijhoff, Dordrecht, 135 pp.
Kachanov, L.M., 1987. Elastic solids with many cracks: a
simple method of analysis. Int. J. Solids Struct., 23: 23-43.
Kirby, S.H., 1980. Tectonic stress in the lithosphere: Constrains provided by the experimental deformation of racks.
J. Geophys. Res., 85: 6353-6363.
Lemaitre, J., 1985. A continuous damage mechanics model for
ductile fracture. J. Eng. Mater. Technol., 107: 83-89.
Lockner, D.A., Byerlee, J.D., Kuksenko, V., Ponomarev, A.
and Sidorin, A., 1991. Quasi-static fault growth and shear
fracture energy in granite. Nature, 350: 39-42,
V. LYAKHOVSKY ET AL.
Lockner, D.A. and Madden, T.R., 1991. A multiple-crack
model of brittle fracture. 1. Non-time-dependent simulation; 2. Time-dependent simulation. J. Geophys. Res., 96:
19,623-19,654.
Lomakin, E.V. and Rabotnov, Yu.N., 1978. Sootnosheniya
teorii uprugosti dla isotropnogo raznomodulnogo tela. Izv.
Akad. Nauk SSSR, Mekh. Tverd. Tela, 6: 29-34 (in Russian).
Lyakhovsky, V.A. and Myasnikov, V.P., 1984. On the behavior
of elastic cracked solid. Phys. Solid Earth, 10: 71-75.
Lyakhovsky, V.A. and Myasnikov, V.P., 1985. On the behavior
of visco-elastic cracked solid. Phys. Solid Earth, 4: 28-35.
Lyakhovsky, V.A. and Myasnikov, V.P., 1987. Relation between seismic wave velocity and state of stress. Geophys. J.
Astron. Soc., 2: 429-437.
Lyakhovsky, V.A. and Myasnikov, V.P., 1988. Acoustics of
rheologically nonlinear solids. Int. J. Phys. Earth Planet.
Inter., 50: 60-64.
Lyakhovsky, V., Podladchikov, Y. and Poliakov, A., 1991.
Rheological model of a fractured solid. UMSI Rep.
91/296, 30 pp.
Malvern, L.E., 1969. Introduction to the Mechanics of a
Continuum Medium. Prentice-Hall, Englewood Cliffs, NJ,
713 pp.
Mints, M.V., Kolpakov, N.I., Lanev, V.S., Rusanov, M.S.,
Lyakhovsky, V.A. and Myasnikov, V.P., 1987. On the
problem of the nature of subhorizontal seismic boundaries
(interpretation of Kola super-deep hole). Rep. Acad. Sci.
USSR, 296: 71-76.
Myasnikov, V.P., Lyakhovky, V.A. and Podladchikov, Yu.Yu.,
1990. Non-local model of strain-dependent visco-elastic
media. Rep. Acad. Sci. USSR, 312: 302-305.
O'Connel, R.J. and Budiansky, B., 1974. Seismic velocities in
dry and saturated cracked solids. J. Geophys. Res., 79:
5412-5426.
Okubo, P.G. and Aki, K., 1987. Fractal geometry in the San
Andreas fault system. J. Geophys. Res., 92: 345-355.
Rabotnov, Y.N., 1969. Creep rupture. Proc. 12th Int. Congt.
Appl. Mech., Springer, Berlin.
Raleigh, C.B. and Paterson, M.S., 1965. Experimental deformation of serpentinite and its tectonic implications. J.
Geophys. Res., 70(16): 3965-3985.
Reynolds, O., 1885. On the dilatancy of media composed of
rigid particles in contact. Philos. Mag. Ser. 5, 20: 469-481.
Robinson, E.L., 1952. Effect of temperature variation on the
long-term rupture strength of steels. Trans. Am. Soc.
Mech. Eng., 174: 777-781.
Sayers, C.M. and Kachanov, M., 1991. A simple technique for
finding effective elastic constants of cracked solids for
arbitrary crack orientation statistics. Int. J. Solids Struct.,
27: 671-680.
Schock, R.N., 1977. The response of rocks to large stresses.
In: D.L. Roddy, R.O. Pepin and R.B. Merrill (Editors),
Impact and Explosion Cratering. Pergamon Press, New
York, NY, pp. 657-688.
Stavrogin, A.N. and Protosenya, A.G., 1979. Plasticity of
Rocks. Nedra, Moscow, 301 pp. (in Russian).
Stavrogin, A.N. and Protosenya, A.G., 1985. Strength of
Rocks. Nedra, Moscow, 450 pp. (in Russian).
Truskinovsky, L., 1991. Kinks versus shocks. In: R. Fosdick, E.
Dumn and M. Slemrod (Editors), Shock Induced Transitions and Phase Structures in General Media. IMA Vol.,
Springer-Verlag, Berlin.
Turcotte, D.L., 1986. Fractals and fragmentation. J. Geophys.
Res., 91: 1921-1926.
Umantsev, A.P. and Roitburd, A.L., 1988. Nonisothermal
relaxation in nonlocal media. Sov. Phys. Solid (FTT),4:
613.
Velde, B., Dubois, J., Moore, D. and Touchard, G., 1991.
Fractal patterns of fractures in granites. Earth Planet. Sci.
Lett., 104: 25-35.
Von Karman, T., 1911. Festigkeits versuche unter all seitigem
Druck. Verh. Dtsch. Ingr., 55: 1749-1757.
Volarovich, M.P. (Editor), 1988. Physical Properties of Rocks.
Handbook, Nedra, Moscow, 255 pp. (in Russian).
Yukutake, H., 1989. Fracturing process of granite inferred
from measurements of spatial and temporal variations in
velocity during triaxial deformation. J. Geophys. Res., 94:
15,639-15,651.
Zienkiewics, O., 1971. The Finite Element Method in Engineering Science. McGraw Hill, New York, NY, 541 pp.
Zobak, M.D. and Byerlee, J.D., 1975. The effect of microcrack dilatancy on the permeability of Westerly granite. J.
Geophys. Res., 80-B: 752-755.