F'ractals, Vol. 2, No. 4 (1994) 567-581
@ World Scientific Publishing Company
FRACTAL PLASTIC SHEAR BANDS
ALEXEI N. B. POLIAKOV* and HANS J. HERRMANN,
PMMH (U.R.A. 857), ESPCI, 10 rue Vauquelin, 75231 Paris, Cedex 05, France
HLRZ, KFA Julich, Postfach 1913, 52425 Jiilzch, Germany
YUFU YU. PODLADCHIKOV*
Department of Sedimentary Geology, Vrije Universiteit,
1081 HV Amsterdam, The Netherlands
STEPHANE ROUX
PMMH (U.R.A. 857), ESPCI, 10 rue Vauquelin, 75231 Paris, Cedex 05, France
Received September 12, 1994; Accepted September 16, 1994
Abstract
We present a numerical study model of shear bands in rocks with a non-associated plastic flow
rule. The system drives spontaneously into a state in which the length distribution of shear
bands follows a power law and where the spatial organization of the shear bands appears to
be fractal. The distribution of local gradients in deviatoric strain rate has different scaling
exponents for each moment which we calculate and discuss. Samples of granodiorite from the
Pyrenees sheared under high confining pressure are analyzed and their properties compared
with the numerical results.
1. INTRODUCTION
a network of thin shear bands crisscrossing each
n i ~ l using Carrara
Already in 1911 von ~ & ~ r showed
marble t h a t when rocks are cornpressed under sufficiently high hydrostatic pressures and sheared, they
undergo a transition from brittle fracture t o plastic deformation characterized by t h e appearance of
other at roughly 45' with respect t o the ~ r i n c i ~ a l
stress axis. IAter the same effect was shown for
other types of rocks as sandstones, ~ i m e s t o n e as
~~?~
well as for soils and sand.4 There are also numerous
observations of plastic shear zones in the field.5v6
"Also affiliated to the Institute of Experimental Mineralogy of RAN, Chernogolovka, Russia.
567
568
A . N . B. Poliakov et al.
Fig. 1 Natural example of ductile shear bands in a rock (very homogeneous granodiorite) from the Central Pyreneas (region
of Cap de Long). The 200 FF bill in the lower left corner serves as scale. The box-counting technique gives a fractal dimension
of 1.58 f 0.08.
Figure 1 shows dark conjugate shear zones that we
obtained from a granitoid of the Neouvielle Massif in the Central Pyrenees consisting of a very homogeneous granodiorite with equiaxes minerals of
an average size 1-2 mrn. This massif has been deformed during Alpine orogenesis, thus developing
evenly distributed conjugate shear zones on different scales ranging from centimeters to kilometers.
We present examples on the scale of several meters
which we obtained in the central part of the massif
along the Cap de Long.
Recently a n analysis of granite using a petrographic microscope7 showed that a cut through a
fault network gives fractal structures with fractal
dimensions depending on pressures and the nature
of the external load. We present in this paper a
numerical simulation of the shear bands of elastoplastic materials on different scales and calculate
their fractal properties. We also measure for the
first time in the field the fractal dimension of shear
bands in granodiorite formed under confining pressures of around 5 kbars.
2. CONSTITUTIVE RELATIONS
It is well known from experiments that rocks increase in volume and dilate, when distorted by
shear. Dilation occurs by tensile cracking and is
necessary to lift sliding blocks over asperities. A
parameter suitable for characterizing a dilatant material is the dilatancy angle $. In t he case of simple
shear, tan($) is the ratio of plastic volumetric strain
rate divided by plastic shear strain rate. Materials
with $ = 0 are plastically incompressible and so
conserve volume. Materials with higher $J dilate
more with plastic strain. This gives the plastic flow
rule which besides the yield criterion characterizes
the type of plasticity.
The plasticity of rocks is quite different from classical plasticity of metals,$ not only because of dilatancy but also because the yield linearly depends
on pressure (Mohr-Coulomb criterion). The onset
of plasticity is well described by a relationg:
where T and cr, are the shear and the normal stresses
resolved on any plane within the material and the
material constants 4 and c are the "friction angle"
and the "cohesion", respectively. In general, the
flow rule of plasticity is non-associated, i.e., the
angle of dilatancy $ under shear is different from
$ - the plastic strain increments are not normal to
the yield hyper-surface. The friction angle $ is typically 30-40' and the dilation angle is smaller than
10' for rocks.
Fmctcsl Plastic Shear Bands 569
A stability analysis10 shows that in some cases
lane
strain) a n important consequence of nonassociativity is t h e localization of initially homogeneous deformations into shear bands even if the material strain hardens. Thus we obtain localization
in t h e pressure-dependent plastic material due to a
rheological instability without softening, heterogenities i n material properties or non-uniform boundary
conditions. More details can be found elsewherea8>11
3. NUMERICAL MODEL
We have used a n explicit Lagrangian technique
similar t o FLAC as developed by Cundall12to simulate an elastoplastic two-dimensional medium under plane-strain condition. FLAC is a very powerful technique to simulate nonlinear rheological
behaviors a t very high resolution due to an explicit
time-marching scheme which does not require to
store large matrices which are typical for implicit
methods.
We applied pure shear on a square sample in two
different ways, either by prescribing fixed velocities
at all boundaries (FBV), as follows13:
or applying periodic boundary conditions (PB)
keeping the spatial average of gradient of velocities
t o a prescribed value:
where a is a constant and i is a strain rate. Both
of these boundary conditions give very similar and
nearly uniform distributions of localization for systems having more than 100 x 100 elements. (Note
t h a t t h e maximum velocity will be at the boundaries a n d will be denoted as Vb,in the following.)
Besides the abovementioned material constants
c, 4 a n d Q we can also change the elastic moduli or
Lam4 constants X and p. All elements were initialized w i t h the same confining pressure p.
0 ur numerical technique has an intrinsic time
step At chosen sufficiently small so that elastic
waves propagate only a fraction 6 of the length
of one element ensuring numerical stability of the
explicit scheme:
and found that the results do not depend o n this
parameter.
Since our initial setup was completely homogeneous the random nature of the shear bands is due
to minute effects in the round-off of the floating
point numbers in the computer. These round-off
errors only trigger the initial position of the shear
bands but have no influence on the further evolution
of their pattern. We checked this by simulating a
system with weak quenched disorder and found the
same result as in the homogeneous case. The only
difference between these two simulations w a s that
localization in the homogeneous case started later
than in the inhomogeneous case. This delay occurs
because in the homogeneous case it takes some time
for the round-off errors to accumulate sufficiently in
order to trigger the instability.
The results presented in this paper are shown after a transient time, sufficiently long to ensure that
the average number of shear bands does not change
anymore with time. The location of the shear bands
and their strength is, however, not constant with
time as we will discuss in the next section,
An important point is worth being mentioned. In
order to reach equilibrium, the algorithm includes
a damping term. The latter rnimicks a viscous effect, but not with a physically fixed viscosity, but
rather proportional to the out of balance force at
each node, as in Cundall original code.12 Therefore,
when the system is in equilibrium, the viscous term
vanishes and thus one obtains the solution looked
for. However, dynamic effects may be sensitive to
the numerical damping. This point is to be considered as we shall see that the structure of t h e shear
banding we will observe is essentially a dynamic effect dependent on the loading velocity. However,
the algorithm used is so much more efficient for
handling large size systems that a comparison with
other methods over such length and time scales is
merely impossible. Therefore, we report the numerical results obtained but we cannot reach a definite
conclusion on the role played by the artificial damping. We however feel that the results we present
below shows such a remarkable agreement w i t h the
field measurements that the numerical results are
worth being reported and analyzed in detail.
4. RESULTS
where Vsound
is the p-wave velocity and p is the density of t h e material. We performed several calculations w i t h different time steps (6 = 0.1, 0.25, 0.5)
4.1 Stress and Strain Rate Distributior
Figure 2 shows a snapshot of the second invariants
of (a) strain rate; (b) deviatoric stress; and ( c ) local
570 A. N. B. Poliakou e t al.
rotations. We visualize in colors t h e local changes
of the second invariant of the strain rate [Fig. 2(a)]
such that in the yellow and red regions there are
strong changes either in the direction or in the magnitude of the motion of the material. The blue
regions are elastic. In Fig. 2(b) high deviatoric
stresses are presented by red color and low ones by
Fig. 2 Snapshots of the (a) strain rate; (b) deviatoric
stresses; (c)local rotations for bi-periodic boundary conditions, using q5 = 40°, $J = OD, c = 0, X = p (Poisson ratio,
u = 0.25), p / p =
or B = p / p . Vsound/Vbc= 1.
Fig. 3 Comparison of a numerical result (accumulated plastic strain) (a), with a natural rock from the Pyrenees (b).
The scale of (b) is about 2 meters (A 5 FF coin can be seen
slightly above the center of (a)).
Fractal Plastic Shear Bands 571
blue color. In Fig. 2(c) local rotations are coded
such that clockwise rotations are red, anticlockwise
ones are blue, and green denotes essentially rotationless regions. In all cases the color scale has
been chosen to match exactly the range of values obt ained in the picture, so that small inhomogeneities
may be exaggerated. We see that spontaneously
shear bands are formed in which the plastic deformation occurs. These bands form an angle 8
with 45' - $12 < B < (45O - q / 2 ) with respect to
the horizontal which is consistent with bifurcation
theory."$14
In Fig. 3 we compare the network of shear bands
of natural granite with the results of a simulation
Fig. 4 Strain rates maps obtained for different friction 4 and dilation .IC, angles. Note that localization is stronger with
increase of non-associativity g5 .IC,. Boundary conditions are FBV, system size is 300 x 300, and the other parameters are the
same as in Fig. 2. Color coding as in Fig. 2(a).
-
with the same parameters as in Fig. 2, except that
the pressure p is increased by 100 (B = 100). If
we identify one discrete element of our simulation
with the size of a grain of the rock (R 5 mm), the
samples of Figs. 3(a) and 3(b) have about the same
size (the number of grains in the sample is of the
same order as the number of elements in our numerical model). A difficulty in comparing the two
pictures arises from the fact that we do not know
the geological history of the rock so that on the one
hand, we cannot compare the time scales, and that
on the other, the natural rock is a cumulation of
all shear bands that have occurred, and one cannot
exclude that during the geological deformation, the
external load changed direction or magnitude. Note
that for comparison we do not show the snapshot
of strain rate as in Fig. 2(a) but the accumulated
plastic strain after l o 4 time steps.
4.2 Dependence on Parameters
First let us discuss the influence of the dilation and
friction angles on the localization. Figure 4 shows
the strain rate for different 4 and $ and other identical parameters. Figures 4(a) and 4(d) demonstrate
that a small difference of these angles (20") gives
thick and regular shear bands. An increase of the
difference between these angles enhances the localization and makes shear zones thinner and more irregular [see Fig. 4(b) with C$ -1C, = 40' and Fig. 4(c)
with q5 - 3 = 60'1. It is interesting to note from
comparing Figs. 4(a) and 4(d), that 4 - is not
sufficient to determine the shape of the shear bands
but that the absolute value of the angles is import ant.
We performed many numerical experiments
studying all possible parameters which can influence the evolution of the shear bands. The parameters of our model are:
e
*
the elastic properties p and A (since we did not
vary the Poisson ratio, one of these two parameters characterizes the elastic stiffness of the
material X = ,u)
the plastic properties which are as discussed
above, gl, $ and c. In the rest of this study, c = 0,
4 = 40" and ll/ = 0'.
the system size. Since there is no intrinsic length
scale in the problem, only the ratio between the
system size and the mesh size is relevant. We call
this ratio L.
the loading characteristics, which contains two elements. The initial hydrostatic pressure p and the
imposed average shear rate. The latter is characterized by the velocity along the boundary Vbc.
Written in dimensionless form, three parameters
appears: the system size L, the ratio between the
hydrostatic pressure and an elastic constant p / p
which can be interpreted as proportional to the elastic deformation under the initial hydrostatic pressure, and finally the ratio of the sound velocity to
the boundary velocity c/Vbc. All other parameters
are kept constant in the rest of this work.
In fact a useful combination of the latter pararneters can be introduced. The shear strain after a
time t can be written as &,t/L. Thus the time
at which the yield limit will be reached is r, =
pL tan(+)/(p&,): The time for an elastic wave to
travel through the system is T,, = Llc. The ratio
of these two intrinsic time scales gives a non dixncnsional number which we define as B:
where we have dropped the tan(&) termed which
will be kept const ant.
We found numerically that the shear band patterns only depends on B and L. Changing the pressure p, the shear rate hc,
the sound velocity c or
the elastic properties ,u, while keeping B unchanged
does not affect the results. This is shown in Fig. 5.
We see that the evolution of the system depends
only on this non-dimensional parameter B by varying confining pressure, elastic modulus and velocity
of loading. Figure 6 shows that the spacing of shear
bands is controlled by this parameter.
The decrease in shear band separation with increasing B can be qualitatively explained the following way: It is l ~ n o w n , that
~ ~ when
~ ' ~ a~band
~ ~ forms
it causes a decrease of mean stress inside of the band
due to elastic unloading. The pressure outside the
band increases, inhibiting the formation of another
parallel band nearby. This change in pressure propagates through the system at the sound velocity and
thus at high B (large c), the band formation will
affect a larger scale. On the contrary, at small B
the sound velocity is so low compared with the loading rate that the shear bands have to be X X ~ O ~ I ?
densely distributed to release the imposed shear.
The dependence on the other properties p, p, . . . cam
be analyzed in similar terms using the construction
of B.
In Fig. 7 we show how the average spacing between the shear bands as measured with a ruler
Fractal Plastic Shear Bands 5
g. 5 Various parameters having the same value of B and
nensionless control parameters. Color coding as in Ipig, 2(a
from Fig. 6 depends on B. As we will see in the
next section t h e spacing also depends critically on
t h e resolution L (fractal character) so that it is important in Fig. 7 to work at constant system size.
W e see t h a t t h e spacing increases - though slowly
- with B , in agreement with the previous discussion. The quantitative analysis of this variation is
.), showing that these two parameters are t h e only t~
to be taken with caution. Indeed, as can be seen
from Fig. 6, when B is less than 1, the s h e a r bands
form domains in which the bands are all parallel.
There are almost no crossing of bands. For larger
B, on the contrary, the two families of conjugated
shear bands (with opposite slopes) are intermingled
and do not form domains. This might explain t h e
Fig;. 6 Dependence of the strain rate on the para]meter B = p / p . V lound/V~c
,
[or systems of size 300 x 300 and periocdic
boulndary conditions. T h e other parameters are the slame as in Fig. 2 an d the color coding is like that in Fig. 2(a).
Fractal Plastic Shear Bands 575
100
1
-4
10
1o
- ~
1 o"
1 o2
1 o4
1 o6
B
Fig. 7 Log-log dependence of the average spacing A S between shear bands as a function of the parameter B, The result
can be approximated by the power law as A S oc B'.~.
apparent discontinuity of the band separation for
small B in Fig. 7. For large B, one begins to be
sensitive to the finite size of the system, and the
effective separation might be significantly altered
by the presence of periodic boundary conditions.
Other boundary conditions are even more strongly
biased because of the constraint of the imposed velocity close to the boundary. A denser distribution
of bands arises in this case. However, for the practical purpose of using this separation, we can fit
roughly a power-law of exponent 0.2 through the
data. It is possible however that this apparent behavior is far from what it would be in an infinite
system size.
Another interesting question is to monitor the
time evolution of shear bands. To study this problem, we make a vertical cut through a plot of strain
rates as shown in Fig. 6 and record the evolution of
this one-dimensional slice with time. Figure 8 shows
a plot where the horizontal axis gives strain-rates
along the cut and the vertical axis is time. Different
time-dependent behaviors are observed depending
on the parameter B. For low B there are regular
and stable shear bands. For high B shear bands
have a strong time-dependent behavior and a fluctuating intensity. The life time is also shorter for
decreasing B. Again this emphasizes the fact that
the structuration of the shear banding is essentially
a dynamical effect of our model.
Figure 9 shows the time evolution of the stress.
Stress components (ox, and oxy) averaged over all
elements. Both values can be calculated analytically for a single element (a,, = -p . (1 f sin($)) c cos($), os, = 0) and we see that in this sense
the total system has the same global perfect-plastic
behavior as the single element. On the contrary,
the stress (pressure and maximum shear stress) in
a single element is strongly fluctuating with time.
a
576 A . N. B. PoliaLov et al.
i
r :
i
-
{j.!/:
Fig. 8 Time evolution of the shear bands for different B. T i ~ n eincreases vertically upwards. Color coding as in f p i g , l ( a ) .
Because the material is closc/or a t the yiclcl, the
pressure and the shear stress fluct.uation of a single
element are autocorrelatecl with each other. Thus
despite t h e overall rcsponse of the sample being sirnple and similar to the characlcrist.ic of a single elcment, there is a strong stress arld st,rain rate hcterogeneity in space and strong fiuct.nations with tilric
for every single elcxncnt.
4.3 Fractal and Mult ifractal
Analysis
In Fig. 10 we see snapshots from the evolution of
the system with different mesh sizes L but the same
physical parameters. Evidently, 1)ccansc. of t.lio 1111dcrlying instability the shear bantls bocolllcb t.llirllichr
when the mesh is refined. This intrillsic. ~tlosllticpendcllce is of course lilrlitcd for real ror-its o r 1 t , l ~
lower end by the size of the gmirls.
We arltllyzccl the scale invari:~llce of t , l l t a ol,sc$rvcci
s11~c2r1)an~lsby cllailgirlg t,ho illcasll siac~or. chcl~iivalenlly t.11~resolution of the systlt:lrl L. 'I'llis t,c,c:hnicpe is equivalenl t o tllc c1assic;~l l,ox-c,olilltilig
r r ~ c t l ~ o d .For
' ~ each grid clcrrlcnt i wc: cousitlrr
ez , the second invariant of the strain r l ~ t tcnsor
,~
and define the measure pi = ei/
e,. Ld, us dcfine their moments as Mq =
py. T h e nornializetl
17
moments mq = ( ~ , / ~ ~ ) ~ ~ % with
c c a thc
l e system
xi
xi
Fractal Plastic Shear Bands
20
i1
1
577
- ox, averaged over all elements
-. .- - oxy averaged over all elements
- - - - pressure i n o n e element
T i n o n e element
TIME
Fig. 9 T i m e evolution of a,, and a,, components averaged over the whole grid and evolution of the pressure and the second
invariant of the deviatoric stress in one specific element (central one).
size. T h e various moments have different scaling
exponents d,, defined as rnq oc LWdq. In Fig. 11
we see how d, varies with q for different values of
23. We conclude that within our numerical accur a c y t h e distribution of local shear rate seems to be
multifract al.
The geometrical fract a1 dimension Do can be exO L - ~ ' . Thus the scalpressed as Mo = L ~ =
i n g exponent of the first normalized moment ml
t a k e n w i t h positive sign should be compared with
t h e geological data on faults. The geometrical Gact a l dimension of the shear band network in our numerical experiment is Do = 1.6f0.1 for B = 1to 10.
T o verify this result, we performed a box-counting
analysis of a single picture from Fig. 10 with the
highest resolution (500 x 500). The zeroth order
m o m e n t (or geometrical fractal dimension) gave the
same fractal dimension as obtained from different
meshes. However, higher moments have different
exponents for these two methods. This shows that
finite size effects are important and that within t h e
range of the sizes that we can simulate, it is n o t
possible t o find reliable values for d,. It is even
possible that the multifractal nature disappears i n
the thermodynamic limit.
For smaller B the network of shear b a n d s becomes dense while for B of the order 100 t h e finite size L of the box is felt and so fewer s h e a r
bands fit into the system so that the fractal dimension actually decreases. This behavior is d u e
to the fact that as seen in Fig. 7, the characteristic length of the problem, namely A S between shear
bands increases with B. We therefore propose a two
578 A . N. B. Poliakov et al.
.,
*:
d .
Fig. 10 Dependence of the strain rate on different mesh sizes for B = 1 and periodic boundary conditions. Color coding as
in Fig. 2(a).
F ~ a c t a lPlastic Shear Bands 579
Fig. 11 Dependence of the rnultifractal scaling exponent dg on q for different values B = 0.1,1, 10,100.
parameter scaling law of the form:
(6)
where Mo is the number of plastic elements and
F is a scaling function. If we combine the latter
expression with a power-law increase of AS cc BY
with y = 0.2, we obtain:
M~ N L - ~ ~ F ( L B - Y )
We actually plotted our data according t o Eq. (7)
and considering that 3 ( x )
xd-dl for small B
we achieved reasonable data collapse for dl smaller
than 1.6 and y larger than 0.2. These discrepancies are apparently due to the rather small range of
system sizes that we can consider.
We scanned our sample of Fig. 1 and analyzed
the digitalized image by special software (UFC). We
obtained a fractal dimension of Do = 1.58 f:0.08.
For similar problems, different fractal dimensions
have been observed in nature depending on rock
type and tectonic evolution. ~ a r t o n analyzed
l~
17
fracture maps at different scales and found dimensions ranging from 1.32 to 1.7 (box-flex method);
the analysis of fault systems in iIapanlg yielded values between 1.06 and 1.6 (box-counting technique)
and sand box experiments2' gave 1.75 f0.1 (capacity method). One should also note that faults o n the
surface of the earth can have different origins. Our
numerical result Do = 1.6 zt 0.1 is consistent with
some of these field observations and experiments. I n
fact, since real faults are three-dimensional, a deviation of our numbers from the fields values would
not be surprising.
We also analyzed the histogram of the strain rate
e by plotting the distribution p ( e ) against e in a
log-log plot dividing both axis through log(L). As
in the electric analog model for perfect plasticity,21
580 A.
N. B. Poliakov
et
al.
Length of shear bond
Fig. 12 Length distribution of shear bands for a system of size 500 x 500 for the same physical parameters as in Fig. 2. The
straight line is a guide t o t h e eye of slope 2.07.
we find that the data collapse for different sizes L
on a single hat-shaped curve. This shows the resemblance between fiacture and shear bands. I t is,
however, important to point out a crucial difference
between the two analyses. In our present analysis,
we consider the strain rates, while in most fracture
models, the stress is analyzed.
We also looked at the length distribution of shear
bands. Numerically the length of the bands was obtained using some burning algorithm.22That means
starting with one end of a shear band, we iteratively
occupy all connected neighboring elements belonging to that shear band. The number of iterations
gives the length of the shear band.
In Fig. 1 2 we see a log-log plot of this distribution. We do not consider lengths smaller than the
width of the bands, i.e., less than 20 lattice units.
Beyond that we see a power-law decay with an exponent of r = 2 . l k 0.1. This value is also in agreement
with length distributions measured in the field and
in experiments.
20923-26
5 . Summary
Summarizing, we have found that the plastic behavior of rocks under high confining pressure gives rise
to the localization of deformation along a network of
shear band with a complex geometry. In spite of the
numerous parameters of the model, we have shown
that the ratio B of confining pressure to the elastic shear modulus multiplied by the ratio of sound
velocity to velocity of loading, represents a charao
teristic nondimensional parameter that controls the
behavior of the system together with the system size
L. We studied the behavior of the system depending
on B and L. A characteristic length scale, namely
the typical spacing between shear bands in a finite
system varies with B.
Moreover, the network of shear bands is fractal,
and their length distribution follows a power-law.
We formulated a two parameter scaling law for the
fraction of plastic regions as functions of L and B.
The shear rate field forms a "multifractal" measure.
Fractal Plastic Shear Bands 581
We also analyzed the fractal dimension of rocks
that were sheared under high confining pressure i n
t h e past and found a n agreement with our numerical
data. Our results are also consistent with previous
field observations of fault networks and laboratory
experiments.
Acknowledgments
We t h a n k Ethan Dawson, Peter Cundall and J. P.
Villote for useful discussions and suggestions.
Raymond Munier generously helped us with analyzing Fig. 1 and calculating fractal dimensions by
t h e box-counting technique. Dr. Joe Hull pointed
out t o us t h e interesting and beautiful geological region in Pyrenees with t h e shear bands from Fig. 1.
This work was completed with t h e support of t h e
Swedish N F R grant G-GU 10517-301 during the
stay of A. Poliakov i n t h e Laboratoire de Physique
e t MBcanique des Milieux HBt6rogBnes (ESPCI).
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25. P. Davy, J. Geophys. Res. 98, 12.141 (1993).
26. G. Ouillon, C. Castaing and D. Sornette, Hierarchical Scaling of Faulting (to be published in Europhys.
Lett. 1994).
4.1 Stress and strain rate distributions
a)
b)
c)
Figure 2: Snapshots of the a) strain rate b) deviatoric stresses c) local rotations for bi-periodic boundary conditions,
using = 40 , = 0 , c = 0, = (Poisson ratio, = 0:25), p= = 10;9 or B = p= Vsound =Vbc = 1
Figure 2 shows a snapshot of second invariants of a) strain rate b) deviatoric stress and c) local rotations. We visualize
in colors the local changes of the second invariant of the strain rate (Fig. 2a) such that in the yellow and red regions
there are strong changes either in the direction or in the magnitude of the motion of the material. The blue regions
are elastic. In Fig. 2b high deviatoric stresses are presented by red color and low ones by blue color. In Fig. 2c local
rotations are coded such that clockwise rotations are red, anticlockwise ones are blue and green denotes essentially
rotationless regions. In all cases the color scale has been chosen to match exactly the range of values obtained in the
picture, so that small inhomogeneities may be exaggerated. We see that spontaneously shear bands are formed in
which the plastic deformation occurs. These bands form an angle with 45 ; =2 < < (45 ; =2) with respect to
the horizontal which is consistent with bifurcation theory?, ?].
a)
b)
Figure 3: Comparison of a numerical result (accumulated plastic strain) a) with a natural rock from the Pyrenees (b).
The scale of (b) is about 2 meters (A 5FF coin can be seen slightly above the center of (a)).
In Fig.3 we compare the network of shear bands of natural granite to the results of a simulation with the same
parameters as in Fig.2 except that the pressure p is increased by 100 (B = 100). If we identify one discrete element of
our simulation with the size of a grain of the rock ( 5mm) the samples of Fig.3a and 3b have about the same size (the
number of grains in the sample is the same order as the number of elements in our numerical model). A diculty in
comparing the two pictures arises from the fact that we do not know the geological history of the rock so that on one
hand we cannot compare the time scales and that on the other hand the natural rock is a cumulation of all shear bands
that have occured and one cannot exclude that during the geological deformation the external load changed direction
or magnitude. Note that for comparison we do not show the snapshot of strain rate as in Fig. 2a but the accumulated
plastic strain after 104 time steps.
4.2 Dependence on parameters
First let us discuss the inuence of the dilation and friction angles on the localization. Fig. 4 shows the strain rate for
dierent and and identical other parameters. Fig. 4a and Fig. 4d demonstrate that a small dierence of these angles
(20 ) gives thick and regular shear bands. An increase of the dierence between these angles enhances the localization
and makes shear zones thinner and more irregular (see Fig. 4b with ; = 40 and Fig. 4c with ; = 60 ). It
is interesting to note from comparing Figs. 4a and 4d that ; is not sucient to determine the shape of the shear
bands but that the absolute value of the angles is important.
We performed many numerical experiments studying all possible parameters which can inuence the evolution of
the shear bands. The parameters of our model are:
3
= 60o = 0o
a)
= 40o = 20o
c)
b)
d)
Figure 4: Strain rates maps obtained for dierent friction and dilation angles. Note that localization is stronger
with increasing of non-associativity ; . Boundary conditions are FBV, system size is 300300 and the other
parameters are the same as in Fig. 2. Color coding as in Fig. 2a.
- the elastic properties and (since we did not vary the Poisson ratio, one of these two parameters characterize the
elastic stiness of the material = )
- the plastic properties which are as discussed above , and c. In the rest of this study, c = 0, = 40 and = 0 .
- the system size. Since there is no intrinsic length scale in the problem, only the ratio between the system size and
the mesh size is relevant. We call L this ratio.
- the loading characteristics, which contains two elements. The initial hydrostatic pressure p and the imposed average
shear rate. The latter is characterized by the velocity along the boundary Vbc .
Written in dimensionless form, three parameters appears: the system size L, the ratio between the hydrostatic
pressure and an elastic constant p= which can be interpreted as proportional to the elastic deformation under the
initial hydrostatic pressure and nally the ratio of the sound velocity to the boundary velocity c=Vbc . All other
parameters are kept constant in the rest of this work.
In fact a useful combination of the latter parameters can be introduced. The shear strain after a time t can be
written Vbc t=L. Thus the time at which the yield limit will be reached is y = pL tan()=(Vbc ). The time for an elastic
wave to travel through the system is ew = L=c. The ratio of these two intrinsic time scales gives a non dimensional
number which we dene as B :
(5)
B = y = Vpc
ew
bc
where we have dropped the tan() termed which will be kept constant.
We found numerically that the shear band patterns only depends on B and L. Changing the pressure p, the shear
rate Vbc , the sound velocity c or the elastic properties while keeping B unchanged does not aect the results. This
is shown in Fig. 5.
We see that the evolution of the system depends only on this non-dimensional parameter B by varying conning
pressure, elastic modulus and velocity of loading. Fig. 6 shows that the spacing of shear bands is controlled by this
parameter.
The decrease in shear band separation with increasing B can be qualitatively explain dthe following way: It is
known?, ?, ?], that when a band forms it causes a decrease of mean stress inside of the band due to elastic unloading.
The pressure outside the band increases inhibiting the formation of another parallel band nearby. This change in
pressure propagates through the system at the sound velocity and thus at high B (large c), the band formation will
aect a larger scale. On the contrary, at small B the sound velocity is so low compared to the loading rate that the
shear bands have to be more densely distributed to release the imposed shear. The dependence on the other properties
, p, ... can be analysed in similar terms using the construction of B .
In Fig. 7 we show how the average spacing between the shear bands as measured with a ruler from Fig. 6 depends
on B . As we will see in the next section the spacing also depends critically on the resolution L (fractal character) so
4
Vbc
Vbc
7
P = 0:1 G = 106 Vsound
Vbc = 10
5
P = 103 G = 108 Vsound
Vbc = 10
Figure 5: Various parameters having the same value of B and L (B = 1) showing that these two parameters are the
only two dimensionless control parameters. Color coding as in Fig. 2a.
that it is important in Fig. 7 to work at constant system size. We see that the spacing increases | though slowly
| with B in agreement with the previous discussion. The quantitative analysis of this variation is to be taken with
caution. Indeed, as can be seen from Fig. 6, when B is less than 1, the shear bands form domains in which bands are
all parallel. There are almost no crossing of bands. For larger B , on the contrary, the two families of conjugated shear
bands (with opposite slopes) are intermingled and do not form domains. This might explain the apparent discontinuity
of the band seperation for small B in Fig. 7. For large B , one begins to be sensitive to the nite size of the system, and
the eective separation might be signicantly altered by the presence of periodic boundary conditions. Other boundary
conditions are even more strongly biased because of the constraint of the imposed velovity closed to the boundary. A
denser distribution of bands arises in this case. However, for the practical purpose of using this separation, we can t
roughly a power-law of exponent 0.2 through the data. It is possible however that this apparent behavior is far from
what it would be in an innite system size.
Another interesting question is to monitor the time evolution of shear bands. To study this problem, we make a
vertical cut through a plot of strain rates as ones of Fig. 6 and record the evolution of this one-dimensional slice in
time. Figure 8 shows a plot where the horizontal axis gives strain-rates along the cut and the vertical axis is time.
Dierent time-dependent behaviours are observed depending on the parameter B . For low B there are regular and
stable shear bands. For high B shear bands have a strong time-dependent behaviour and a uctuating intensity. The
life time is also shorter for decreasing B . Again this emphasizes the fact that the structuration of the shear banding
is essentially a dynamical eect of our model.
Figure 9 shows the time evolution of the stress. Stress components (xx and xy ) averaged over all are almost.
Both values can be calculated analytically for a single element (xx = ;p (1 + sin()) ; c cos() xy = 0) and we
see that in this sense the total system has the same global perfect-plastic behaviour as the single element. On the
contrary the stress (pressure and maximum shear stress) in a single element is strongly uctuating in time. Because
the material is close/or at the yield the pressure and the shear stress uctuation of a single element are autocorrelated
to each other. Thus despite the overall response of the sample is simple and similar to the characteristic of a single
element, there is a strong stress and strain rate heteroginity in space and strong uctuations in time for every single
element.
4.3 Fractal and multifractal analysis
In Fig. 10 we see snapshots from the evolution of the system with dierent mesh sizes L but the same physical
parameters. Evidently, because of the underlying instability the shear bands become thinner when the mesh is rened.
5
B=1
B = 100
a)
B = 10
c)
B = 10000
e)
b)
d)
f)
Figure 6: Dependence of the strain rate on the parameter B = p= Vsound =Vbc for systems of size 300300 and periodic
boundary conditions. The other parameters are the same as in the Fig 2 and the color coding is like in Fig. 2a.
This intrinsic mesh dependence is of course limited for real rocks on the lower end by the size of the grains.
We analyzed the scale invariance of the observed shear bands by changing the mesh size or equivalently the resolution
of the system L. This technique is equivalent to the classical box-counting method ?]. For
grid element i we
Pieach
consider ei , the second
invariant
of
the
strain
rate
tensor
and
dene
the
measure
p
=
e
=
e
.
Let
us dene their
i
i
i
P
moments as Mq = i pqi . The normalized moments?] mq = (Mq =M0 )1=q scale with the system size. The various
moments have dierent scaling exponents dq , dened as mq / L;dq . In Fig. 11 we see how the dq vary with q for
dierent values of B . We conclude that within our numerical accuracy the distribution of local shear rate seems to be
multifractal.
The geometrical fractal dimension D0 can be expressed as M0 = LD0 = L;d1 . Thus the scaling exponent of the rst
normalized moment m1 taken with positive sign should be compared to the geological data on faults. The geometrical
fractal dimension of the shear band network in our numerical experiment is D0 = 1:6 0:1 for B = 1 to 10. To
verify this result, we performed a box-counting analysis of a single picture from Figure 10 with the highest resolution
(500500). The zeroth order moment (or geometrical fractal dimension) gave the same fractal dimension as obtained
from dierent meshes. However, higher moments have dierent exponents for these two methods. This shows that
nite size eects are important and within range of sizes that we can simulate, it is not possible to nd reliable values
for dq and it is even possible that the multifractal nature disappears in the thermodynamic limit.
For smaller B the network of shear bands becomes dense while for B of the order 100 the nite size L of the box is
felt and so fewer shear bands t into the system so that the fractal dimension actually decreases. This behaviour is due
to the fact that as seen in Fig. 7 the characteristic length of the problem, namely S between shear bands increases
with B . We therefore propose a two parameter scaling law of the form
M0 / L;d1 F (L=S (B ))
6
(6)
B = 10.
a)
B = 100.
c)
b)
d)
Figure 8: Time evolution of the shear bands for dierent B. Time increases vertically upwards. Color coding like in
Fig. 2a.
We also analyzed the fractal dimension of rocks that were sheared under high conning pressure in the past and
found an agreement with our numerical data. Our results are also consistent with previous eld observations of fault
networks and laboratory experiments.
6 Acknowledgements
We thank Ethan Dawson, Peter Cundall and J.P. Villote for useful discussions and suggestions. Raymond Munier
generously helped us analysing Fig 1. and calculating fractal dimensions by the box-counting technique. Dr. Joe Hull
pointed out to us the interesting and beautiful geological region in Pyrenees with the shear bands from Fig.1. This
work was completed with the support of the Swedish NFR grant G-GU 10517-301 during the stay of A. Poliakov in
the Laboratoire de Physique et Mecanique des Milieux Heterog"enes, (ESPCI).
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8
L=100
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a)
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c)
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e)
b)
d)
f)
Figure 10: Dependence of the strain rate on dierent mesh sizes for B=1 and periodic boundary conditions. Color
coding as in Fig. 2a.
Figure 11: Dependence of the multifractal scaling exponent dq on q for dierent values B = 0:1 1 10 100.
10