Earth and Planetary Science Letters 259 (2007) 469 – 485
www.elsevier.com/locate/epsl
Nature of stress accommodation in sheared granular material:
Insights from 3D numerical modeling
Karen Mair a,⁎, James F. Hazzard b
a
Physics of Geological Processes, University of Oslo, Norway
b
RocScience Inc., Toronto, Canada
Received 13 February 2007; received in revised form 4 May 2007; accepted 4 May 2007
Available online 10 May 2007
Editor: R.D. van der Hilst
Abstract
Active faults often contain distinct accumulations of granular wear material. During shear, this granular material accommodates
stress and strain in a heterogeneous manner that may influence fault stability. We present new work to visualize the nature of
contact force distributions during 3D granular shear. Our 3D discrete numerical models consist of granular layers subjected to
normal loading and direct shear, where gouge particles are simulated by individual spheres interacting at points of contact
according to simple laws. During shear, we observe the transient microscopic processes and resulting macroscopic mechanical
behavior that emerge from interactions of thousands of particles. We track particle translations and contact forces to determine the
nature of internal stress accommodation with accumulated slip for different initial configurations. We view model outputs using
novel 3D visualization techniques. Our results highlight the prevalence of transient directed contact force networks that
preferentially transmit enhanced stresses across our granular layers. We demonstrate that particle size distribution (psd) controls the
nature of the force networks. Models having a narrow (i.e. relatively uniform) psd exhibit discrete pipe-like force clusters with a
dominant and focussed orientation oblique to but in the plane of shear. Wider psd models (e.g. power law size distributions D = 2.6)
also show a directed contact force network oblique to shear but enjoy a wider range of orientations and show more out-of-plane
linkages perpendicular to shear. Macroscopic friction level, is insensitive to these distinct force network morphologies, however,
force network evolution appears to be linked to fluctuations in macroscopic friction. Our results are consistent with predictions,
based on recent laboratory observations, that force network morphologies are sensitive to grain characteristics such as particle size
distribution of a sheared granular layer. Our numerical approach offers the potential to investigate correlations between contact
force geometry, evolution and resulting macroscopic friction, thus allowing us to explore ideas that heterogeneous force
distributions in gouge material may exert an important control on fault stability and hence the seismic potential of active faults.
© 2007 Elsevier B.V. All rights reserved.
Keywords: numerical modeling; force chains; fault gouge; earthquake mechanics
1. Introduction
⁎ Corresponding author.
E-mail addresses: karen.mair@fys.uio.no (K. Mair),
hazzard@rocscience.com (J.F. Hazzard).
0012-821X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2007.05.006
Faults in nature often have significant accumulations
of granular fault gouge. The presence and evolution
state of this gouge affects frictional strength and
stability. Both these properties in turn determine the
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K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
essential to investigate processes that may be operating
in faults.
There is growing evidence (Cates et al., 1998;
Howell et al., 1999) that force distributions in sheared
granular materials are highly heterogeneous i.e. particles
do not all carry the same load. Direct observations of
force chains in 2D photo elastic shearing experiments
conducted at low stresses (e.g. Oda et al., 1982; Howell
et al., 1999) indicate that enhanced load is preferentially
carried on a limited number of particles that set up a
network of force chains. Between these chains are
shielded regions where particles carry reduced load. 2D
numerical modeling (Cundall et al., 1982; Morgan and
Boettcher, 1999; Aharonov and Sparks, 1999) confirms
the prevalence of these features.
The presence of force chains have been invoked to
help explain the comminution of granular fault gouge
(Sammis et al., 1987) and they offer a convenient way
to interpret results from recent 3D laboratory shearing experiments at geophysically relevant conditions
mechanical nature of slip along a given fault. To
understand the processes that may be operating in faults
with gouge it is useful to investigate sheared granular
material. At present there are 2 main approaches: laboratory friction experiments; and numerical modeling.
Laboratory friction experiments are generally conducted either at high stresses (MPa) (e.g. Karner and
Marone, 2001) relevant to geophysical conditions or at
relatively low stresses (Pa) (e.g. Losert et al., 2000).
Despite the difference in conditions, the relatively low
stress experiments may help elucidate particular micro
processes that are relevant to higher stress experiments
and hence natural fault systems. With some notable
exceptions much of the numerical modeling work in this
field has been conducted in 2D (Mora and Place, 1998;
Aharonov and Sparks, 1999; Morgan and Boettcher,
1999). Recent work (Hazzard and Mair, 2003; Abe and
Mair, 2005) has revealed the importance of out-of-plane
motions and grain fracture in sheared granular systems
and demonstrated that 3D numerical modeling is
Table 1
Numerical simulations
Simulation
psd
Diameter (std)
d/H
tn021g
tn027g
tn028g⁎
tn029g ⁎⁎
tn030g
tn031g
tn051g+
tn035g2
tn036g2
tn044g
tn048g
tn049g
tn040g
tn041g
tn037g
tn038g
tn045g
tn050g
tn052g
tn046g
tn047g
tn014f
tn032f
tn033f
tn053f
tn034f
tn054f
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Powerlaw D = 2.6
Powerlaw D = 2.6
Powerlaw D = 1.6
Powerlaw D = 1.6
Powerlaw D = 0.8
Powerlaw D = 0.8
254 (22)
254 (22)
254 (22)
254 (22)
254 (22)
254 (22)
254 (22)
254 (44)
254 (44)
254 (22)
254 (22)
254 (22)
400 (44)
400 (44)
500 (44)
500 (88)
500 (44)
500 (44)
500 (44)
600 (60)
600 (60)
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.068
0.108
0.108
0.135
0.135
0.135
0.135
0.135
0.162
0.162
μ (mean)
μ (std)
dil rate (std)
0.3538
0.00946
0.0115
0.0106
0.3573
0.3591
0.3539
0.35728
0.00967
0.01184
3.21
2.85
3.31
0.3585
0.3611
0.3553
0.3603
0.354
0.3612
0.3536
0.3517
0.3639
0.337
0.3608
0.3488
0.3618
0.3377
0.348914
0.347329
0.371007
0.364849
0.35881
0.35361
0.01286
0.0127
0.01138
0.01219
0.0117
0.01759
0.02355
0.0219
0.02833
0.02445
0.02413
0.03344
0.0308
0.03074
0.0090
0.0102
0.0157
0.0227
0.0165
0.0214
3.68
3.19
3.41
2.68
2.5
5.74
5.87
8.72
15.2
9.0
6.71
8.14
16.5
17.8
3.86
4.86
4.77
0.0198
0.0283
0.0287
0.0302
Fmax
3.13
7.74
Modeled particles have microproperties as follows: shear modulus 22 GPa; Poisson's ratio 0.25; and an inter-particle friction of 0.5. Normal stress is
5 MPa. Particle size distribution is Gaussian or Power Law. Mean and standard deviation (std) of particle diameter are given in μm. Initial layer thickness is
3.7 mm. d /h is scaled grain size where d = mean diameter and H is initial layer thickness. Dil rate is standard deviation (std) in dilatancy rate. Boundary is
rough and 1 particle wide (except in the case where it is 2 particles wide = ⁎; 4 particles wide = ⁎⁎, or consists of a smooth boundary = +).
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
(Karner and Marone, 2001; Mair et al., 2002; Anthony
and Marone, 2005). Although force chain networks
are generally believed to exist in 3D sheared granular
material, a key question concerns their morphology
since they have not been directly observed to date.
Blair et al. (2001), Mueth et al. (1998) and others
present laboratory evidence for heterogeneous load
distributions in static 3D granular assemblages. In these
experiments, the non-homogeneous distribution of
normal load at the boundary of a static pack (e.g. the
stress minimum at the center of a sand pile, Vanel et al.,
1999) suggests that force must be distributed unevenly throughout the pack. However, the morphology of
force chain distributions is not directly apparent from
these measurements and furthermore it is not clear
how this result translates to 3D granular systems under
shear.
We can then ask the following questions: Are
directed force networks (or chains) the dominant load
bearing structure in sheared 3D granular systems? What
are the key morphological features of these networks?
Do they form linear elements, similar to those in 2D, or
occur as more complicated surfaces? What influences
their occurrence, abundance and nature? For example
how do grain scale characteristics, properties and
interactions influence the morphology of chains? Does
the development, interaction and breakdown of force
chain networks influence the macroscopic stress state
and hence potentially control frictional strength and
stability?
Here we build on recent 3D numerical simulations
demonstrating the importance of out-of-plane behavior
(Hazzard and Mair, 2003) to investigate the nature of
stress accommodation in sheared 3D granular materials.
We focus on i) the visualization of contact force distributions in 3D to characterize their morphology; ii) the
dependence of force distributions on particle size
distribution, and strain history; and iii) the influence of
force chains on macroscopic stress and hence friction.
Our results show that directed force networks do exist
in 3D numerical models of granular shear, their morphology reflects specific characteristics of the granular
assemblage and their development and breakdown may
be directly linked to changes in macroscopic sliding
friction.
471
or field investigations of natural faults. Here we use the
distinct element method (e.g. Cundall and Strack, 1979)
in 3D to model shear in granular materials. Using this
method, an assembly of spherical particles (perfect and
indestructible) is considered. There are no bonds
between particles, and they interact only at points of
contact. This approach is particularly useful in modeling
granular systems since individual grains can be represented by discrete particles in the model (e.g. Mora and
Place, 1998; Morgan, 1999; Aharonov and Sparks,
1999). We prescribe particle and contact properties
(Table 1) then load the numerical system. An explicit
solution scheme updates particle positions and contact
forces each time step. Under load, particles interact
according to simple laws, however, the interactions
2. Method
Numerical modeling is a very useful tool for
investigating dynamic fault processes since it enables a
degree of visualization and analysis of evolving systems
that is not generally possible in laboratory experiments
Fig. 1. Schematic of 3D numerical model at a) initial conditions prior to
shear, and b) after 100% strain. Top and bottom boundary particles are
shaded. A vertical marker band (shaded) is displaced to the right after
shearing (shear direction indicated by arrow). The top boundary layer
has been removed in (b) to view the marker band. The x, y, z coordinate
system is shown — initial layer thickness in z direction is 3.7 mm.
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K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
between thousands of particles result in a complicated
emergent macroscopic response. For the research
presented here, we use Particle Flow Code in 3 dimensions (PFC3D, 1999, Potyondy and Cundall, 2004) with
a Hertz–Mindlin contact model (Cundall, 1988). The
shear force at each individual contact is limited by the
coefficient of interparticle friction we specify. If shear
force is exceeded, then stable sliding occurs at that
contact.
Granular layers were generated by filling a prism
with a random assembly of particles (generated from a
‘seed’ condition). Particle microproperties are assigned
then the assembly is compacted to the desired stress
state (here 5 MPa). The top and bottom boundaries are
composed of a controlled layer of particles to provide
roughness (Fig. 1a). We apply normal stress (in the z
direction) to a frictionless wall that confines the particles
at the top of the model. A servo-control mechanism is
activated to adjust vertical (z direction) wall velocity in
such a way to maintain the desired normal stress (total
contact force on the wall divided by wall area). This is a
similar mechanism to the one used in laboratory friction
experiments. The lower boundary is fixed, the left and
right sides are periodic boundaries and the front and
back are frictionless walls. Shear stress is applied by
driving the top boundary particles in the positive x
direction at a constant velocity (see arrow Fig. 1b). Particles in the top boundary layer are free to move and
rotate in all other directions. This applied stress
generates shear in the granular layer. A marker band,
originally vertical (Fig. 1a) illustrates particle displacement in the layer (Fig. 1b) after 100% strain.
We monitor transient microscopic processes (e.g.
particle motion, contact forces) as well as macroscopic
properties (e.g. friction, dilation) for different initial
configurations and applied loading conditions. Particle
displacements, velocities and rotations are measured
continuously during numerical simulation of shearing
to allow visualization of dynamic particle interactions.
The contact forces between adjacent particles are monitored allowing us to track the nature of internal stress
accommodation for different initial configurations and
as a function of loading. Total contact forces are plotted
as 3D cylinders where thickness and shading is proportional to the magnitude of the total contact force and
length indicates the distance between particle–particle
centers.
A suite of numerical simulations were carried out for
the conditions specified in Table 1. Loading geometry
and particle properties were chosen to be comparable to
recent laboratory experiments (Mair et al., 2002). The
simulations reached deformations up to 500% shear
strain. The specific simulations carried out were motivated by the following goals: i) to investigate the style
of stress accommodation during shear in the model
layers, the sensitivity of this to particle characteristics
Fig. 2. Different particle size distribution (psd) configurations used in numerical simulations: a) Gaussian; b) power law D = 2.6; c) power law D = 1.6;
d) power law D = 0.8.
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
473
Fig. 3. (a) Friction versus shear strain data plotted for several model runs (tn027g, tn021g, tn030g) having the same Gaussian particle size distribution,
with mean particle radius 125 μm (scaled grain size: d / H = 0.068, where d is diameter and H is layer thickness), identical applied loading conditions
but a distinct starting (seed) model. This demonstrates the reproducibility of the model results. (b). Friction versus shear strain plotted for Gaussian
and power law particle size distributions having exponents D = 2.6, 1.6 and 0.8. (c) Friction versus shear strain for Gaussian size distribution models
having mean particle radii r = 125, 200, 250, 300 μm (corresponding to d / H = 0.068, 0.108, 0.135, 0.162 respectively).
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K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
and the influence on resultant macroscopic behavior; ii)
to determine the reproducibility of our model results for
identical runs using different starting layers (seeds).
The particle size distributions (psd) used in our
simulations are illustrated in Fig. 2. A Gaussian size
distribution (Fig. 2a) is obtained by having a standard
deviation about a mean particle diameter. For the small
standard deviation we choose, this results in a fairly
narrow particle size range that approximates the size
distribution used in recent laboratory experiments (Mair
et al., 2002). Several different mean particle diameters
(for a given layer thickness) were chosen in different
models to investigate the influence of number of particles across a sheared granular layer (see Table 1). An
approximation to a power law size distribution (Fig. 2b, c
and d) is obtained by varying the relative abundances of
four particle size fractions having diameters equal to
62.5 μm, 125 μm, 250 μm, 500 μm). The distribution is
defined using the power law:
Ni ¼ Nmax ⁎ðRmax =Ri ÞD
where Ni and Ri are the incremental number (i.e.
abundance) and particle radius respectively. Nmax and
Rmax are abundance and radius of the maximum size
fraction (i.e. 250 μm), and D is the power-law exponent
(often referred to as the fractal dimension). The
approximation of a power law size distribution by this
method has been previously used by Morgan (1999) and
others. In this way we approximate distributions with 3D
power law exponent D = 0.8, 1.6, 2.6. These power law
exponents are chosen to represent different stages of a
fault maturity. D = 0.8 is indicative of an immature
coarse granular breccia having a grain supported texture,
whereas D = 2.6 is characterized by a fines dominated,
matrix supported texture. Exponent D = 2.6 is an often
cited distribution for mature natural fault gouges (e.g.
(Sammis et al., 1987; An and Sammis, 1994) and possibly representative for a zone of recurring earthquakes.
The influence of particle size distributions on macroscopic friction and contact force distributions are now
presented in Section 3, along with the reproducibility of
the numerical simulations.
3. Results
3.1. Macroscopic observations
3.1.1. Friction
The macroscopic frictional response of several simulations, all having the same particle size distribution
(psd) but different ‘seeds’ (or starting models), is shown
as a function of shear strain in Fig. 3a where friction =
shear stress / normal stress measured at the upper and
lower boundaries. All models have a fairly stable friction with small, high frequency fluctuations about a
mean level of ≈ 0.35. The characteristics of the macroscopic data are comparable for all the modeling runs
shown, indicating good reproducibility of the basic
mechanical data for simulations having the same loading conditions but different ‘seed’ or starting models.
The friction responses for models with different
particle size distributions are compared in Fig. 3b. We
show friction curves for models having Gaussian size
Fig. 4. Close up of dilatancy rate dh / dx and friction plotted as a function of shear strain. dh / dx is calculated from layer thickness (h) and shear
displacement (x) data using a moving window of 100 points.
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
distributions and power law size distributions (with
exponent D = 2.6, 1.6, 0.8). Note that the mean friction
level is comparable for all the particle size distributions
studied indicating that first order friction is insensitive to
particle size distribution. Friction fluctuations about the
mean level, however, are enhanced for the power law
particle size distributions. This effect is relatively minor
for the power law size distribution models with D = 2.6,
however, fluctuations in friction are markedly enhanced
for power law models with exponent D = 1.6 and
D = 0.8. In addition to having higher amplitude, the
friction fluctuations also have longer wavelength than in
the Gaussian case.
475
In Fig. 3c we plot friction responses for several
simulations with the same particle size distribution
(Gaussian) but different mean particle diameters. This
effectively compares models with different numbers of
particles across a given layer width. The mean level of
friction is similar for all datasets (approximately 0.35)
however the fluctuations in friction about this mean
level increase directly with the mean particle size (i.e.
with reduced numbers of particles across a layer). In
general, the fluctuations in friction appear symmetric,
perhaps suggesting geometrical causes, rather than true
stick slope motion that would tend to produce sawtooth
curves.
Fig. 5. (a) Total particle displacements in x direction (m) versus particle position in layer (in z direction in m) after 100% shear strain for entire
models. Curves (smoothed) are compared for Gaussian and power law D = 2.6 particle size distribution models. (b). Total particle displacements (m)
versus particle position in layer (in z direction in m) after 100% shear strain for power law D = 2.6 simulation shown in Fig. 5a. Thin x–z slices
(0.25mm wide in y-dimension) from y-positions y = +2 mm, 0 mm and − 2 mm are shown to highlight any variability with y-position in the model.
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The size of fluctuations in friction is characterized
by calculating the standard deviation about a mean
friction level over a shear displacement interval of several mm (2–10 mm unless otherwise stated) in each
numerical model. Standard deviation in friction as a
function of particle size is summarized in Table 1. There
is a direct correlation between standard deviation in
friction i.e. size of the friction fluctuations, and the mean
particle size for a given layer thickness. That is, the
Gaussian models exhibit increasing fluctuations for less
particles across a granular layer (see Fig. 3c). Similarly,
the power law size distribution models show larger
fluctuations for models with a higher proportion of
larger particles (i.e. lower power law exponents of
D = 0.8, 1.6, see Fig. 3b). This case also corresponds to a
smaller number of particles making up a granular layer.
These observations highlight the influence of the system
size of the model on the results we present.
3.1.2. Assemblage dilatancy
During shear, we continuously monitor changes in
layer thickness of the model (in z direction) required to
maintain constant normal stress. These dilation and
compaction events are macroscopic reflections of dynamic particle motions associated with frictional sliding.
Fig. 4 illustrates a good positive correlation between
dilatancy rate (dh / dx) and friction fluctuations for a
portion of the slip. This relationship is apparent in all
simulations. Dilatancy rate is calculated using a moving
window of 100 points where dh is change in layer
thickness (h) and dx is change in slip (x) in the direction
of shear. Fluctuations in dilatancy rate are further
characterized by taking the standard deviation of dh / dx
for a displacement interval 2–10 mm slip. These data,
summarized for several model runs, indicate a clear
direct correlation between the size of fluctuations and
the mean particle radius (and hence system size) in
Table 1.
3.2. Particle motions
A 3D model sheared to 100% strain is shown in
Fig. 1b. Displacement of a marker band suggests that
particle motions decrease gradually from the top, to the
bottom of the layer. This is the typical output from all the
modeling runs. The cumulative displacements of
individual particles are plotted as a function of position
(z-coordinate) as smoothed curves in Fig. 5a. Both
Gaussian and power law psd particle models have largest
displacements in the upper region of the layer, gradually
decreasing with depth. Localization of slip within the
layers would be apparent on this plot as kicks or
discontinuities disrupting the smooth curve. Neither
model shows serious discontinuities hence we infer there
is no evidence for persistent localization of slip within
the layers. Since we have plotted entire layers in Fig. 5a,
Fig. 6. Histogram showing distribution of contact force magnitudes for the different particle size distribution models indicated. Plot shows the
probability density function P( f ) of forces f = F / Fmean i.e. contact force magnitude F normalised to mean contact force Fmean. The main plot is log–
log. We use adaptive binning where each bin hosts a constant number for data points. Inset shows semi-log plot of the same data highlighting the
behaviour near f = 1. Solid black line in inset is a fit of equation P( f ) ∝ exp(− Bf ) with slope B = 1.17.
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
it is possible that the spread of data, particularly in the
power law psd case, may mask expressions of localized
slip at a particular y-position. We have therefore also
analyzed a series of thin x–z slices (0.25 mm thick in the
y direction) from three different y-positions throughout
the deformed layers. Examples are shown in Fig. 5b.
477
These three slices show a slightly different particle
displacement characteristics as one moves through the
layer (in the y direction), however, they show no clear
indication of persistent slip localization. We conclude
that the accumulated strain in the granular layers is
essentially distributed and not persistently localized.
Fig. 7. Contact forces plotted for a Gaussian psd model run (tn045) after 100% strain. Three different views are shown: a) xz-transport plane (front view); b)
perpendicular xy plane (top view); c) xz-transport plane (rear view). Shearing direction is indicated by grey arrows in plots a) c), and normal force is applied
vertically i.e. in the z direction. The bounding box of the model is indicated as white outline. The width and shading of contact force cylinders are
proportional to the magnitude of total contact force (in Newtons) acting between adjacent particles. The length of the cylinders represents the particle–
particle centers. Animations: Fig. 7-dynamic.avi (Appendix) shows the same figure initially in position a) and rotating clockwise about the z-axis.
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3.3. Contact forces
An overview of contact forces i.e. total force between
two adjacent particles, for different models is presented
in Fig. 6. This figure shows a probability density function P( f ) of contact forces F normalised to mean force
Fmean (i.e. f = F / Fmean). Data are shown as log–log (main
plot) and semi-log (inset) graphs. There is a clear
influence of psd in the different models. The Gaussian
model shows a crossover around f = 1, see inset to Fig. 6.
Large forces f N 1 are fit reasonably well by an exponential ‘tail’ and small forces f b 1 show different behavior
consistent with expectations from previous work (e.g.
(Aharonov and Sparks, 2002, Aharonov and Sparks,
2004). The power law D = 2.6 model shows rather different behaviour not fit by an exponential relationship
(the D = 1.6 and D = 0.8 cases exhibit intermediate
behaviour). The observation that the different psd
model curves do not collapse onto a single line indicates
that their force distributions are intrinsically distinct.
An example of the spatial distribution of contact
forces for a Gaussian psd model (tn045) after ≈ 100%
shear strain is shown in Fig. 7. Three views of the
complete model are shown to highlight the 3D nature of
the contact force population. The width and shading of
the 3D cylinders represents the magnitude of total
contact forces and the length illustrates the distance
between particle–particle centers. Orientation of the
cylinders represents the orientation of total force
between adjacent particles. Individual particles are not
shown. There is preferred orientation of the larger
contact forces oblique to the shearing direction, particularly well illustrated in Fig. 7c. It appears that the
larger contact forces create a directed force network that
preferentially carries stress across the sheared layer.
Particles are clearly also supported by many contacts in
Fig. 8. Contact force orientations weighted by force magnitude are presented as polar histograms for a Gaussian psd model (black) and a power law
D = 2.6 model (grey) after 100% shear strain. Data are projected onto transport xz plane (diagram a, c) and the perpendicular xy plane (diagram b, d).
Data are shown as a) and b) large contact forces, i.e. those with F / Fmean N 1; and c) and d) small contact forces i.e. those with F / Fmean b 1. Data are
binned and smoothed using a 10 point average. The bin value is plotted as radius for a given angle. Wall forces are not included in this plot.
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
the xy plane perpendicular to shear (Fig. 7b), however,
there is an absence of a directed network of larger forces
acting in the y-orientation. Animation: Fig. 7-dynamic
(Appendix) shows the same model output as shown in
Fig. 7, the model is gradually rotated about the z-axis to
further highlight the 3D spatial distribution of contact
forces.
The dominant orientations of contact forces weighted
by magnitude are presented as rose diagram in Fig. 8.
Data are shown for Gaussian and power law D = 2.6
models. The role of contact force magnitude on the
preferred orientation is examined by decomposing the
dataset into large and small forces. Given the crossover
in the probability distribution of forces at f = 1 (Fig. 6
479
and previous work) we use this to discriminate small
and large forces. When large (F N Fmean) contact forces
are considered with respect to the transport xz plane
(Fig. 8a), we see that both models show a peak force
orientation at ≈50° to shearing direction. The Gaussian
size distribution model is fairly focussed whereas the
power law D = 2.6 model shows a wider range of
orientations (highlighted by the ‘fat waist’ and reduced
peak of the power law polar histogram plot). In the xy
plane (Fig. 8b), the large contact forces in both models show a preferential orientation subparallel to shear
(i.e. x direction). Interestingly, the small contact forces
(F b Fmean) show a preferential orientation perpendicular
to shearing (Fig. 8d) that is strongest in the Power Law
Fig. 9. Interacting large contact forces are plotted in 3D model space as cylinders where width and shading scale with force magnitude. Three
projections xz plane, xy plane and yz plane are shown for Gaussian model a) b) and c) and power law D = 2.6 d) e) and f ) models respectively. Both
models have reached 100% shear strain. Sense of shear is indicated by arrows in diagrams a) and d) and × ○ mark c) and f ) signifying shear into (×)
and out of (○) the page respectively. Animations: Fig. 9a-dynamic.avi and Fig. 9b-dynamic.avi (Appendix) show Gaussian and power law D = 2.6
models respectively rotated about the z-axis such that the 3D connectivity of the contact forces can be better visualized.
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D = 2.6 case. This indicates the possibility of a secondary
supporting network of smaller contact forces that may
be important to determining that overall contact force
distributions.
3.4. Force networks
To highlight the spatial distribution and connectivity
of the main load bearing elements in the granular
models, we analyzed the occurrence of the largest
contact forces with proximity to each other. We use an
algorithm to find the largest contact force, then check
the neighboring particles for the next largest force, and
move to that particle. The algorithm then checks the
new neighboring particles for the next largest force and
moves on etc until a boundary is reached. This procedure is repeated traveling in the opposite direction
from the maximum force. The whole process is
repeated 10–15 times to alleviate the potential problems caused by a spurious high contact force isolated
near an edge. The result is a 3D spatial distribution of
connected contact forces (Fig. 9) highlighting the
manner in which stress is transmitted across the
sheared granular layer. In keeping with recent literature, we call these interconnected high contact force
features ‘force networks’.
Fig. 10. Snapshots of contact forces of a Gaussian size distribution model (tn045) at increasing strain are shown for a thin 2D slice of the 3D model. The
orientation of the xz plane is shown and sense of shear is indicated by arrows on first plot. The width and color of force cylinders scales to their magnitude.
A subplot of friction (0.25–0.43) versus shear strain (50–100%) is shown for each snapshot, the orange dots indicating the current position on the friction
strain curve. Areas of interest are circled in white. The gradual buildup of a directed force network in a) and b) corresponds to increasing friction level. A
reduction in friction c) correlates to breakdown of a high force network. A force network then builds d) and breaks down e) associated with an increase and
decrease in friction level respectively. Finally a directed force network starts to build in new location consistent with friction increase. Animation: Fig. 10dynamic.avi (Appendix) shows animation of force networks and macroscopic friction data evolving with increasing strain.
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
The force networks developed after 100% shear
strain for Gaussian (Fig. 9a, b, c) and D = 2.6 power
law (Fig. 9d, e, f) psd models are presented as plots
of the xz-, xy- and yz planes respectively. Animations
(Fig. 9a-dynamic and Fig. 9b-dynamic (Appendix))
show the force networks for the two psd models rotated
about the z-axis to highlight the 3D structure and
connectivity. The Gaussian model (Fig. 9a) shows force
components directed obliquely to shear direction in the
xz-transport plane. From above (Fig. 9b) and viewed
along the shearing direction (Fig. 9c), we can clearly
discern ≈4 discrete pipe like clusters of connected
forces located near the corners of the model. They have
little obvious component of out of plane force (y direction) and appear to have limited interaction with each
other (although clearly they may be connected or supported by small forces). The power law D = 2.6 model
(Fig. 9d, e, f) also shows connected force elements
oriented oblique to shear direction (Fig. 9d) as expected
from Fig. 8. Viewed from above (Fig. 9e) we see a
greater degree of connectivity of the force elements
perpendicular to shearing direction (y direction). Fig. 9f
also highlights significant linkage of the contact force
elements out-of-the plane and an absence of the spatially
concentrated clustering shown by the Gaussian model.
This results in a more sheet like or tree like high force
network compared to the Gaussian psd model.
In summary, Gaussian models appear to have several,
spatially discrete pipe-like force clusters, whereas the
power law size distribution D = 2.6 models develop
more sheet like 3D force networks that have a higher
degree of out-of-plane linkage.
3.5. Evolution of force networks with strain
Fig. 10a–f shows a series of snapshots of contact
force networks as a function of increasing strain from
50–100% for a Gaussian psd particle model (tn045).
Animation: Fig. 10-dynamic (Appendix) shows a movie
of this record. Total contact forces are plotted for a thin
x–z slice of a 3D sheared granular layer. The subplots
show friction versus shear strain for 50–100% shear
strain, with the red dots indicating the stage reached on
the friction curve for each snapshot. We have highlighted, using circles, the important contact force networks
and interpreted them in terms of frictional behavior as
follows. Fig. 10a shows friction increasing and a strong
force system is developing oblique to the shearing
direction; b) friction has just reached local maximum and
the force element seen in previous snapshot is strengthened; c) friction has dropped to a local minimum value
and the central part of the high force network has
481
significantly reduced in magnitude and perhaps broken
this chain; d) friction is on the increase again and two
subparallel high force elements have developed; e) a
drop in friction appears to coincide with the breakdown
or weakening of the contact force network highlighted; f)
shows a slight increase in friction associated with the
possible development or at least strengthening of a new
(high magnitude) force network, oblique to shearing
direction located to the left side of the layer.
We suggest that these observations highlight a
potentially important link between development and
strengthening of strong force networks and an increase in
macroscopic friction. Conversely we seem to see a reduction in friction when strong force networks are
reduced or broken. The transition of the force network to
a new orientation (Fig. 10f) highlights the transient nature of individual networks but the persistence of a
pattern of strong force networks directed obliquely to the
shearing direction.
4. Discussion
4.1. Friction and dilatancy
The macroscopic friction values generated in our
numerical particle models are comparable to those found
in previous 3D numerical modeling (Hazzard and Mair,
2003). They are also approximately equivalent to the
friction levels measured in laboratory shearing experiments carried out on idealized spherical particles in a nondestructive loading regime (Mair et al., 2002). The friction
levels we record in 3D are higher than those generally
found in previous 2D numerical models (e.g. (Morgan,
1999, Hazzard and Mair, 2003) and in idealized 2D
experimental work (Frye and Marone, 2002). Frye and
Marone (2002) showed that 3D granular friction exceeds
2D friction by an amount that is equal to the interparticle
friction on the extra out-of-plane contacts that don’t exist
in 2D. So the enhanced friction in our models (compared
to 2D models) is likely due to extra surface friction
interactions in the 3rd dimension. The friction level we
measure is lower than laboratory values for sheared
angular gouge material as is expected due to the roughness
effects of angular grains. The influence of grain shape on
macroscopic friction is discussed in detail in Mair et al.
(2002) and in Anthony and Marone (2005). Our results
are entirely consistent with those laboratory results.
The fluctuations about the mean level of friction we
see in 3D numerical models are significantly smaller in
amplitude than for the 2D models (e.g. Morgan, 1999,
Hazzard and Mair, 2003). Fluctuations are also closely
and positively correlated to dilatancy rate (Fig. 4).
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K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
Plausible explanations (Frye and Marone, 2002, Hazzard
and Mair, 2003) involve particle reorganization mechanisms associated with shear. In 2D, all the particle motion
needed to accommodate shear is constrained to be inplane (i.e. in z direction) resulting in large dilations in the
z direction. In 3D, grain reorganizations are accommodated by in-plane motions as well as out-of-plane motion
in the y direction (Hazzard and Mair, 2003), hence the
dilation component normal to the shear direction (in z
direction) and therefore the dilatancy rate (dh / dx) is
markedly reduced. Since dilatancy rate is correlated to
friction (Morgan, 1999, Frye and Marone, 2002, this
study Fig. 4), we can completely explain the reduced
friction fluctuations in 3D.
Friction fluctuations, in our 3D models, are systematically smaller for increasing power law exponent
D = 0.8, 1.6, 2.6 (Fig. 3c and Table 1). This implies a
microstructural response to shearing (likely particle
reorganization) that is sensitive to particle size (i.e. the
number of particles across a granular layer) or particle
size distribution. An increase in mean particle size (for a
constant Gaussian psd model and a constant gouge
layer thickness) produces systematically larger friction
fluctuations (Table 1) as well as an increase in dilatancy
rate fluctuations. This system size sensitivity (equivalent to the inverse of d / H in Table 1) is sufficient to
explain the power law psd observations since a reduction in exponent D, leads to an abundance of largest
size particles, a greater importance of interactions between large particles and an overall effective increase in
mean particle size for the model layer of a given
thickness (i.e. essentially a reduction in number of particles occurring across a layer). A similar observation
was made in 2D by (Morgan, 1999) however, the
friction fluctuations for a given size distribution in 2D
were much larger.
4.2. Particle motions
Particle displacements are accommodated throughout
the particle layer for all the model configurations we
consider. Even when thin slices of the entire model layer
are analyzed, the cumulative particle motion decays
more or less monotonically from the upper sheared
region to the lower fixed boundary, with no convincing
evidence for persistent localized slip planes developing.
This suggests that, the entire layer actively participates in
shear. Individual particle motions depend mainly on
particle location with respect to the z-axis and appear to
be relatively insensitive to individual particle size or
particle size distribution (at least for the conditions we
have investigated).
This observation concurs with 2D numerical results
(Morgan and Boettcher, 1999) showing that after
substantial slip, finite strain was distributed throughout
the layer. We note however, that Morgan and Boettcher
(1999) also describe transient episodes of strain
localization during their simulations that alternate
between multiple and single slip planes. It is possible
that transient episodes of local slip exist in our
simulations but they are not discernible from our current
analysis. Aharonov and Sparks (2002) also show
evidence for fluctuations between distributed and localized slip from instantaneous velocity profiles, in 2D
simulations of dense granular shear. A direct comparison
with this study is not possible from our current measure
of accumulated shear however this will be investigated in
our future work.
In the laboratory, strain localization is common and
usually inferred from a transition to velocity weakening
behavior (Beeler et al., 1996; Mair and Marone, 2000)
or the occurrence of comminution bands interpreted as
evidence of concentrated shear. We note however that
experiments conducted at low (non-grain fracture) stress
(Mair and Marone, 2000) maintain velocity strengthening behavior (interpreted as distributed shear) throughout and show no organization of gouge layers. This does
not of course preclude short transient episodes of
localized slip but results in cumulative behavior that is
qualitatively consistent with our numerical results.
These experiments (Mair and Marone, 2000) are indeed
closer to our numerical conditions where grain fracture
is not permitted.
4.3. Contact forces and force networks
We demonstrate that networks, characterized by an
organized set of highly forced contacts, exist and appear
to be prevalent in 3D granular shear simulations. These
features have been commonly observed in 2D numerical
models (Morgan and Boettcher, 1999; Aharonov and
Sparks, 1999; Aharonov and Sparks, 2004) and 2D
photoelastic experiments (e.g. Howell et al., 1999) of
granular shear. Our analyzes indicate a highly heterogeneous distribution of forces with a subset of contacts
preferentially carrying higher than average force that
represent the dominant load bearing structure in the
sheared 3D granular layer. In addition to the large
contact forces that transmit load preferentially in an
orientation ≈ 50° to the shearing direction, we show
evidence for a support network of small forces oriented
perpendicular to shear. These contacts may be termed
spectator particles although their influence on the
overall force network is likely significant. These
K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
observations are qualitatively consistent with those from
2D studies.
The morphology of the connected high magnitude
contact forces depends on the particle size distribution
of models as well as mean particle size (i.e. system size
of the model). A narrow (Gaussian) particle size distribution gives rise to a set of discrete pipe-like clusters
containing high contact forces. These clusters plunge at
≈ 50° to shear and are not obviously linked in the y
direction by highly forced contacts however we see
evidence for low contact forces oriented in appropriate
directions to provide an out of plane support network. A
wide (power law) size distribution granular model results in more distributed tree or sheet-like force networks that have significant out-of-plane linkage and
appear to enjoy a wider spread of orientations. When we
consider large contact forces (e.g. Fig. 8a), we see that
individual contact forces for power law D = 2.6 models
enjoy a wider spread of orientations than Gaussian psd
models. Power law D = 2.6 models also show a directed
force network of small forces perpendicular to shear that
is more developed than in the Gaussian models.
We show that high load force networks evolve with
strain and their individual elements may rapidly
disintegrate and reform, although their overall pattern
(i.e. spatial distribution and dominant orientation)
appears to be persistent. These characteristics are consistent with 2D simulations (Morgan and Boettcher,
1999; Aharonov and Sparks, 1999) however the
morphology of our 3D force networks is unsurprisingly
more complicated. We show a link between evolution
of contact force distributions and macroscopic friction
(Fig. 10). It appears that the development or enhancement of a contact force network element that spans a
granular layer (from top to bottom) is associated with an
increase in macroscopic friction. In contrast, when a set
of linked contact forces breakdown and a chain becomes
disrupted, macroscopic friction level is decreased. This
observation has potentially important implications for
fault zone stability, if we believe that force networks
indeed develop in natural faults containing granular
gouge material.
In our models, grain fracture is not permitted, however, the distribution of high magnitude contact forces
and force networks may indicate the ‘fracture potential’ for different systems. According to the constrained
comminution model of Sammis et al. (1987) neighboring
grains of equivalent size are most likely to fracture. Our
Gaussian psd models, having many similar sized
particles, have many similar sized contact forces giving
rise to significant ‘fracture potential’. In contrast, for
power law D = 2.6 distributions, the smaller grains form
483
a matrix supported texture which effectively cushions
the large grains. The wide distribution of contact forces
and in particular the abundance of small forces reduces
the fracture potential of neighbors. We suggest this may
be a more stable configuration (Sammis et al., 1987)
where breakage of neighboring grains is less likely. This
is entirely consistent with the Sammis et al. (1987)
hypothesis and results of Morgan and Boettcher (1999).
On considering the mechanical implications of
different force chain morphologies, we speculate that
spatially discrete, pipe-like force networks (typical in
Gaussian psd models) may be indicative of a ‘fragile
network’. This means that if one contact fails, the entire
linear cluster is at risk of failing. Since a single linear
cluster accommodates a significant proportion of the
total stress, it seems reasonable that this action may
result in a macroscopic stress drop. By contrast, tree-like
force networks, typical for power law psd models, may
be more resilient. In their case, it is unlikely that failure
of a single contact would cause complete network
failure (or a macroscopic stress drop) since there is
significant out-of-plane communication and support. In
addition, the individual contacts have a wide range of
sizes and orientations, hence there are a wider range of
grains and contacts that can potentially offer support.
4.4. Comparison with laboratory predictions
A relation between particle size distribution and the
nature of stress accommodation was predicted from
laboratory studies (Mair et al., 2002). From interpretations of laboratory friction data, Mair et al. (2002)
suggested that narrow size distribution (i.e. Gaussian)
granular systems have discrete, focused, well organized
force chains, whereas wide psd (power law) systems
have more spread out force chains. Our 3D numerical
results completely support these predictions and indicate
hybrid behavior for intermediate psd systems.
Unlike laboratory data that commonly show highly
repetitive stick–slip behavior (e.g. Karner and Marone,
2000; Mair et al., 2002) there is a notable lack of
repetitive stick–slip in our 3D numerical simulations.
This discrepancy can be accounted for by the constant
velocity boundary condition used in simulations. This
effectively simulates an infinite stiffness loading frame,
whereas a more compliant loading frame would allow
stick slip motion. An adapted boundary condition (other
than a constant velocity boundary condition) would be
required to reproduce laboratory-like stick slip (e.g.
Aharonov and Sparks, 2004).
Another relevant consideration, given the strong
laboratory evidence for time dependent healing between
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K. Mair, J.F. Hazzard / Earth and Planetary Science Letters 259 (2007) 469–485
dynamic slip events (Dieterich, 1978; Mair et al., 2002),
is the lack of time dependent contact processes operating
in our model. This is a clear limitation of the model since
without time dependent contact evolution, stick slip
would be purely geometric and not particularly realistic.
2nd order time dependent processes are therefore extremely important and will be added to numerical simulations in future. Importantly, the nature of contact
forces and their persistent use of individual particles will
very likely influence contact healing mechanisms thus
understanding force chain morphology is an essential
first step in the quest for more realistic time dependent
models.
4.5. Application, limitations and future work
Numerical simulations on sheared assemblies of solid
spherical particles that don’t break or evolve, are clearly
a marked simplification of natural fault zones. To
properly model fault gouge, we need rough, 3D particles
that are permitted to break and whose contact properties can evolve with time to simulate contact healing
processes. Additional complexity is necessary in numerical models however it must be added gradually with
each additional level of intricacy being validated through
comparison with laboratory experiments or field observations so that we fully appreciate the relative importance and interaction of the individual process.
In this manuscript we have concentrated on visualizing contact force distributions produced under 3D
granular shear. We did not attempt to simulate grain
breakage, but see Abe and Mair (2005) for recent 3D
grain fracture modeling results. Although there is no
gouge evolution per se, we justify our non-fracture
approach by investigating different ‘end member’
particle size distribution models that may reveal the
behavior that is dominant at different stages of gouge
evolution.
We successfully simulated 3D granular shear, producing first order results that are validated by nonfracture laboratory tests (on idealized granular materials) confirming the mechanical behavior produced by
our simulations. Although our 3D numerical approach
was inherently simple, and certainly lacked 2nd order
complexities, it represents an important step forward
from 2D simulations and highlights the necessity of 3Dmodels for investigating frictional shear. We show
diversity of stress accommodation during 3D shear and
provide a benchmark database for future 3D granular
shear simulations that add complexities such as e.g.
time dependent contact evolution, grain fracture, grain
angularity.
5. Conclusions
Our 3D numerical simulations of sheared granular
material demonstrate the prevalence of 3D directed
force networks that preferentially support load across
the sheared granular layers. We show that force network
morphologies are sensitive to particle characteristics, in
particular grain size distribution, confirming recent
predictions based on laboratory friction experiments.
Models having a narrow (i.e. relatively uniform) psd
exhibit pipe-like force clusters with a persistent orientation oblique to shearing, whereas wider psd models
(e.g. power law D = 2.6 size distributions) show tree-like
force networks that have abundant out-of-plane linkage
and take a wider range of orientations. Highly localized
discrete force clusters may be more ‘fragile’ than treelike force networks, with the failure of an individual
contact more likely to lead to sample failure. Macroscopic friction level is insensitive to the nature of stress
accommodation, however, force network morphology
and evolution appears to be linked to fluctuations about
this mean level of macroscopic friction. We therefore
conclude that heterogeneous force distributions, if they
exist in natural fault gouge material, may exert an
important control on fault stability and hence the seismic
potential of active faults.
Acknowledgements
We thank A. Heath, R.P. Young and Itasca Consulting for European Commission funded software development through the SAFETI project (Nuclear Fission
Program). We are very grateful to M. Dabrowski and E.
Jettestuen (both PGP) for assistance in visualization and
plotting of force chain distributions. We appreciate
reviewers’ insights that helped improve the final manuscript. K. Mair was funded by a Royal Society Dorothy
Hodgkin Fellowship held at University of Liverpool and
more recently by the Center for Physics of Geological
Processes (PGP) at the University of Oslo.
Appendix A. Supplementary data
Supplementary data associated with this article can
be found, in the online version, at doi:10.1016/j.epsl.
2007.05.006.
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