GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 13, 1708, doi:10.1029/2003GL017534, 2003
The importance of the third dimension in granular shear
James F. Hazzard and Karen Mair1
Lassonde Institute, University of Toronto, Canada
Received 15 April 2003; revised 29 May 2003; accepted 5 June 2003; published 11 July 2003.
[1] We present new numerical results showing the
importance of the third dimension in simulations of
granular shear. Our 3-D numerical models of gouge layers
under shear exhibit friction levels notably higher than 2-D
models and values that approach those of recent laboratory
experiments on spherical beads. Fluctuations in friction are
directly related to grain reorganization normal to shear. In
3-D, these fluctuations are reduced due to significant particle
motion in the third dimension, leading to a steady frictional
sliding curve. Since these fluctuations are small compared
with 2-D models, our 3-D simulation offers potential to
investigate small amplitude but extremely important second
order rate and state friction effects that fundamentally
INDEX
control the strength and stability of fault zones.
TERMS: 3210 Mathematical Geophysics: Modeling; 5112 Physical
Properties of Rocks: Microstructure; 7209 Seismology: Earthquake
dynamics and mechanics. Citation: Hazzard, J. F., and K. Mair,
The importance of the third dimension in granular shear, Geophys.
Res. Lett., 30(13), 1708, doi:10.1029/2003GL017534, 2003.
1. Introduction
[2] Earthquakes commonly occur on faults containing
gouge; therefore investigating the processes operating in
gouge sheared under stress is extremely relevant to earthquake and fault mechanics. Laboratory shearing experiments [e.g., Marone, 1998; Beeler et al., 1996] conducted
on thin granular layers have been useful in elucidating how
loading conditions and perturbations influence friction (see
Scholz [1998] and Marone [1998] for recent summaries).
[3] Similar studies can be conducted using discontinuum
numerical models in which the shear zone is represented by
an assembly of particles that can move independently
[Aharonov and Sparks, 2002; Antonellini and Pollard,
1995; Mora and Place, 1998; Morgan and Boettcher,
1999; Scott, 1996]. These models can yield informative
insights into micromechanical processes operating in simulated fault zones by allowing direct observation of particle
motions, stresses and interactions during shearing. The
models also help isolate the influence of material characteristics such as interparticle friction, particle size distribution
[Morgan, 1999], effect of normal stress [Aharonov and
Sparks, 1999] or particle angularity [Mora and Place,
1998].
[4] Validation of numerical models through laboratory
experiments is essential to determining whether the model
behavior approaches reality [Wawersik, 2000]. Only after
1
On leave from Department of Earth Sciences, University of Liverpool,
Liverpool, U.K., L69 3GP.
Copyright 2003 by the American Geophysical Union.
0094-8276/03/2003GL017534$05.00
41
models have been shown to accurately reproduce laboratory
results can they be used to simulate more complex, fieldscale geosystems such as tectonic faulting and earthquakes.
[5] Recent attempts have been made to bridge the gap
between numerical models and the natural fault systems by
conducting highly idealized laboratory investigations (e.g.
perfectly spherical particles or two-dimensional rods [Frye
and Marone, 2002; Mair et al., 2002]). These experiments
have highlighted the importance of particle characteristics
and dimensionality in frictional response.
[6] We present new results from 3-D numerical modeling
of shear in granular materials. We show that 3-D numerical
models exhibit higher friction with smaller fluctuations than
those in 2-D. Results agree well with recent laboratory work
conducted under idealized conditions, validating the general
aspects of our approach and offering the potential for
incorporating further complexity.
2. Method
[7] The distinct element method [Cundall and Strack,
1979] has been used extensively to model granular shear
zones. This approach considers an assembly of perfect,
indestructible cylindrical (2-D) or spherical (3-D) particles
that interact only at points of contact. An explicit solution
scheme is used that updates particle positions and contact
forces each time step.
[8] Particle and contact micro-properties are prescribed
and the resulting macroscopic response is measured. The
laws governing interactions between particles are simple,
however interactions between large numbers of particles
lead to interesting complex behavior.
[9] For the research presented here, the Particle Flow
Code in 3 Dimensions (PFC3D) is used [Itasca Consulting
Group, 1999] with a Hertz-Mindlin contact model [Cundall,
1988]. The shear force at a contact is limited by specified
coefficient of friction. If the shear force is overcome then
stable sliding occurs at the contact. The Hertz-Mindlin
contact model was thought to be a more realistic representation of contact behavior than a linear, Hookian model,
however with the values of normal stress and contact
stiffness presented here, the difference in observed macro
frictional behavior is minimal.
[10] The simulated layers are created by filling a 6-sided
shape with a random assembly of particles (see Figure 1).
Particle microproperties are assigned (see Table 1), then the
assembly is compacted to the desired isotropic stress state of
5 MPa (Figure 1a). Particle properties and the loading
geometry are comparable to recent laboratory experiments
[e.g., Frye and Marone, 2002].
[11] Shearing is horizontal and the top and bottom
boundaries are comprised of a controlled layer of particles
to provide roughness (Figure 1). A servo-control mecha- 1
41 - 2
HAZZARD AND MAIR: 3-D MODEL OF GRANULAR SHEAR
Figure 1. (a) A 3-D numerical model (T001) after
isotropic compaction (5 MPa) but prior to shear. A vertical
marker band is shown. The layer is 3.7 mm thick (z
dimension) by 6.9 mm in the x and y dimensions. The
average particle diameter is 254 mm. Shear stress is applied
by moving the top layer in the positive x direction. The
bottom layer is fixed. (b) The model after 100% shear strain.
The top boundary layer has been stripped off to enable
viewing of the marker band.
nism is activated at the top boundary particles to adjust
vertical velocities thereby maintaining the desired (constant)
normal stress. The bottom boundary is fixed. During a
numerical experiment, the top boundary is driven at a
constant horizontal velocity to apply shear to the particle
layer. The two side boundaries perpendicular to the direction of shear are periodic such that particles exit from one
side and reappear on the other. The other two boundaries
(parallel to the direction of shear) are fixed, frictionless
walls.
[12] A suite of numerical models were run, including
several 2-D simulations, to test the robustness of the results
and the sensitivity to a range of input parameters. The
influence of the 3rd dimension and the effect of using
different particle sizes are described in section 3.
3. Results
[13] A 3-D model after 100% shear strain (Figure 1b)
shows a continuous marker band indicating that strain is
evenly distributed throughout the layer without any obvious
localization features. The frictional response of this model
and a similar 2-D model are presented in Figure 2a. For the
Table 1. Laboratory and Numerical Experiments
Experiment
T007
T001
T009
T012
m349
m301
T102
T103
T101
T104
m518
Description
3-D
3-D
3-D
3-D
3-D
3-D
2-D
2-D
2-D
2-D
2-D
model
model
model
model
lab study (glass beads)
lab study (angular sand)
model
model
model
model
lab study (glass rods)
Mean Particle
Diameter (mm)
(±1 std. dev)
Mean macro
friction
(±1 std. dev)
191 (±16.5)
254 (±22)
381 (±33)
508 (±44)
105 – 149
50 – 150
127 (±11)
191 (±16.5)
254 (±22)
381 (±33)
1000
0.403 (±0.015)
0.407 (±0.010)
0.409 (±0.021)
0.404 (±0.017)
0.473a
0.644a
0.292 (±0.037)
0.322 (±0.072)
0.287 (±0.057)
0.313 (±0.089)
0.274 (±0.053)
Modeled particles and glass beads have a shear modulus of 22.0 GPa,
Poisson’s ratio of 0.25 and a Gaussian size distribution. Interparticle friction
of modeled particles is 0.5. Normal stress is 5 MPa for all tests except
m518, for which sn = 1 MPa. Experimental data is from [Mair et al., 2002]
and [Frye and Marone, 2002].
a
Peak values given instead of mean.
Figure 2. (a) Friction versus shear displacement for 2-D
and 3-D models of shear in granular layers. Laboratory data
is also shown for spherical beads and angular sand from
Mair et al. [2002]. (b) Friction and dilatancy rate (dh/dx
where h is the layer thickness and x is the shear
displacement) for the 2-D and 3-D models.
2-D numerical model, sliding friction has mean value of 0.3
and exhibits large fluctuations (between 0.2 and 0.38) as a
function of sliding. These observations agree well with
previous 2-D numerical results [e.g., Morgan, 1999], and
with 2-D laboratory experiments [Frye and Marone, 2002].
[14] In the 3-D model, sliding friction is enhanced to a
mean level of 0.4. This supports the hypothesis that
friction is higher in 3-D than 2-D due to an extra dimension
of grain interaction (sliding, rolling, dilation and compaction) during shear [Frye and Marone, 2002].
[15] Variations in friction in the 3-D model are significantly smaller than in the 2-D case, ranging between 0.38
and 0.43. These fluctuations are associated with dilation and
compaction of the granular layer during shear. Figure 2b
highlights the clear connection between the amplitude of
friction fluctuations, and dilatancy rate (dh/dx where h is the
layer thickness and x is the shear displacement) for the 2-D
and 3-D models.
[16] Figure 3 summarizes the macroscopic friction characteristics for a series of 2-D and 3-D models with different
particle sizes (see Table 1). The systematic differences
between 2-D and 3-D models, discussed above, are clear
and reproducible and the models compare favorably to
laboratory results. For a given case (2-D or 3-D), particle
size has little effect on average friction although smaller
particles tend to produce reduced frictional fluctuations. The
influence of adding a 3rd dimension to the models is clearly
much stronger than that induced by changes in particle size
(at least over the range shown).
HAZZARD AND MAIR: 3-D MODEL OF GRANULAR SHEAR
Figure 3. Mean friction and one standard deviation plotted
against particle size for 2-D and 3-D models. Friction values
represent the model response between 1 and 2.5 mm of
shear. Laboratory studies are shown for comparison.
[17] The observed fluctuations are likely related to the
number of layers of free grains. Clearly 3-D models will
have significantly more free layers for the same particle
size. It is possible that a 2-D model with extremely small
particles could yield small fluctuations like the 3-D models,
however this would reproduce the observed macro results
without capturing the important micromechanisms producing the observations.
[18] Figure 2 shows that the reduced friction fluctuations
in 3-D are due to smaller changes in layer thickness required
to accommodate shear. The third dimension means that
particles have greater freedom to re-organize ‘out-of-plane’.
Figure 4 demonstrates this, showing velocities of individual
particles during shearing (viewed from above). There is
significant motion of particles in directions not parallel to
the shearing direction indicating ‘out-of-plane’ motion is a
common phenomenon for accommodating the shear strain.
This motion is clearly not permitted in the 2-D case.
[19] To quantitatively examine this effect, the average x,
y and z particle velocities were calculated for each of the
models at 100% strain. It was found that the average
velocities perpendicular to shear (RMS of y and z velocities) were 25 –45% of the average velocity in the shear
direction (x-velocity) for both 2-D and 3-D models. In 2-D,
all of the shear-perpendicular strain is the z-direction.
Whereas in 3-D, this strain is being accommodated (roughly
equally) in each of the directions (y and z) perpendicular to
shear.
41 - 3
[21] When angular gouge is sheared in the laboratory
under similar conditions, no stick-slip motion is observed
(see Figure 2). Therefore certain special conditions are
required for the stick-slip phenomenon including the use
of spherical grains with a narrow size distribution. These
conditions were replicated closely in the numerical experiments however stick-slip motion was not observed.
[22] One possible explanation is that the numerical models are loaded by a constant velocity boundary, which
essentially simulates a loading frame of infinite stiffness.
It is well known that a more compliant loading frame
increases the potential of observing stick-slip motion.
Therefore, to reproduce the stick-slip behavior observed in
the laboratory, more realistic boundary conditions may be
required.
[23] Another contributing factor is likely the lack of timedependent processes in the model. In the laboratory, the size
of stress drops increase systematically with recurrence
interval (ie stick time between events) indicating that a time
dependent healing process is operating between dynamic
slip events [see Mair et al., 2002]. Our numerical model has
no time dependent processes (e.g., contact healing) therefore it is not surprising that it does not imitate this behavior.
The current model is more representative of a fault being
sheared at infinite velocity, hence laboratory and model
results would converge (i.e. exhibit stable sliding) at very
high loading rates [see Berman et al., 1996; Karner and
Marone, 2000].
[24] Stick slip motion associated with first order, purely
geometric effects has been produced in 2-D numerical
models when clusters of bonded particles make up the shear
zone [Mora and Place, 1998] or when rolling of particles is
inhibited [Morgan, 2003]. Particle geometry and resulting
distribution of stress across 3-D granular layers is clearly
also an important factor influencing stick slip in the laboratory [Mair et al., 2002]. However, the time dependent
nature of stick slip cycles emanating from extremely simple
3-D geometries (unbroken spherical beads) in the laboratory
4. Discussion
[20] Figures 2 and 3 show graphical comparisons of 3-D
modeling and laboratory results. The level of friction
attained in the 3-D numerical model approaches the frictional strength of 3-D spherical beads sheared in the
laboratory under similar loading conditions. However, unlike the laboratory data that show highly repetitive stick slip
[Mair et al., 2002], our numerical results indicate stable
sliding at a relatively constant level. Increasing particle size
in the models (see Figure 3) produced larger friction
fluctuations but the magnitude and character of these
fluctuations still differed significantly from the laboratory
stick-slip observations.
Figure 4. Total velocity of each particle in a horizontal slice
of the 3-D model (T001) at 50% strain as viewed from above.
The slice shows approximately one quarter of the shear plane
(see inset). The velocity vectors (length and width) are scaled
to a maximum equal to the shear loading velocity.
41 - 4
HAZZARD AND MAIR: 3-D MODEL OF GRANULAR SHEAR
suggests that 2nd order grain-scale healing processes are
fundamentally important [Mair, in prep 2003].
[25] Attempts have recently been made to add 2nd order
rate and state dependent effects to 2-D numerical models
[Abe et al., 2002; Igami et al., 2002; Morgan, 2003]. This is
a welcome step forward and has been successful in qualitatively reproducing certain aspects of laboratory rate and
state friction. One issue however, is that these subtle second
order effects are masked by the large fluctuations in sliding
friction caused by geometric interactions as grains bounce
over each other. In the 3-D model presented here, natural
variations in macroscopic friction are small (compared to
the 2-D case) therefore we have the potential to isolate 2nd
order changes in friction associated with rate and state
effects and thereby assess the relative importance of 1st
order geometry and 2nd order healing mechanisms. We
propose therefore that 3-D models are essential for investigation of these important 2nd order frictional effects.
project (Euratom Program), and the Royal Society for enabling collaboration between K. Mair and the University of Toronto. We also wish to thank
the reviewers, whose comments helped improve the manuscript. K. Mair is
supported by a Royal Society Dorothy Hodgkin Fellowship. J. Hazzard is
supported by the Lassonde Institute.
References
[26] New 3-D numerical modeling has revealed that outof-plane particle motion is extremely relevant to understanding the mechanical behavior of granular material under
shear. In 3-D models, levels of friction are higher than in
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[29] Acknowledgments. We wish to thank C. Marone, K. M. Frye
and R. P. Young for helpful discussions, Itasca Consulting and A. Heath for
European Commission funded software development through the SAFETI
J. Hazzard and K. Mair, Lassonde Institute, University of Toronto,
Toronto, Canada, M5S 3E3. (j.hazzard@utoronto.ca; k.mair@utoronto.ca)
5. Conclusions