Macroscopic Friction Response of Rotational and Non-rotational Lattice
Solid Gouge Models in 2D and 3D
S. Latham*, S. Abe & P. Mora
Australian Computational Earth Systems Simulator (ACcESS) MNRF,
The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia
QUAKES, Earth Systems Sciences Computational Centre,
The University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia
Traditionally, 2D Discrete Element Models (DEMs) have been preferred over 3D models, for fault gouge simulations, because of the computational cost of solving 3D problems. In order to realistically simulate fault gouge
processes it is important to characterise differences between 2D and 3D models and be able to assess whether
2D models are adequate for approximating 3D gouge dynamics. In this paper, 2D and 3D fault gouges are simulated as two rectangular elastic blocks of bonded particles, separated by a region of randomly sized non-bonded
spherical gouge particles, sheared in opposite directions by normally-loaded driving plates. The dynamic behavior of multiple model parameterisations is analysed by examining instantaneous macroscopic fault friction (µI )
statistics. The response of the mean macroscopic friction is characterised for varying values of inter-particle
(microscopic) friction µP in 2D and 3D and for non-rotational and rotational particle dynamics. In the nonrotational models, realistic angular gouge mean macroscopic friction values (E[µI ] ≈ 0.6) are obtained in the
simulations for a 2D inter-particle friction value of µP = 0.3 and 3D value of µP = 0.2. The rotational models
exhibit mean macroscopic friction values of E[µI ] = 0.3 (in 2D) and E[µI ] = 0.38 (in 3D) for inter-particle
friction values µP ≥ 0.3. The 2D rotational macroscopic friction values are in close agreement with comparable
2D glass-rod (E[µI ] ≈ 0.3) laboratory experiments. In the 3D case, the simulated mean macroscopic friction
values are lower than those of 3D spherical bead laboratory experiments where 0.4 < E[µI ] ≤ 0.45.
1 INTRODUCTION
The presence of gouge particles plays a fundamental
role in influencing the macroscopic behavior of faults
and shear zones. In order to gain a greater understanding of earthquake nucleation processes in fault gouge
zones, it is important to characterise the relationships
between the microscopic and macroscopic mechanics. Computational simulation has played an important role in the analysis of complex granular materials, allowing researchers to vary micro-mechanical
parameters and observe the resulting influence on the
macro-mechanics of the computational model.
The Lattice Solid Model (LSM) (Mora & Place
1994; Place & Mora 1999) is a variant of the DEM
(Cundall & Strack 1979) which has been used to
model the dynamics of fault gouge processes. Typical 2D fault gouge models, using the LSM, have involved tens of thousands of particles. For comparable
3D problems, particle numbers can readily increase
into the millions. These large 3D problems have remained intractable for serial implementations of the
LSM. Parallel computing architectures, such as the
SGI Altix 3700 super-cluster, provide the opportunity
to solve large computational problems. In order to
take advantage of high performance systems, a Message Passing Interface version of the LSM has been
implemented (Abe et al. 2004). Recent benchmarks
demonstrated an 80% parallel efficiency for the parallel LSM on 128 processors of the SGI Altix 3700
(Latham et al. 2004).
Idealized laboratory experiments (Frye & Marone
2002; Mair et al. 2002) provide the opportunity to
compare computational experiments with real granular shear data (Hazzard & Mair 2003) in order to
validate numerical models and/or calibrate simulation
parameters. This paper describes results from simulations of 2D and 3D gouge regions, using the parallel implementation of the LSM, and compares macroscopic friction (also termed the effective or fault friction) statistics with those from comparable laboratory
experiment. Section 2 gives a brief overview of the
LSM and a description of the fault gouge model setup.
Macroscopic friction statistics produced in 2D and 3D
gouge simulations, for varying values in inter-particle
The LSM (Mora & Place 1994; Place & Mora 1999)
is a particle based model similar to the DEM (Cundall
& Strack 1979). The model consists of spherical particles which are characterized by their radius, mass,
position and velocity. The particles interact with their
nearest neighbours by imparting elastic and frictional
forces.
In the gouge simulations, particles were restricted
to interact in one of two ways. A pair of particles
could be involved in either a bonded interaction or in
a frictional interaction with one another. A volume of
bonded particles simulates a linear elastic solid within
the model. Particles within the gouge region are nonbonded and undergo frictional interactions. An artificial viscosity is also present in the model to prevent the buildup of kinetic energy in the closed system. The amount of viscous damping has been chosen such that the stick-slip dynamics are not significantly influenced (Mora & Place 1994). In the nonrotational models, the bonded and frictional interactions are as described in Latham et al. (prep). In the
rotational models, additional linear elastic shearing
forces are applied at particle boundaries which generate moments.
The simulated fault gouge is represented as two
rectangular elastic blocks of bonded particles, with
a rough fault zone separated by a region of randomly sized non-bonded gouge particles. The elastic
blocks are sheared in opposite directions by normallyloaded driving plates. Figures 1 and 2 illustrate the
the gouge region of 2D and 3D models, respectively.
The block particles are uniformly sized with radius
R0 and bonded in a regular 3D lattice. The roughness
particles range in radial size from 0.4R0 to R0 in the
3D models and from 0.165R0 to R0 in the 2D models. The block and roughness particles are bonded to
neighbouring block and roughness particles.
The gouge particles are non-bonded, and interact
with neighbouring gouge and roughness particles via
repulsive linear elastic and Coulomb frictional forces.
In the 3D model, the gouge particles range in radial
size from 0.4R0 to R0 and in the 2D model range in
size from 0.165R0 to R0 .
An x-z layer of particles at the top of the upper
block and an x-z layer of particles at the bottom of
the lower block are elastically bonded to walls (not
shown in Figures 1 and 2) which lie in the x-z plane.
These walls apply compressive forces in the positive
and negative y directions and are also sheared in opposite x directions at constant velocity. The lower
block moves in the positive x direction while the upper block moves in the negative x direction.
A circular boundary condition occurs on y-z planes
at the left (x=0) and right (x = 34R0 ) extents of the
particle domain (a particle exiting the right hand side
of the model reappears at the left side and vice–versa).
In the 3D configuration, frictionless elastic confining
walls in the x-y planes prevent particles from being
“squeezed” in the z direction out of the gouge region.
The 2D and 3D models have the same x dimension and similar y dimension, with the difference in
y dimension due to the different circle and sphere
regular–lattice packings used in 2D and 3D, respectively. The same constant pressure was applied to all
models by the uppermost wall (which moves in the y
dimension to maintain constant pressure) and the bottom wall remained fixed in the y dimension.
The 2D model contains 1701 particles (713 gouge
particles) and the 3D model contains 10690 particles (4123 gouge particles). The 2D and 3D simulations were run to 500% strain, with the driving walls
Figure 1. Gouge region of the 2D model.
Figure 2. Gouge region of the 3D model.
friction and for rotational and non-rotational particles,
are presented in Section 3.
2 FAULT GOUGE MODEL
sheared at speed 0.001R0 /T , where T is model time
unit. The P-wave speed in each model is ≈ R0 /T .
3 MACROSCOPIC FRICTION RESULTS
Figures 3 and 4 plot the instantaneous macroscopic
friction statistics (mean, standard deviation, minimum
and maximum) versus inter-particle friction µP . The
instantaneous macroscopic friction µI is measured by
dividing the shear stress (in the x direction) on the
driving walls by the normal stress (y direction) on the
driving walls. Macroscopic friction values from the
initial “load-up” phase in the simulations are excluded
when computing the µI statistics.
In both the 2D and 3D non-rotational models of
Figure 3, the mean macroscopic friction E[µI ] increases monotonically with the increasing value of
inter-particle friction µP . Morgan (2004) has shown
similar 2D friction responses using non-rotational
models. The 2D mean macroscopic friction values
remain below those of the 3D model for an equivalent value of inter-particle friction. The macroscopic
friction variance also increases monotonically with
increased µP . In the 2D case, the variance V [µI ]
is greater than the 3D variance for the corresponding value of µP . Similarly, the range (max(µI ) −
min(µI )) of instantaneous macroscopic friction values is greater in 2D than 3D. Laboratory gouge experiments with angular sand typically produce mean
macroscopic friction values of E[µI ] ≈ 0.6 (Mair
et al. 2002). The 2D model attained E[µI ] = 0.65 for
µP = 0.3 while the 3D model produced E[µI ] = 0.60
for µP = 0.2. While the mean macroscopic friction
values agree closely with that of angular sand for
these
values of
p
p values of µP , the standard deviation
V [µI ] = 0.07 in the 2D model and V [µI ] = 0.02
in the 3D model appear much greater than the negligible standard deviation values produced in the angular sand laboratory experiments. The non-rotational
models more readily produce stick-slip behaviour, especially for higher values of µP , where as stick-slip
was absent from the stable-sliding angular sand laboratory experiments.
Simulations performed by Mora et al. (2000) which
used 2D angular aggregates of bonded spherical particles, generated realistic macroscopic friction values
of E[µI ] = 0.6. This work also demonstrated that the
mean macroscopic friction of 0.6 was insensitive to
the level of inter-particle friction in the presence of
rotational dynamics as a result of self-regulation between the amount of slipping and rolling in the gouge
layer.
In the 2D and 3D rotational models of Figure
4, the mean macroscopic friction remains approximately constant for values of µP > 0.3. For these
higher
p values of µP , the 2D model has E[µI ] ≈ 0.3
V [µI ] ≈ 0.05, while in 3D E[µI ] ≈ 0.38 and
and
p
V [µI ] ≈ 0.02. As for the non-rotational cases, the
mean macroscopic friction is higher in the 3D model
Figure 3. Non-rotational model µI statistics versus µP for 2D
(top) and 3D (bottom) cases.
Figure 4. Rotational model µI statistics versus µP in the 2D (top)
and 3D (bottom) cases.
than in the 2D model, with the 2D model having a
greater variance and range of instantaneous macroscopic friction values than the 3D model.
Laboratory experiments with glass rods (Mair et al.
2002) produced 2D mean macroscopic
friction valp
ues of E[µI ] = 0.274 with V [µI ] = 0.053 (Hazzard & Mair 2003). Laboratory rod-shearing experiments in 1D produced friction values in the range
0.14 ≤ E[µI ] ≤ 0.19 (Frye & Marone 2002). The measured 1D E[µI ] value is equivalent to the physical coefficient of friction for the substance being sheared
and corresponds to the µP friction coefficient used
in the simulations. The 2D macroscopic and interparticle friction simulation values, of E[µI ] = 0.274
for µP = 0.15, agree well with the 2D laboratory
experiments. The macroscopic
pstandard deviation in
the glass rod experiment is pV [µI ] = 0.053 while
the 2D simulation produced V [µI ] = 0.033. This
lower standard deviation value is likely to be due to
the greater particle size distribution (PSD) used in the
simulations. As suggested by Mair et al. (2002), it is
the uniformity of sliding which is sensitive to PSD
rather than mean frictional strength. A narrow PSD
promotes stick-slip (and large instantaneous friction
variance), while a wide PSD exhibits stable sliding.
Laboratory experiments with glass beads produced
3D mean macroscopic friction values of E[µI ] = 0.45
(Mair et al. 2002) in the stable sliding case of wide
PSD. The narrow PSD experiments with stick-slip behaviour, gave a maximum macroscopic friction value
of max(µI ) = 0.473 (Hazzard & Mair 2003) and a
mean macroscopic friction value of E[µI ] ≈ 0.41 (estimated from plots in Mair et al. (2002)). The 3D simulations produced mean macroscopic friction values
no greater than 0.39, with maximum friction values
of max(µI ) = 0.45. The higher friction values given
by the laboratory experiments are possibly due to the
presence of non-spherical beads. Mair et al. (2002)
point out that the frictional strength is sensitive to particle angularity. Therefore, the presence of a small
percentage of non-spherical or angular particles is
likely to increase the frictional strength by inhibiting
rolling.
4 CONCLUSION
The rotational simulations have produced macroscopic friction values which agree quantitatively with
laboratory experiment, especially in the 2D case. The
simulated macroscoptic friction in 3D was higher than
that in 2D which concurs with similar computational
and laboratory results (Hazzard & Mair 2003). The
3D simulations had a slightly lower macroscopic friction response than that of spherical bead experiments.
Models using a mix of spherical particles and nonspherical aggregates may be a more accurate model
of the glass bead experiment and yield higher mean
friction. The significant differences between the friction response of 2D and 3D gouges, in both the sim-
ulations and laboratory experiments, lend strong support for 3D computational modeling of fault gouge
dynamics in order to conduct quantitative analysis of
3D stick-slip processes.
ACKNOWLEDGMENTS
Funding support for this work is gratefully acknowledged. Project work is supported by the ACcESS
MNRF, The Uni. of Queensland and SGI. The ACcESS MNRF is funded by the Australian Commonwealth Government and participating institutions
(Uni. of Queensland, Monash Uni., Melbourne Uni.,
VPAC, RMIT) and the Victorian State Government.
Computations were performed on the ACcESS SGI
Altix 3700, which was funded by the QLD State Government Smart State Research Facility Fund and SGI.
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