GEOPHYSICAL RESEARCH LETTERS, VOL. 32, L05305, doi:10.1029/2004GL022123, 2005
Grain fracture in 3D numerical simulations of granular shear
Steffen Abe
Australian Computational Earth Systems Simulator (ACcESS), University of Queensland, Brisbane, Australia
Karen Mair
Physics of Geological Processes, University of Oslo, Norway
Received 1 December 2004; revised 16 January 2005; accepted 4 February 2005; published 9 March 2005.
[1] We present a new method to implement realistic grain
fracture in 3D numerical simulations of granular shear. We
use a particle based model that includes breakable bonds
between individual particles allowing the simulation of
fracture of large aggregate grains during shear. Grain
fracture simulations produce a comminuted granular
material that is texturally comparable to natural and
laboratory produced fault gouge. Our model is initially
characterized by monodisperse large aggregate grains and
gradually evolves toward a fractal distribution of grain sizes
with accumulated strain. Comminution rate and survival of
large grains is sensitive to applied normal stress. The fractal
dimension of the resultant grain size distributions (2.3 ± 0.3
and 2.9 ± 0.5) agree well with observations of natural
gouges and theoretical results that predict a fractal
dimension of 2.58. Citation: Abe, S., and K. Mair (2005),
Grain fracture in 3D numerical simulations of granular shear,
Geophys. Res. Lett., 32, L05305, doi:10.1029/2004GL022123.
1. Introduction
[2] Natural faults often contain a layer of fragmented
rock known as fault gouge that develops as rough fault
surfaces grind over each other. The evolution of this
material influences the frictional strength and stability and
hence earthquake potential of faults, therefore, it is important to understand how gouge develops. Observations of
natural and laboratory fault gouges indicate that particle size
distribution changes with accumulated shear due to grain
fracture. In mature fault zones, or sheared laboratory samples, gouge commonly develops a wide size distribution,
best characterised by a fractal distribution of dimension
2.6 [Sammis et al., 1987; Marone and Scholz, 1989;
Blenkinsop, 1991].
[3] Recent laboratory investigations of granular shear
highlight the dependence of mechanical behavior on particle
shape, roughness, particle size distribution and dimensionality [Losert et al., 2000; Mair et al., 2002; Frye and Marone,
2002]. Discontinuum numerical models of granular shear can
yield important insights into micro-processes operating in
fault zones and offer a degree of visualization not generally
afforded by laboratory experiments. They are well suited to
tracking dynamic particle interactions, however most simulations to date have been conducted in 2D [e.g., Mora and
Place, 1998; Aharonov and Sparks, 2002; Morgan, 1999]
and only few simulations include gouge evolution [Place and
Mora, 2000; Abe et al., 2002]. To the authors knowledge, no
Copyright 2005 by the American Geophysical Union.
0094-8276/05/2004GL022123$05.00
attempts have been made to implement gouge evolution in
3D although a 3D approach is essential in modeling these
granular systems [Hazzard and Mair, 2003].
[4] We present new results from 3D numerical simulations
of granular shear where we implement gouge evolution
through realistic fracture of aggregate grains. We show that
comminution of aggregate grains with accumulated strain
produces a realistic gouge texture having a broad fractal size
distribution and a mix of angular and spherical grains. Our
results agree well with Sammis et al. [1987], constrained
comminution model for the self similar development of
fault gouge.
2. Simulation Approach
[5] Particle based simulation methods like the discrete
element method (DEM) [Cundall and Strack, 1979], and the
lattice solid model (LSM) [Mora and Place, 1994; Place
and Mora, 1999] have been used extensively to investigate
the dynamics of fault gouge. In this paper, an extended
version of the 3D LSM has been used that includes
rotational degrees of freedom for the particles, and shear
and rotational breaking mechanisms for bonds between
particles (Y. Wang et al., Implementation of particle-scale
rotation in the 3-D lattice solid model, submitted to Pure
and Applied Geophysics, 2004, hereinafter referred to as
Wang et al., submitted manuscript, 2004).
[6] The model consists of spherical particles characterized
by their radius, mass, position and linear and angular
velocity. They interact with nearest neighbors by elastic
and frictional forces. The particles can be linked together
by breakable elastic bonds. At contacts between unbonded
particles, the normal forces are determined by a linear elastic
contact law and the tangential forces are calculated by a
Coulomb friction law. The forces and moments between
bonded particles are calculated according to the normal,
shear, bending and torsional stiffness of the bonds as
described by Wang et al. (submitted manuscript, 2004). If
the forces acting on a bond exceed the bond strength, it is
broken and the particles interact by elastic and frictional
forces afterward. The stiffness parameters and the fracture
criterium for the bonds are chosen so the macroscopic elastic
and fracturing behavior of a block of material (formed by
particles bonded together) closely resembles that of a linear
elastic solid (Wang et al., submitted manuscript, 2004).
3. Model Setup
[7] Our model setup consists initially of a number of
large, roughly spherical aggregate grains between two
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ABE AND MAIR: 3-D SIMULATION OF GRAIN FRACTURE
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Figure 1. (a) Initial configuration of model B001a and
B001b. (b) Model B001b after compaction and 45% shear
strain.
blocks of solid material. The aggregate grains and solid
boundary blocks are formed by many bonded randomly
sized spherical particles arranged in dense packing
(Figure 1a). The random sphere packing is generated using
the algorithm by Place and Mora [2001]. The use of equal
sized particles to construct the aggregate grains would have
been computationally efficient, however, would have lead
to unrealistic fracturing, preferentially aligned with the
lattice direction of the resulting regular arrangement of
particles within the grains [see Donzé et al., 1993; Mora
and Place, 1993]. Particle sizes range from 0.2– 1.0 model
units and the aggregate grains themselves have a diameter
of 4 model units (B002) and 8 model units (B001a,b
and B003). The aggregate grains interact as unbonded
particles (see section 2). Model setup parameters are
presented in Table 1.
[8] A force is applied to the solid boundary blocks in
y-direction to maintain a constant normal stress. The outer
edges of both blocks are then driven at constant velocity in
the +x and x direction to generate shear in the model.
Circular boundary conditions are applied in the x-direction
i.e. particles leaving the model on one side return at the other
side. In the z-direction, fixed frictionless boundaries are
used. To understand the state of the gouge during the
simulation it is necessary to find out which groups of
particles are still connected by bonds and therefore form
an aggregate grain. We determine this by investigating
whether one particle can be reached from the other by
following a continuous chain of intact bonds. If they are
connected, the particles belong to the same grain. To acquire
this information, an undirected graph [see, e.g., West, 1996]
is constructed where particles are mapped to the nodes of the
graph and the bonds between the particles are mapped to the
edges. From this graph the connected components, which
Table 1. Parameters of Model Setups
Model
Nr.
Particles
Grains
Particles
Per
Grain
B001a
B001b
B002
B003
287538
287538
54792
296456
28
28
52
29
8000
8000
700
8000
Elastic
Modulus
Breaking
Strength
Normal
Stress
25
12
12
12
300
300
300
300
36
17
17
17
GPa
GPa
GPa
GPa
MPa
MPa
MPa
MPa
MPa
MPa
MPa
MPa
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Figure 2. Number of surviving large grains with increasing strain in two models with different normal stress.
represent the aggregate grains existing at that time, can be
extracted using a Depth First Search (DFS) [Sedgewick,
2002].
4. Results
[9] A typical 3-D aggregate grain simulation is shown
in Figure 1. The initial configuration, prior to loading
(Figure 1a) highlights the uniform size distribution of
aggregate grains, each comprised of many bonded spherical
particles. After normal compaction and shear strain of 45%
(Figure 1b), significant fracture of aggregate grains has
occurred, reducing the overall grain size, and decreasing
the porosity of the layer. Despite the pulverization of some
aggregate grains (Figure 1b, I), other grains survive intact or
as large aggregate fragments (Figure 1b, II).
[10] In Figure 2 we plot the number of surviving large
aggregate grains (diameter > 6 model units) as a function of
shear strain for two simulations. The number of ‘‘survivor
grains’’ decreases with accumulated strain, the rate of
attrition being sensitive to normal stress. Importantly, several large aggregate grains survive to relatively large shear
strains. For example, in the higher normal stress simulation,
the rate of removal of large grains is significantly reduced
after 25% strain.
[11] We characterize the comminution rate as the rate of
particle bond breakage during shear. In simulations, the
comminution rate in general decays with increasing strain
(Figure 3). At higher normal stress, comminution is associated with initial compression, prior to shear. In contrast, at
lower normal stress, the comminution rate increases only
after a finite shear has occurred. Occasional spikes in
the curves indicate short lived enhanced bond breakage
that may represent aggregate grains breaking apart by an
avalanche or zipper type mechanism.
[12] Cumulative particle density (by number) versus
equivalent grain diameter is summarized in Figure 4. The
aggregate grains (i.e. composed of two or more particles)
and single particles are distinguished. The particle size
distribution is well described by 2 straight sections in a
log-log plot. One section represents the distribution of the
small, mainly single particle, grains and the other represents
the distribution of the larger, aggregate grains. To compare
our results with previous published work [e.g., Sammis et
al., 1987; Marone and Scholz, 1989] we calculate fractal
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ABE AND MAIR: 3-D SIMULATION OF GRAIN FRACTURE
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Simulation B001b shows a fractal dimension DB001b =
2.3 ± 0.3 which is still rising after 45% strain.
5. Discussion
Figure 3. Rate of bonds breaking during shear. The rate is
calculated as the difference between the number of bonds
remaining intact between consecutive snapshots of the
simulation. The snapshots are taken every 0.00125% strain.
dimension as a function of increasing strain (Figure 5). We
obtain the fractal dimension from the grain size distribution
using a logarithmic binning procedure [Corral, 2004]. The
fractal dimension is then calculated from a least squares fit of
a power law to the data. The size distribution of small grains
(diameter <1.0 model units) is dominated by single particle
grains and is therefore an artifact of the initial particle size
distribution and not grain fracture. Thus we use only the
distribution of large grains (diameter >1.0 model units) for
the determination of the fractal dimension.
[13 ] Data from all simulations are reasonably well
described by a fractal distribution. Although we calculate
fractal dimensions for a relatively limited range of grain
sizes (1.0 – 8.0 model units for B001a and B001b, 1.0–
4.0 model units for B002) this is comparable to previous
work [e.g., Biegel et al., 1989]. Fractal dimension D
increases systematically with strain. Simulations B001a
and B002 reach apparently stationary distributions with
fractal dimensions DB001a = 2.6 ± 0.3 after 35% strain and
DB002 = 2.9 ± 0.5 after 45% strain respectively. The
difference between D obtained from the two simulations
is within the error of each model and therefore doesn’t
represent a significant difference between the models.
Figure 4. Grain size distribution for different strains in
model B001b (see Table 1).
[14] Our 3D numerical simulations of grain fracture
during shear produce a material having a comminution
texture that is qualitatively comparable to natural and
laboratory produced fault gouges [Sammis et al., 1987;
Marone and Scholz, 1989]. The evolution of grain size
distribution in simulations is a direct function of accumulated strain and applied normal stress, consistent with
previous laboratory results [Biegel et al., 1989; Morrow
and Byerlee, 1989; Marone and Scholz, 1989].
[15] From an initial uniform size distribution of spherical
aggregate grains, we produce a wide grain size distribution
that is described by a fractal distribution. This agrees well
with field observations that show mature fault gouges tend
to be described by fractal distributions of dimension D 2.6
[e.g., Blenkinsop, 1991; An and Sammis, 1994]. This
observation is also consistent with Sammis et al. [1987]
constrained comminution model where fracture probability
of grains is dominated by the relative size of their nearest
neighbors rather than intrinsic grain strength. This consistency is reassuring, given the match between our simulation
boundary conditions and those required by the Sammis et al.
[1987] model i.e. a uniform initial grain size and grain
size independent strength. Although additional grain size
reduction mechanisms may operate in nature, e.g. at high
strains, high confining pressure or for very small grains, the
constrained comminution model describes well the fragmentation processes in our simulations.
[16] The evolution of fractal size distribution with shear
produced in our simulations is very similar to that produced
in laboratory studies [Marone and Scholz, 1989] suggesting
similar grain fracture processes are operating and offering a
first order validation of the numerical fracture technique.
Changes in grain shape are not quantified however we note
a qualitative increase in the abundance of angular fragments
with shear as well as survival of some spherical (or near
spherical) original grains. Laboratory studies of initially
spherical grains sheared under grain fracture conditions
[Mair et al., 2002] produce a similar mix of angular and
spherical fragments.
Figure 5. Evolution of the estimated fractal dimension of
the grain size distribution with increasing strain. The error
bars are representative for each model.
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ABE AND MAIR: 3-D SIMULATION OF GRAIN FRACTURE
[17] The results of our simulations are encouraging,
one limitation, however, is that the initial configuration
contains relatively few aggregate grains. Clearly, it would
be preferable to simulate a larger number of aggregate
grains, however, there is a limit to the number of spherical
particles that can be tracked to large strains in a reasonable
simulation time. We could alternatively simulate a larger
number of small aggregate grains (provided they were
composed of fewer particles) however, we believe the
current model set-up (with aggregate grains composed of
many sub-particles) provides the best approximation to real
grain fracture. Another potential artifact of the current
model is the fact that the smallest possible fragments
produced by grain fracture are spherical particles of diameter 0.2– 1 model units. Whilst ideas of a ‘‘grinding limit’’
(where particle size reduction stops) have been invoked to
explain natural comminution textures [An and Sammis,
1994], the spherical shape of the smallest grains in our
simulations may influence the mechanical properties of the
gouge. We could address this issue by restricting the
breakdown of small aggregate grains, however, this would
artificially bias the grain sizes produced by fracturing unless
significantly larger models were used. Our future numerical
work will focus on the mechanical and structural behavior
of comminuted gouge for a range of initial configurations
and loading conditions.
6. Conclusions
[18] We present a new method to implement realistic
grain fracture in 3D numerical simulations of granular
shear. Our simulations succeed in producing a granular
material that is texturally and quantitatively similar to
mature natural fault gouge and gouge produced in the
laboratory. Results are consistent with Sammis et al.
[1987] constrained comminution model of fault gouge
development. Our method offers the potential to investigate
the particle dynamics and mechanical behavior of evolving
gouge during slip on simulated 3D faults.
[19] Acknowledgments. Support is gratefully acknowledged by the
Australian Computational Earth Systems Simulator Major National
Research Facility, the Queensland State government, The University of
Queensland, and SGI. The ACcESS MNRF is funded by the Australian
Commonwealth Government and participating institutions (Univ. of
Queensland, Monash Univ., Melbourne Univ., VPAC, RMIT, ANU) and
the Victorian State Government. Computations were made using the
ACcESS MNRF supercomputer, a 208 processor 1.1 TFlops SGI Altix
3700 which was funded by the Queensland State Government Smart State
Research Facility Fund and SGI. We thank Dion Weatherley for the use of
his software to analyse particle size distributions.
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S. Abe, ACcESS, University of Queensland, Brisbane, Qld 4067,
Australia. (steffen@esscc.uq.edu.au)
K. Mair, Physics of Geological Processes, University of Oslo, P.O. 1048
Blindern, N-0316 Oslo, Norway. (karen.mair@fys.uio.no)
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