A Granular material is an assembly of many discrete solid particles interacting

2. Introduction

A
Granular material is an assembly of many discrete solid particles interacting
with each other due to dissipative collisions and is dispersed in a vacuum or
an interstitial fluid. Granular particles can be considered as the fourth state
of matter very different from solids, liquids or gas.
The
force interactions between the particles play a key role in defining the
mechanics of granular flows. Several forms of granular flows exist in nature
and in industrial processes ranging from avalanches in the form of landslides
to powder mixing in chemical industry. Granular materials cover a broad area of
research at the intersection of different scientific fields including soft
matter physics, soil mechanics, powder technology and geological processes.
Despite the wide variety of properties, the discrete granular structure of
these materials leads to a rich generic phenomena, which has motivated research
for its fundamental understanding.
In
the past, experimental studies have been carried out to study the behaviour of
granular particles. In recent years, due to the advancement in computer
processing speed numerical simulation of such granular flow is seen as an
effective alternative tool to study and understand the behaviour of granular
flows. Since a granular system is
composed of individual particles and each particle moves independently of each
other, it is difficult to predict the behaviour of granular system using
continuous models. The discrete approach developed for particle scale numerical
modelling of granular materials has become a powerful and reliable tool. This discrete approach is called as Discrete
Element Method (DEM). The philosophy behind the DEM simulation of granular
flows is to model the system at microscopic level or particle level and study
their behaviour including the detection and collision between particles and
their environment. DEM can efficiently and effectively model the dynamics of
assemblages of particles. Technically, the discrete approach requires a
time-discretised form of equations of motion governing particle displacements
and rotations. DEM is particularly useful in modelling materials that undergo
discontinuous deformations because of the contact with other particles in the
system, breakage of contact bonds and compaction of broken fragments.
Numerous models that have been created in the
past to study the different factors affecting granular friction are mostly
based on the DEM model. In most of the studies that were conducted to study
granular friction, it was assumed that the granular particle shape is circular
or spherical. These models have been unable to reliably predict what is
observable in real granular materials. Direct numerical simulations show that
non-sphericity of the particles hampers their flow and considerably increases
stresses. It is therefore conceivable that non-sphericity may impact upon
friction behaviour as well. Studies
which investigate the effect of particle shape on granular friction have been
noticeably lacking. In a study conducted by Clearly and Sawley (2002), they
investigated the effect of granular particle shape and concluded that it had
significant impact on granular flows.

This study aims at investigating granular
friction in the case of non-circular particles. The study was conducted on a
single layer of particles in order to provide baseline friction results.
Existing DEM models of granular materials have very high computational
requirements. A number of the studies that were reviewed required
supercomputers to complete the simulations. Clearly with such high
computational requirements, we could not use DEM for the purpose of this study.
The simple model proposed in this project involves Monte-Carlo style computer
modelling using MATLAB. We anticipate that the model proposed in this thesis
can be used for complex DEM modelling in the future that uses similar
particles.
3. Literature Review

A
wide variety of behaviours are demonstrated by active faults that range from
stable creep to stick-slip motion. The frictional strength (described by rate and
state-dependent friction laws) determines the shear along a fault. The physical
mechanisms responsible for the friction behaviour are not very well understood
even though the empirical rate and state-dependent laws have been successfully
applied to many types of laboratory data and field observations in the past
(Marone 1998 &Scholz 1998).
Static
friction is defined as the ratio of shear stress to normal stress. For two
surfaces to begin sliding, static friction is necessary. It generally increases
linearly with respect to the time that the surfaces are in stationary contact
with one another. On bare surfaces, normal and shear loads are supported by the
roughness of the “asperities” (Rabinowicz, 1951), such that the real area of
contact between two surfaces is only a small fraction of the apparent area of
contact. Frictional interfaces strengthen with time, as contact junctions
either increase in area through asperity deformation (Dieterich& Kilgore,
1994; 1996) or undergo an increase in bond strength between the surfaces (Rice,
1976; Hirth and Rice, 1980; Michalske&Fueller, 1985).
Frictional sliding of brittle shear zones at
shallow depth is commonly accompanied by production, accumulation, and
evolution of non-cohesive wear detritus with angular forms called fault gouge.
It has long been recognized that the presence of fault gouge has significant
effects on the mechanical behaviour of fault zones (Byerlee, 1967; Byerlee and
Summers, 1976; Scholz et al., 1972), and the stability of natural faults
(Scholz et al., 1969; Marone and Scholz, 1988). Therefore a full understanding
of fault strength, dynamic behaviour, and earthquake mechanisms requires a
complete knowledge of the frictional properties of fault gouge under a wide
range of conditions.

A considerable body of experimental work has
been carried out to identify the effects of a variety of gouge materials on
frictional behaviour of fault zones under various experimental conditions.
Early work was focused on the effects of gouge on base level rock friction, as
the production of gouge during slip of initially bare rock surfaces was found
to result in a decrease in frictional resistance (e.g., Byerlee, 1967; Scholz
et al., 1972). Studies on gouge rock friction (Engelder et al., 1975; Byerlee
et al., 1978; Logan et al., 1979; Moore et al., 1988; Tullis et al., 1989)
indicated that the accumulation of gouge, and development of shear localization
features within gouge, may influence fault constitutive behaviour and stability
of sliding. Further studies (Dieterich, 1979; Ruina, 1983; Rice and Ruina,
1983; Chester and Higgs, 1992; Reinen et al., 1994; Perrin et al., 1995)
demonstrated that second-order variations in friction of simulated fault zones
can be described by several constitutive laws in terms of slip rate and state
of the frictional shear zone. The Dieterich-Ruina rate/state constitutive law,
currently in best agreement with experimental results (Beeler et al., 1994;
Nakatani, 2001), provides the basis for predicting the frictional response to a
change in slip rate (i.e., velocity strengthening and velocity weakening).

To validate and constrain the constitutive
laws for frictional evolution under a wide range of conditions, laboratory
experiments investigating the effects of variations in extrinsic factors on the
frictional strength of simulated fault gouge have been carried out. These
factors include normal load (Linker and Dieterich, 1992; Richardson and Marone,
1999), shear load (Nakatani and Mochizuki, 1996; Nakatani, 1998; Karner and
Marone, 1998; Olsen et al., 1998), loading velocity (Mair and Marone, 1999),
normal stress (sn) vibrations (Richardson and Marone, 1999), shear load
perturbations (Karner and Marone, 2001), shear displacement (Beeler et al.,
1996), and hydrothermal conditions (Karner et al., 1997). These studies have
led to empirical descriptions of friction in terms of the various extrinsic
factors. Nonetheless, none of existing friction laws can completely describe
the observed frictional behaviour of fault gouge, due in part to the complexity
and evolution of the topography of the contacting surfaces (e.g., Karner and
Marone, 1998; Richardson and Marone, 1999, Karner and Marone, 2001). Recently,
attention has been focused on the influences of various intrinsic factors, such
as mineralogy (Olsen et al., 1998; Saffer et al., 2001)), grain shape, size,
size distribution, and surface roughness (Mair and Marone, 2000; Frye and
Marone, 2002; Mair et al., 2002), on shear zone strength and sliding behaviour.
As an example of these influences, recent investigations have demonstrated that
the coefficient of sliding friction for gouge made of spherical glass beads is
markedly lower than for angular quartz sand (Mair et al., 2002).

Although
faults are often thought of as two bare surfaces, natural and laboratory faults
develop a zone of granular material, or fault gouge, with displacement (Sammis
et al., 1986; Wong et al., 1992). Frictional behavior of granular shear zones
is influenced by the development of shear fabric and the localization of strain
rate onto these shear planes (Logan et al., 1979; Yund et al., 1990;
Mair&Marone, 1999). Further, in response to loading perturbations, granular
layers compact and dilate much more than bare surfaces, and the density changes
within the layer may affect the frictional strength of the material (Mead,
1925; Marone& Kilgore, 1993; Segall& Rice, 1995; Mair&Marone, 1999;
Losert et al., 2000).
Laboratory experiments have made significant
advances in understanding the effects of gouge on rock friction, but as shown
above, the underlying micromechanical processes of these effects remain
elusive. Poor understanding of the micromechanics of gouge deformation has
obviously impeded the development of the next generation of constitutive laws
built upon a micromechanical framework to quantitatively interpret friction
data. With laboratory experiments, it is difficult to set up experiments under
identical boundary conditions in order to directly correlate variations in
gouge friction to variations of extrinsic factors, intrinsic factors, and
corresponding deformation mechanisms during simulated gouge deformation.
Therefore many important problems related to micromechanics of gouge cannot be
fully addressed by current laboratory experiments. For example, outstanding
questions include how deformation is partitioned among mechanisms such as grain
rolling, sliding, and fracturing, and how deformation mechanisms affect both
base level and second-order frictional strength of fault gouge, and evolve with
grain size, shape, size distribution, configuration, and microstructure.

Ideally,
laboratory experiments would identify the fundamental, crystal lattice-scale
and meso-scale mechanisms of friction under the range of geological conditions
relevant to the seismic cycle. However, our current level of understanding of
fault material properties together with technical limitations in the laboratory
does not allow that level of generality. Thus, an important tool in scaling
laboratory friction behavior to natural fault zones is the numerical model.
Numerical simulations of the frictional behavior of fault zones represent an
alternative approach for exploring micromechanisms of shear zone deformation,
and offer an outstanding opportunity to extend our knowledge of fault mechanics
beyond the domain accessible to laboratory observations (Mora and Place, 1998,
1999; Morgan and Boettcher, 1999; Morgan, 1999; Place and Mora, 2000). Unlike
many continuum numerical models based on the macroscopic and continuous media
in which rheology is assumed in advance (i.e., intrinsic properties are
averaged) (Day, 1982; Fukuyama and Madariaga, 1995; Madariaga et al., 1997;
Tang, 1997), the distinct element method (DEM) (Cundall and Strack, 1979)
provides a way to study the dynamic behavior of discontinuous granular
materials, and therefore fault gouge, as a function of intrinsic variables and
contact physics. This technique has been successfully employed to reproduce
characteristic shear fracture arrays commonly observed in naturally and
experimentally deformed gouges (Morgan and Boettcher, 1999).
Models
that use rate and state-dependent friction are effective at reproducing the
behavior of earthquake faults (Rice et al., 2001; Marone 1998). However,
discrete element and finite element models of granular friction behavior
produce data quite different from laboratory observations of friction (Mora
& Place, 1998; 1999; Aharonov& Sparks, 1999; Morgan and Boettcher,
1999). While laboratory measurements of sliding friction for sand are 0.6,
sliding friction is as low as 0.3 in numerical models. The low friction values
from numerical models are sometimes used to attribute the inferred weakness of
some crustal faults to the presence of fault gouge (Mora and Place, 1998;
1999). The discrepancy between numerical model and laboratory observations may
be attributable to the different initial and boundary conditions. Because of
computational limitations, numerical models of granular friction often rely on
several simplifying boundary conditions, such as smooth, round, 2-D, non-breaking
particles. To investigate the effect of these conditions on friction, Frye
&Marone (2002) studied the friction of smooth, round, non-breaking spheres
and also investigated the effect of particle dimensionality by simulating 2D
conditions with glass and pasta rods.
Serrano and Rodriquez-Ortiz (28) developed a numerical model for
assemblies of discs and spheres. Contact forces and displacements are
calculated for equilibrium conditions assuming that increments of contact
forces are determined by incremental displacements of the centers of the
particles. A major draw-back of the method used to solve the equations is that
only a relatively small number of particles can be processed.

Bazant (31) recently used a simplified
numerical method to study the microstructureand crack-growth in geomaterials.
This method is somewhat related to the discrete elementmethod as the frictional
interaction is replaced by a force-displacement relation with a tensile
strength limit. However block motion and the on-off nature of contacts were
incompletely treated.

One limitation in many of the recent DEM
simulations, however, has been the use of circular gouge grains. Base friction
values obtained from two-dimensional (2-D) DEM simulations are about 0.3 (e.g.,
Morgan, 1999), significantly lower than the base level friction predicted by
Byerlee’s law, raising concerns that the simulations represent nonphysical
results. Laboratory experiments carried out on real materials with equally
simple 2-D geometries, e.g., quartz rods and even pasta, however, have yielded
remarkably similar friction data to numerical simulations of 2-D circular
particles, thereby confirming the first-order DEM friction values (Frye and
Marone, 2002). Further idealized laboratory studies of 3-D spherical particles
have demonstrated slightly higher friction values, approaching 0.45, but also
show that friction increases with particle angularity (Mair et al., 2002).
These laboratory results serve to validate the DEM simulations, but show that
circular particles are too simple to represent real fault gouge. Natural fault
gouge is usually composed of angular-shaped grains that are thought to exhibit
significantly less grain rolling and more grain interlocking than particle
dynamics simulations (Mora and Place, 1998, 1999; Morgan and Boettcher, 1999;
Mair et al., 2002).

Laboratory experiments on angular sand also
show that σn
plays an important role in the development of microstructures in fault gouge by
controlling the active deformation mechanisms, i.e., rolling and sliding
dominant at non-fracture regime, and fracture and grain size reduction more
active at higher normal stresses (Mair et al., 2002). The results suggest that
active deformation mechanisms are not only dependent on grain shape but also on
σn.
Therefore σn
may also lead to additional variations of friction. In fact, second-order
effects on friction associated with variations in σn have been shown to produce a
decreasing trend in friction observed in many laboratory experiments (e.g.,
Maurer, 1965; Murrell, 1965; Byerlee, 1967, 1968; Handin, 1969; Jaeger, 1970;
Edmond and Murrell, 1971; Saffer et al., 2001; Saffer and Marone, 2003), but
the dependency of friction on σn
has not been well studied in previous numerical simulations on rock friction.
As it is often difficult to conduct laboratory experiments under the exact same
conditions to examine the effect of a single variable on the variation of rock
friction, numerical experiments can provide a better understanding of effect of
sn on active deformation mechanisms and friction.

In order to study the variation of frictional
strength and dynamic behavior of fault zones as a result of changing grain
shape and σn,
Guo& Morgan (2004) carried out DEM simulations using grains constituted of
bonded circular particles under a range of normal stresses from 5 to 100 MPa.
In this way, arbitrarily shaped grains were generated to reproduce more
realistic fault gouge morphology, and they could quantify the effects of gouge
grain shape and σn
on the friction of simulated granular assemblages. Simulations were carried out
in 2-D for comparison with previous modeling studies. The results showed that
angular grain assemblages are stronger than rounded grain assemblages, as
observed in laboratory experiments of 3-D materials (Mair et al., 2002).
Frictional strength of simulated granular assemblages increases nonlinearly
with decreasing σn,
and follows an inverse power law that is identical in form to a theoretical
friction law based on Hertzian contact model (Bowden and Tabor, 1964; Jaeger
and Cook, 1976; Villaggio, 1979). The decreasing trend of sliding friction with
increasing σn is comparable to recent laboratory observations (Saffer et
al., 2001; Saffer and Marone, 2003). Grain shape irregularity is also observed
to affect the strength of granular assemblages. It determines the rate of
change in friction by increasing both the coefficient and exponent of the
friction law. Their results demonstrated that DEM simulations can appropriately
represent the characteristic mechanical behavior of irregular granular gouge,
bringing us one step closer to understanding the micromechanics of fault slip
and friction.

Investigation into the effect of non-spherical
particles on simulations of friction of granular materials has been limited.
Cleary &Sawley (2002) studied the effect of particle shape on granular
flows, focusing on geometrically-complex industrial applications, in particular
the hopper discharge. They observed both hopper flow rate and flow patterns at
various levels of blockiness and at different aspect ratios. They found out
that the blockiness of the material made little difference to the actual
patterns of the particle flow. The effect of particle shape was found to be
significant with flow rates reducing by around 28% as the particles shape
changed from circular to square and increasing by a similar amount as the
aspect ratio was increased from 0.2 to 1. For example, when the aspect ratio
was reduced to 0.2 by changing from circular to elliptical particles, it was
observed that the particles flow rate decreased and substantial changes in the
structure of the particle flow were also observed.

Cleary (2008) went on to use the same model to
study the effect of particle shape on shear flows. The impact of particle shape
on continuum flow properties was explored. He investigated the effect of
particle shape on the distribution of velocity, volume fraction, stress tensor
and granular temperature distributions within the shear layer. He established
that the effects of shape variation were extremely strong and of very high
importance for all the research to follow.

Cleary and Sawley (2002) and Cleary (2008)
recognized how significant the effects of particle shape on granular flow are.
Therefore, it’s rational to assume that different particle shapes will also
have a significant effect on frictional sliding in granular materials. The
present research into friction of granular materials has not yet resulted into
a complete model due to the lack of research on this topic in the past. The
effect of particle shape on friction in granular materials is largely absent in
the literature, although the few studies which have investigated into it have
established it to have a significant effect. The aim of this study is to
improve the knowledge of this shape-effect by creating a simple model for
granular materials using non-spherical particles.

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