A Granular material is an assembly of many discrete solid particles interacting with each other due to dissipative collisions, dispersed in a vacuum or an interstitial fluid. These particles are considered as the 4th state of matter other than solids, liquids and gas.
Force interaction between the various particles defines the mechanics of the granular flows. Different forms of granular flows are present in nature like avalanches and in industrial processes like powder mixing. They cover a large area of research with intersection from different scientific fields like soft matter physics, soil mechanics, powder technology and geological processes. Due the wide variety of properties, the discrete granular structure of these materials leads to a complex mechanical phenomena, resulting in motivated research for its fundamental understanding.
In past, experimental studies have been done to study behaviour of the granular particles. Recently, with advancement in computer processing speed, numerical simulation of such granular flow has become an alternative yet effective tool to study and understand the behaviour of granular flows. Granular system consists of individual particles and each particle moves independently. Thus it is difficult to predict the behaviour of granular system using continuous models. This discrete approach is called as Discrete Element Method (DEM) and has become a powerful and reliable tool. The philosophy is to model the system at microscopic level and study their behaviour including the detection and collision between particles and their environment. DEM can effectively and efficiently model the dynamics of an assembly of particles. The discrete approach requires a discrete time equations of motion governing particle displacements and rotations. DEM is very useful for modelling materials which undergo discontinuous deformations due the contact with other particles, breakage of contact bonds and broken fragments compacting together.
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Various models that have been used in the past to study the different factors affecting granular friction are mostly based on the DEM model. Most of these studies assume that the granular particle shape is circular or spherical which restricts them to reliably predict what is observable in real granular materials. Direct numerical simulations show that non-sphericity of the particles hinders their flow and considerably increases stress. Thus it safe to assume that non-spherical nature of these particles may also impact upon friction behaviour. There are very few studies which investigate the effect of particle shape on granular friction. In one such study is conducted by Clearly and Sawley (2002), in which investigation of the effect of granular particle shape on granular flows has been done.
The study here aims at investigation of granular friction in the case of non-circular particles. The study has been conducted on a single layer of particles to provide baseline friction results. The already existing DEM models for granular materials have high computational needs like requirement of even supercomputers. Thus it is not practical to envisage such models for this study. Rather a simple model is proposed in this project which 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
Various active faults demonstrate a wide variety of behaviours such as starting from stable creep to the stick-slip motion etc. As described by rate and state-dependent friction laws, the frictional strength determines the shear along a fault. Though the physics or the mechanisms of a frictional behaviour is not very well known the empirical rate & state-dependent laws are successfully applied to many types of laboratory data and field observations in the past (Marone 1998 & Scholz 1998).
Static friction is the ratio of shear to normal stress. Static friction is required for two surfaces to begin sliding. It increases linearly with time when surfaces are in stationary contact with one another. In case of bare surfaces, roughness of the "asperities" (Rabinowicz, 1951) counter-act the normal and shear loads. Thus the real contact area between two surfaces is a small fraction of the apparent area of contact. Frictional interfaces grow strong with time due to either increase in area of contact junctions through asperity deformation (Dieterich & Kilgore, 1994; 1996) or increase in bond strength between the surfaces (Rice, 1976; Hirth and Rice, 1980; Michalske & Fueller, 1985).
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Non-cohesive wear detritus with angular forms called fault gouge often get produced, accumulate and evolve during frictional sliding of brittle shear zones at shallow depth. Fault gouges have remarkable impacts on the mechanical behaviour of fault zones (Byerlee, 1967; Byerlee and Summers, 1976; Scholz et al., 1972) and stability of natural faults (Scholz et al., 1969; Marone and Scholz, 1988). Thus knowledge of the frictional properties of fault gouge under a wide range of conditions is very much required for a full understanding of fault strength, dynamic behaviour, and earthquake mechanisms.
Fairly large numbers of experiments have been carried out under various experimental conditions to identify the effects of a variety of gouge materials on frictional behaviour of fault zones. Earlier focus was on the effects of gouge on base level rock friction, as the production of gouge during slip of initially bare rock surfaces resulted decrease in frictional resistance (e.g., Byerlee, 1967; Scholz et al., 1972). Other studies on gouge rock friction indicated that the accumulation of gouge and development of shear localization features within gouge may influence fault constitutive behaviour and stability of sliding (Engelder et al., 1975; Byerlee et al., 1978; Logan et al., 1979; Moore et al., 1988; Tullis et al., 1989). Further studies 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 (Dieterich, 1979; Ruina, 1983; Rice and Ruina, 1983; Chester and Higgs, 1992; Reinen et al., 1994; Perrin et al., 1995). The Dieterich-Ruina rate/state constitutive law which is 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).
Laboratory experiments investigating the effects of variations in extrinsic factors like normal load (Linker and Dieterich, 1992; Richardson and Marone, 1999), loading velocity (Mair and Marone, 1999), shear load (Nakatani and Mochizuki, 1996; Nakatani, 1998; Karner and Marone, 1998; Olsen et al., 1998), normal stress (Richardson and Marone, 1999), shear displacement (Beeler et al., 1996), shear load changes (Karner and Marone, 2001) and hydro-thermal conditions (Karner et al., 1997) on the frictional strength of simulated fault gouge have been carried out to validate the constitutive laws for frictional evolution under a wide range of conditions which has led to empirical descriptions of friction in terms of the various extrinsic factors. But none of existing friction laws can completely describe the observed frictional behaviour of fault gouge basically due 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, focus has been shifted on the influences of various intrinsic factors like 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. An example of these influences is best demonstrated by Mair et al., 2002, where it has been observed that the coefficient of sliding friction for gouge made of spherical glass beads is markedly lower than for angular quartz sand.
Faults are mostly thought of as two bare surfaces but as indicated by (Sammis et al., 1986; Wong et al., 1992) natural and laboratory faults develop a zone of granular material/fault gouge, with displacement. Development of shear fabric and the localization of strain rate onto the shear planes influence the frictional behaviour of granular shear zones (Logan et al., 1979; Yund et al., 1990; Mair & Marone, 1999). Moreover with changes in loading patterns, the granular layers compact and dilate much more than bare surfaces and the resulting 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).
Significant advances in understanding the effects of gouge on rock friction has been possible through various laboratory experiments, but as indicated above, the micro-mechanical processes responsible for these effects are still unknown. Very less knowledge of micro-mechanics of gouge deformation has hindered the development of the next generation of constitutive laws based on the micro-mechanical framework to quantitatively interpret and analyse the friction data. 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 and questions related to micromechanics of gouge like "how are deformations partitioned among mechanisms such as grain rolling, sliding & fracturing" and "how various mechanisms of deformation affect both base level and second-order frictional strength of fault gouge that evolve with grain size, shape, configuration, distribution, structure etc. cannot be fully addressed by current laboratory experiments.
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Ideally, the fundamental, crystal lattice-scale and meso scale mechanisms of friction under the range of geological conditions relevant to the seismic cycle shall be identifiable by laboratory experiments. However, with current level of knowledge and technical limitations, it is not feasible. Thus, numerical modelling is the alternative tool in scaling laboratory friction behaviour to natural fault zones which offers approach for exploring micro-mechanisms of shear zone deformation (Mora and Place, 1998, 1999; Morgan and Boettcher, 1999; Morgan, 1999; Place and Mora, 2000). Unlike many numerical models based on the macroscopic and continuous media where rheology is assumed in advance (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 behaviour of discontinuous granular materials and therefore fault gouge, as a function of intrinsic variables and contact physics. This technique has been successfully envisaged to reproduce characteristic shear fracture arrays commonly observed in naturally and experimentally deformed gouges (Morgan and Boettcher, 1999).
Some models use the rate and state-dependent friction and are very effective at reproducing the behaviour of earthquake faults (Rice et al., 2001; Marone 1998). However as indicated by (Mora & Place, 1998; 1999; Aharonov & Sparks, 1999; Morgan and Boettcher, 1999), discrete element and finite element models of granular friction behaviour produce data quite different from laboratory observations of friction. As an example, laboratory measurements of sliding friction for sand is 0.6 while the same is as low as 0.3 with numerical models. The low values from numerical models are sometimes used to take care of the inferred weakness of some crustal faults to the presence of fault gouge (Mora and Place, 1998; 1999). 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 which causes the discrepancy. 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.
A simplified numerical method to study the micro-structure and crack-growth in geo-materials was used by Bazant (31). In this method frictional interaction is replaced by a force-displacement relation with a tensile strength limit which relates it to the discrete element method. But incomplete treatment of the block motion and the on-off nature of contacts are the drawbacks.
A numerical model for assemblies of discs and spheres has been developed by Serrano and Rodriquez-Ortiz (28). At equilibrium conditions contact forces and displacements are calculated assuming that increments of contact forces are determined by incremental displacements of the centres of the particles. The method used to solve the equations can only process a relatively small number of particles which is a major draw-back.
In many of the recent DEM simulations, one of the limitations have been the use of circular gouge grains. Base friction values from 2-D DEM simulations are about 0.3 (e.g., Morgan, 1999), much lower than the base level friction predicted by Byerlee's law, indicating nonphysical results from simulations. However, laboratory experiments carried out on real materials with equally simple 2-D geometries yields remarkably similar friction data to numerical simulations of 2-D circular particles (Frye and Marone, 2002). Further laboratory studies under idealized conditions of 3-D spherical particles have demonstrated slightly higher friction values, approaching 0.45, but also indicate an increase in friction with particle angularity (Mair et al., 2002). The laboratory results above validate the DEM simulations, but also indicate that the circular particles are too simple to represent real fault gouge. Natural fault gouge consist of angular-shaped grains that are conceived to show 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 Ïƒn carries out an vital role in developing the 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 Ïƒn which may also lead to additional variations of friction. In fact, according to (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), second-order effects on friction associated with variations in Ïƒn have been shown to produce a decreasing trend in friction observed in many laboratory experiments, but the dependency of friction on Ïƒn 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, numerical experiments can provide a better understanding of effect of Ïƒn on active deformation mechanisms and friction.
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, nn order to study the variation of frictional strength and dynamic behaviour of fault zones as a result of changing grain shape and Ïƒn,. Randomly shaped grains were created to produce real like fault gouge. It was possible to quantify the effects of gouge grain shape and Ïƒn on the friction of simulated granular assembly. 2D simulations were done to compare with previous modelling studies. Results revealed the angular grain assembly to be much stronger than rounded grain assembly as observed in laboratory experiments of 3D materials (Mair et al., 2002). Frictional strength of simulated granular assembly increases nonlinearly with decreasing Ïƒn, 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 trend of decreasing sliding friction with increasing Ïƒn is comparable to recent laboratory observations (Saffer et al., 2001; Saffer and Marone, 2003). Effect of grain shape irregularity on the strength of granular assembly is also observed. It determines the rate of change in friction by increasing both the coefficient and exponent of the friction law. Results indicate that DEM simulations can appropriately represent the characteristic mechanical behaviour of irregular granular gouge thus resulting in a better understanding of micromechanics of fault slip and friction.
Although the study on the effects of non-spherical particles on simulations of friction in granular materials has been very less, one of such studies is what done by Cleary & Sawley (2002) where the effect of particle shape on granular flows, focusing on geometrically-complex industry applications have been investigated. Observations on both hopper flow rate and flow patterns at various levels of blockiness and at different aspect ratios were done. Results revealed that the blockiness of the material made little difference to the actual patterns of the particle flow but the particle shape was found to have significant effect on the same 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 found that the particles flow rate decreased and with substantial changes in the particle flow structure.
With the same model, study on the effect of particle shape on shear flows was done by Cleary (2008). The impact of particle shape on continuum flow properties was explored along with study on the effect of particle shape on the distribution of velocity, volume fraction, stress tensor and granular temperature distributions within the shear layer. The effects of shape variation were extremely strong and of very high importance for all the research to follow.
The significance of the effects of particle shape on granular flow was clearly recognized by Cleary and Sawley (2002) and Cleary (2008). Thus, it is safe 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.