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 L05305 1 of 4 ABE AND MAIR: 3-D SIMULATION OF GRAIN FRACTURE L05305 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 L05305 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 2 of 4 L05305 ABE AND MAIR: 3-D SIMULATION OF GRAIN FRACTURE L05305 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. 3 of 4 L05305 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. References Abe, S., J. Dieterich, P. Mora, and D. 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