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. REFERENCES Abe, S., Place, D., & Mora, P. 2004. A parallel implementation of the lattice solid model for the simulation of rock mechanics and earthquake dynamics. Pure Appl. Geophys. 161: 2265–2277. Cundall, P. & Strack, O. 1979. A discrete numerical model for granular assemblies. Geótechnique 29: 47–65. Frye, K. M. & Marone, C. 2002. The effect of particle dimensionality on granular friction in laboratory shear zones. Geophys. Res. Lett. 29(19). doi:10.1029/2002GL015709. Hazzard, J. F. & Mair, K. 2003. The importance of the third dimension in granular shear. Geophys. Res. Lett. 30(13). doi:10.1029/2003GL017534. Latham, S., Abe, S., & Davies, M. 2004. Scaling evaluation of the lattice solid model on the SGI Altix 3700. In HPCAsia2004, Proceedings of the 7th International Conference on High Performance Computing and Grid in the Asia Pacific Region: 226–233. Latham, S., Abe, S., & Mora, P. in prep. Parallel 3d simulation of a fault gouge using the lattice solid model. Pure Appl. Geophys. Mair, K., Frye, K. M., & Marone, C. 2002. Influence of grain characteristics on the friction of granular shear zones. J. Geophys. Res. 107(B10). doi:10.1029/2001JB000516. Mora, P. & Place, D. 1994. Simulation of the stick-slip instability. Pure Appl. Geophys. 143: 61–87. Mora, P., Place, D., Abe, S., & Jaumé, S. 2000. Lattice solid simulation of the physics of earthquakes: the model, results and directions. In J. Rundle, D. Turcotte, & W. Klein (eds), GeoComplexity and the Physics of Earthquakes: Geophysical Monograph series, no. 120: 105–125. Washington, DC: American Geophys. Union. Morgan, J. K. 2004. Particle dynamics simulations of rate- and state-dependent frictional sliding of granular fault gouge. Pure Appl. Geophys. 161: 1877–1891. Place, D. & Mora, P. 1999. The lattice solid model to simulate the physics of rocks and earthquakes: Incorporation of friction. J. Comp. Physics 150: 332–372.