3.1. Discrete Element Method
 We use a two-dimensional discrete element method [cf. Cundall and Strack, 1979] to simulate the behavior of a simplified layer of grains. This method treats individual grains as inelastic disks which undergo linear and rotational accelerations due to grain interactions and external forces. The mathematical model and numerical techniques used are very similar to those described in our two previous papers [Aharonov and Sparks, 1999, 2002]. The separation of the centers of two disks, i and j, is described by the vector rij. When the distance between the centers is less than the sum of the radii, Ri + Rj (i.e., the grains overlap), an interaction force is exerted on each grain at the point of contact. The grains are noncohesive, so this force vanishes when the overlap reaches zero. The contact force has shear, Fs, and normal, Fn, components. The normal component consists of a linear elastic repulsive force and a damping force dependent on the relative grain velocities, ij:
Here = (rij · , rij · )/rij is the unit vector parallel to the contact; kn is the normal spring stiffness, mij is the harmonic mean of the two grain masses, and γ is a damping coefficient. The spring stiffness is taken to vary inversely with grain size,
where is the average grain diameter in the system. This variable stiffness is introduced so that grains of all sizes obey a stress-strain relationship with uniform stiffness.
 Shear forces on contacts are determined using an elastic/friction law [Cundall and Strack, 1979]
where = (rij · , −rij · )/rij is the unit vector tangent to the contact, μ is the surface friction coefficient, ks is the shear stiffness, and Δs is the shear displacement since the formation of the contact. Equation (12) implies that relative motion of the grains in the tangential direction at the contact point is initially opposed by an linear elastic force, which increases with sliding until it reaches a maximum value of μFijn. The two spring constants, kn and ks, are chosen so that the overlap of the grains and the elastic sliding distance will be small compared to the grain size.
 Energy is dissipated through two mechanisms. The last term in equation (10) introduces a viscous damping that ensures that collisions between grains are inelastic. In the simulations presented here, most of the dissipated energy is lost during frictional sliding between grains. The coefficient of surface friction used in this study was held fixed at μ = 0.5.
 We nondimensionalize this system to facilitate computation and to the highlight the important interplay of physical processes. Distance is scaled by the average grain diameter, , stress by Young's modulus, E, and mass by the mass of a grain with the average diameter, . We also introduce characteristic scales for force, F0, spring stiffness, k0, and time, t0:
The choice for the timescale was guided by the vibrational period of a purely elastic interaction between two characteristic grains. Velocities are scaled by /t0, approximately equivalent to the acoustic wave speed in a grain. The resulting force equations, with dimensionless primed variables, are
The dimensionless damping parameter is Γ = γ/E2. The simulations presented here are heavily damped, with Γ = 1; this corresponds to a restitution coefficient of about 0.3 for a collision of two grains of diameter (see Aharonov and Sparks  for details); ks′ is the ratio ks/kn. The value of this stiffness ratio is taken to be 0.5, within the range of values that give reasonable agreement with particle collision experiments [Schafer et al., 1996].
 We limit the applied nondimensional confining stresses to less than 0.01, so that the most highly stressed contacts will have overlaps of only 1–2% of a grain diameter. Assuming quartz grains, with a Young's modulus of 60 GPa, our simulations were conducted at confining stresses between 18 and 1800 MPa.
 The contact forces on each grain are summed to give the total linear and rotational accelerations of the grains. There is no gravity applied to this system, so there are no body forces. The accelerations are integrated through time to calculate grain position, and linear and rotational velocity, using a “leapfrog”-like Verlet algorithm [Allan and Tildesley, 1987]. Stability and accuracy requires a time step small enough to resolve the evolution of forces during a collision of grains. In the simulations we present, the grain velocities are always a small fraction (<10−2) of the acoustic wave speed, so that motion of a grain during a time step is very small. The exact choice of time step does not affect the outcome of the runs, but many time steps (typically 105–107) are required to do significant rearrangement of the grains. As a result, dimensionless loading velocities were kept above 10−4, and the total shear strain achieved in the slowest cases was only about 2.
3.2. Model of a Granular Layer
 Simulations were performed in roughly square systems with about 550 (24 × 24) grains. For simplicity, the grains were all of a similar size. Since a uniform grain size will introduce ordering effects, the grain diameters were drawn from a Gaussian distribution with a standard deviation of 0.5 , clipped at plus/minus one standard deviation. For convenience we define “horizontal” and “vertical” directions, although there is no gravity vector to orient the system. The top and bottom boundaries were composed of grains of different sizes glued together along their centerlines to form rigid rough walls. The system is periodic in the horizontal direction, analogous to a laboratory rotary shear apparatus [Mandl et al., 1977; Savage and Sayed, 1984; Hanes and Inman, 1985; Beeler et al., 1996]. The systems were initiated as tall loosely packed boxes, which were compacted vertically by gradually applying normal stress to the top walls. A typical compressed system of grains is shown in Figure 1. Because the system is periodic in the horizontal direction, vertical stresses applied to the top and bottom walls create a confining pressure, N, that is isotropic if averaged over the whole system. The bottom wall was held fixed, while a horizontal force, Q(t), was applied to the top wall, introducing right lateral shear stress. This force mimics that of a compliant spring (Figure 1), increasing as one end of the spring is pulled, and decreasing as the wall moves:
Vsp is the constant loading velocity on the spring and Vwall is the horizontal velocity of the grains of the top wall. The first term on the right-hand side emulates the buildup of tectonic shear stresses on a natural fault or the applied shear stresses in a laboratory friction experiment. In the models presented here, ksp was taken to be 3 × 10−5k0. As in the sliding block model (equation (9)), the transition between stable and unstable sliding will be controlled by a parameter that combines ksp, Vsp and N. Because there are limits to the computationally accessible ranges of driving velocities and confining stresses, ksp was chosen so that the transition to unstable sliding could be investigated efficiently.
Figure 1. Configuration of packed circular grains in a 24 × 24 horizontally periodic granular layer. Shaded circles are grains, and black lines show the orientations of contact normal forces, with the thickness of the lines proportional to the total force on the contact. The top and bottom walls are composed of semicircular grains of different radii, glued together along a line. The top wall is connected to a spring with stiffness ksp, which is pulled at its other end at a constant velocity Vsp. A constant normal stress N is applied to the horizontal boundaries.
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