A Granular Physics-Based View of Fault Friction Experiments

Abstract

Rate- and state-dependent friction (RSF) equations are commonly used to describe the time-dependent frictional response of fault gouge to perturbations in sliding velocity. Among the better-known versions are the Aging and Slip laws for the evolution of state. Although the Slip law is more successful, neither can predict all the robust features of lab data. RSF laws are also empirical, and their micromechanical origin is a matter of much debate. Here we use a granular physics-based model to explore the extent to which RSF behavior, as observed in rock and gouge friction experiments, can be explained by the response of a granular gouge layer with time-independent properties at the contact scale. We examine slip histories for which abundant lab data are available and find that the granular model (1) mimics the Slip law for those loading protocols where the Slip law accurately models laboratory data (velocity-step and slide-hold tests) and (2) deviates from the Slip law under conditions where the Slip law fails to match laboratory data (the reslide portions of slide-hold-slide tests), in the proper sense to better match those data. The simulations also indicate that state is sometimes decoupled from porosity in a way that is inconsistent with traditional interpretations of “state” in RSF. Finally, if the “granular temperature” of the gouge is suitably normalized by the confining pressure, it produces an estimate of the direct velocity effect (the RSF parameter a) that is consistent with our simulations and in the ballpark of lab  data.

Key Points

• We examined the behavior of a sheared granular layer with time-independent contact-scale properties at and away from steady state
• Like gouge samples in the lab, the layer mimics the rate-state friction Slip law in velocity-step and slide-hold (but not reslide) tests
• A normalized granular temperature can be used to estimate the amplitude of the direct velocity dependence of friction in the gouge layer

1 Introduction

Models for estimating the length and time scales of earthquake nucleation rely on a mathematical description of the evolution of local fault friction with time (Dieterich, 1992; Dieterich & Kilgore, 1996). The commonly accepted framework for modeling this behavior, at least at sliding speeds too small for thermal effects to become important, is “rate- and state-dependent friction,” or RSF (Dieterich, 1978, 1979, 1981, 1994; Marone, 1998b; Ruina, 1983). The RSF framework embodies the notion that frictional strength depends upon a nebulous property termed “state,” a function of recent slip history, and the current slip rate. Several versions of RSF laws exist, but the two most popular ones are the slip-dependent “Slip law,” which does a better job matching lab data, and the time-dependent “Aging law,” which matches less data (Bhattacharya et al., 2015, 2017) but which has more published theoretical justifications (e.g., Baumberger & Caroli, 2006). However, none of the existing RSF laws reproduce all of the robust features of available laboratory data (Bhattacharya et al., 2017; Kato & Tullis, 2001). This shortcoming, coupled with the largely empirical nature of RSF, severely limits our ability to apply laboratory-derived friction laws to fault slip in the Earth.

In this paper, we adopt the working hypothesis that rock friction is governed by the behavior of a granular gouge with constant Coulomb friction at grain-grain contacts. Note that by not considering time-dependent plasticity or chemical reactions at the contact scale, we are throwing out what is traditionally thought to be the source of the rate and state dependence of friction (e.g., Baumberger & Caroli, 2006; Dieterich & Kilgore, 1994); all the relevant time dependence in our simulations arises from momentum transfer between the gouge particles, even at very low slip speeds. We use the discrete element method (DEM) to investigate the behavior of a 3-D granular layer sheared at constant normal stress between two rigid and parallel blocks. The model geometry and loading conditions are designed to mimic laboratory rock and gouge friction experiments (we note that laboratory experiments on even initially bare rock surfaces develop, through mechanical wear, either a granular powder or a granular gouge layer, depending upon the total slip distance and that the phenomenology of RSF is common to both those experiments that start with bare rock and those where gouge is used as the starting material (Marone, 1998b)). In this paper we emphasize velocity-step tests, employing a range of shearing velocities (10⁻⁵ to 2 m/s) and confining pressures (1–25 MPa) to model steps of ±1–3 orders of magnitude. These velocity steps are supplemented by a small number of slide-hold and slide-hold-slide tests designed to allow additional comparisons to laboratory experiments and provide further insight into the gouge behavior.

Consistent with RSF and several earlier numerical studies of sheared granular layers, we find that in response to imposed velocity steps there is an immediate “direct velocity effect” (e.g., an increase in friction in response to a step velocity increase), followed by a more gradual “state evolution effect” where the sign of the friction change is reversed (Abe et al., 2002; Hatano, 2009; Makse et al., 2004; Morgan, 2004). Furthermore, the magnitudes of these direct and evolution effects are proportional to the logarithm of the velocity jump, with implied values of the relevant RSF parameters (a and b) that are not far from lab values.

Perhaps our most significant finding is that the granular flow model mimics the Slip state evolution law for those sliding protocols where the Slip law does a good job matching laboratory experiments, and deviates from the Slip law, in the proper sense to better match lab data, for those sliding protocols where the Slip law does a poor job. The former category includes both velocity-step tests (Bhattacharya et al., 2015; Blanpied et al., 1998; Ruina, 1980; Marone, 1998a; Rathbun & Marone, 2013; Ruina, 1983; Tullis & Weeks, 1986) and slide-hold tests (Bhattacharya et al., 2017). Consistent with both lab experiments and the Slip law, and unlike the Aging law, following a simulated velocity-step friction approaches its future steady-state value over slip distances that are independent of both the magnitude and sign of the step (a few grain diameters, in our simulations, or strains of ∼15 %). And consistent with lab experiments, during the hold portion of simulated slide-hold tests stress decays in a manner consistent with the Slip law using RSF parameters not far from those derived from the velocity-step tests, whereas the Aging law, with its time-dependent healing, underestimates the stress decay. Moreover, during the simulated hold, the gouge layer compacts roughly as the logarithm of hold time, similar to lab experiments. This is despite the fact that the stress decay, being well modeled by the Sip law, implies a lack of state evolution. Because state evolution in RSF is traditionally thought to involve the “mushrooming” of contacting asperities and porosity reduction, this indicates that in both the granular simulations and the lab, state is decoupled from gouge thickness (porosity) in a way that is inconsistent with most current interpretations of  RSF.

The granular flow model differs from the Slip law prediction during the reslides following holds, in that the Slip law parameters that fit the hold well underestimate the peak stress upon the reslide. Qualitatively, this is the same way in which the Slip law fails to match laboratory data (Bhattacharya et al., 2017). Collectively, our results hint that the physics-based granular flow model may do a better job of matching the transient response of laboratory rock and gouge friction experiments than any existing empirical RSF constitutive law. This is despite having apparently fewer tunable parameters. Although the model contains a large number of dimensionless parameters, most of these are fixed by the boundary conditions and the elastic moduli of the gouge particles, and the remainder seems to exert very little influence on the frictional behavior of the system. An exception is the grain size distribution; we find that a quasi-normal distribution gives rise to steady-state velocity-strengthening behavior, whereas a quasi-exponential distribution is close to velocity neutral, perhaps transitioning from velocity-weakening to velocity-strengthening behavior with increasing slip speed. Grain shape may also play a significant role, but only spherical grains are employed  here.

The granular model is also well suited to allow us to explore the microphysical origins of its RSF-like behavior. In section 5.4 we begin to address this question, by measuring the kinetic energy of the gouge layer for a range of shear velocities, confining pressures and system sizes. By assuming that this kinetic energy plays the role of temperature in the classical understanding of the rate dependence of friction as a thermally activated Arrhenius processes (Chester, 1994; Lapusta et al., 2000; Nakatani, 2001; Rice et al., 2001), we obtain an estimate of the magnitude of the direct velocity effect (the RSF parameter a) that is close to that determined by fitting the simulated velocity steps.

In exploring the granular model, our intent is not to imply that time-dependent contact-scale processes do not contribute to laboratory friction. Clear evidence of time-dependent contact plasticity comes from the see-through experiments of Dieterich and Kilgore (1994), and evidence of the importance of chemistry and time-dependent interfacial chemical bond formation comes from, among many other studies, the humidity-controlled gouge experiments of Frye and Marone (2002) and the atomic force single-asperity slide-hold-slide experiments of Li et al. (2011). It is not yet clear, however, under what conditions such effects dominate the transient frictional strength of interfaces. Nearly all papers that justify a state evolution law on physical grounds do so for the Aging law (e.g., time-dependent plasticity increasing contact area as log time; Baumberger & Caroli, 2006; Berthoud et al., 1999), even though this law reproduces relatively little laboratory friction data. An exception is Sleep (2006), who proposed that the Slip law arises from the highly nonlinear stress-strain relation at contacting asperities. Here we explore a physics-based model that may do a better job of matching (room temperature and humidity) laboratory rock and gouge friction data than any constitutive law currently in use, and that simultaneously allows one to investigate the attributes of the model that give rise to this behavior.

2 RSF Background

RSF laws treat friction as a function of the sliding rate, V, and the “state variable”, θ. θ has traditionally been thought of as a proxy for true contact area on the sliding interface (Nakatani, 2001), but it has recently been shown that under some circumstances time-dependent contact quality can be the dominant contributor to the evolution of state (Li et al., 2011). In its simplest form, RSF is described by two coupled, first-order, ordinary differential equations. The first describes the relation between friction μ, defined as the ratio of shear stress to normal stress, and the RSF variables:

(1)

where μ_∗ is the nominal steady-state coefficient of friction at the reference velocity V_∗ and state θ_∗. The coefficients a and b control the magnitude of velocity and state dependence of the frictional strength, respectively. The second equation describes the evolution of the state variable θ, the two most widely used forms being

(2)

(3)

with D_c being some characteristic slip distance (Dieterich, 1979; Ruina, 1983). Equation (2) is often referred to as the Aging law since state can evolve with time in the absence of slip; equation (3) is referred to as the Slip law since state evolves only with slip ( when V=0).

It is well established that neither the Aging law nor the Slip law adequately describes the full range of laboratory friction experiments (Beeler et al., 1994; Kato & Tullis, 2001). Laboratory experiments show that in a sufficiently stiff system, for both initially bare rock samples and gouge, following a step change in load point velocity friction approaches its new steady-state value quasi-exponentially over a characteristic slip distance that is independent of both the magnitude and the sign of the velocity step (Bhattacharya et al., 2015; Blanpied et al., 1998; Marone, 1998b; Ruina, 1983). This is precisely the Slip law prediction of state evolution (Nakatani, 2001). The Aging law, on the other hand, predicts a slip-weakening distance that increases as the logarithm of the velocity jump for step velocity increases, and, owing to the approximately linear increase of state with time, exceedingly small slip distances for frictional strength recovery following large step velocity decreases. Both behaviors are completely inconsistent with laboratory data (Nakatani, 2001).

In contrast, conventional wisdom holds that slide-hold-slide experiments are better explained by the Aging law. In part, this stems from the work of Beeler et al. (1994), who ran experiments on initially bare granite surfaces at two different machine stiffnesses and hence two different amounts of slip during the load point holds. They found that the rate of healing, as inferred from the peak stress upon the reslide, was independent of stiffness and hence independent of the small amount of interfacial slip during the load point holds, seemingly consistent with the Aging law and inconsistent with the Slip law. However, Bhattacharya et al. (2017) showed that the Beeler et al. peak stress data could be fit about as well by the Slip law as by the Aging law, and moreover that the stiffness-dependent stress decay during the load point holds could be well modeled by the Slip law, although with a slightly different value of (a−b) than was determined from contemporaneous velocity steps, and was completely inconsistent with the Aging law. The property of the Aging law that prevents it from matching the stress decay during the holds is precisely its time-dependent nature: The gouge strengthens too much to allow any more slip. The rock friction community is thus left in the awkward position that while most theoretical justifications for state evolution are designed to explain the time-dependent healing of the Aging law (e.g., Baumberger et al., 1999), this law seems to explain rather little laboratory rock friction  data.

3 The Computational Model

Our DEM simulations are performed using the granular module of lammps (Large-scale Atomic/Molecular Massively Parallel Simulator), a multiscale computational platform developed and maintained by Sandia National Laboratory (https://lammps.sandia.gov). What we will refer to as the “default” model consists of a packing of 4,815 grains: 4,527 in the gouge layer and 288 in the top and bottom rigid blocks. The grains in the gouge layer have a polydisperse normal-like size distribution (Figure 1b), with a diameter range d=[1:5] mm and average diameter D_mean=3 mm (Figure 1a). The granular gouge is confined between two parallel and rigid plates that are constructed from grains with diameter d=5mm. Grain density and Young's modulus are chosen equal to properties of glass beads (Table 1). The model domain is rectangular with periodic boundary conditions applied in the x and y directions. The size of the system in each direction is L_x=L_y=1.5L_z=20D_mean.

Figure 1

(a) A visualization of the “default” granular gouge simulation. A normal grain size distribution is used, with mean grain diameter D_mean=3  mm. Colors show the velocity of each grain in the x direction, averaged over an upper-plate sliding distance of D_mean during steady sliding at a driving velocity of V_lp=2×10⁻⁴ m/s. The actual velocity profile, averaged over 400 planes normal to z, is shown to the right (black dots). (b) The size distribution of grains in the gouge layer in the default model.

Table 1. DEM Simulation Parameters

Parameter	Value
Grain density, ρ	2,500 (kg/m^3)
Young's modulus, E	50 (GPa)
Poisson ratio, ν	0.3
Grain-grain friction coefficient, μg	0.5, 1.0, 5.0
Confining pressure, σn	1, 5, 25 (MPa)
Coefficient of restitution, ϵn	0.98, 0.82, 0.25, 0.01, 0.003
Time step, Δt	2×10^−8 (s)

• Note. The “default model” values, where multiple values are given, are in bold  font.

The system is initially prepared by randomly inserting (under gravity) grains in the simulation box with a desired initial packing fraction of ∼0.5. The system is then allowed to relax for about 10⁶ time steps, after which three initially identical and relaxed realizations are subjected to confining pressures σ_n=[1,5,25]  MPa. The confining pressure is applied for 1 min, by which time the fast phase of compaction is completed. These confined gouge samples are then subject to shearing at a desired driving velocity imposed by the top rigid plate, while the vertical position of the top wall is adjusted by a servo-control system to maintain the specified (constant) confining pressure. We find that the servo-control system keeps the normal stress constant to within about ±0.1% of the desired value at slip speeds of 0.1 m/s (see supporting information Figure S2 for an example of the servo control during and following a velocity step), and that the variation about the desired value is reduced by about a factor of 5 at slip speeds 5 times smaller. The nondefault systems are prepared using an identical protocol at a confining pressure of σ_n=5  MPa. The driving velocity is applied to the system via a linear spring with a default stiffness of 10¹⁴ N/m attached to the top plate; for practical purposes, this stiffness can be considered to be infinite, in that changes in load point velocity are transferred nearly instantaneously to the upper plate. The grains are modeled as compressible spheres of diameter d that interact when in contact via the Hertz-Mindlin model (Johnson, 1985; Landau & Lifshitz, 1959; Mindlin, 1949).

For two contacting particles {i,j}, at positions {r_i,r_j}, with diameters d_i and d_j, velocities {v_i,v_j} and angular velocities {ω_i,ω_j}, the force on particle i is computed as follows: The normal compression δ_ij, relative normal velocity , and relative tangential velocity are given by

(4)

(5)

(6)

where r_ij=r_i−r_j , n_ij=r_ij/r_ij, with r_ij=|r_ij|, and v_ij=v_i−v_j . The rate of change of the elastic tangential displacement , set to zero at the initiation of a contact, is given by

(7)

where the second term in equation (7) comes from the rigid body rotation around the contact point. Its implementation is there to insure that always locates in the local tangent plane of contact (Silbert et al., 2001). The normal and tangential forces acting on particle i are then given by

(8)

(9)

where k_n and k_t are the normal and tangential stiffness, given by k_n=(2/3)E/(1−ν²) and k_t=2E/(1+ν)(2−ν) (Mindlin, 1949), with E being Young's modulus and ν Poisson's ratio, and m_eff=m_im_j/(m_i+m_j) is the effective mass of spheres with masses m_i and m_j (we note that the most appropriate value of k_t seems to be a matter of some debate, with Shäfer et al. (1996) suggesting values roughly 1,000 times smaller). γ_n and γ_t are the normal and tangential damping (viscoelastic) constants, respectively; we maintain the default lammps option of γ_t=0.5γ_n. As indicated by equations (8) and (9), the model implements damping for both normal and tangential contacts as a spring and dashpot in parallel. Note that the Hertzian normal force given by (8) increases nonlinearly with grain compression δ_ij (equation (4)), as in the absence of damping, consistent with the elastic deformation of contacting spheres.

In a gravitational field g, the translational and rotational accelerations of particles are determined by Newton's second law, in terms of the total forces and torques on each particle, i:

(10)

(11)

The grain-grain coefficient of friction, μ_g, is the upper limit of the tangential force through the Coulomb criterion F_t ≤ μ_gF_n. The tangential force between two grains grows according to the nonlinear Hertz-Mindlin contact law until F_t/F_n=μ_g and is then held at F_t=μ_gF_n until either F_t ≤ μ_gF_n or the grains loose contact.

The amount of energy lost in collisions is characterized by the coefficient of restitution. The values of restitution coefficients, ϵ_n and ϵ_t for the normal and tangential directions respectively, are related to their respective damping coefficients γ_n,t and contact stiffness k_n,t. The restitution coefficient for the normal direction can be calculated by solving the following equation that describes the normal component of the relative motion of two spheres in contact:

(12)

with the initial conditions and δ(0)=0. In this equation, , and d_eff=d_id_j/(d_i+d_j) is the effective diameter for spheres of diameters d_i and d_j. The normal component of the coefficient of restitution can be obtained from the ratio of normal velocity of grains at the end of the collision, defined as , to their initial normal impact velocity: . The collision time t_col is determined by solving equation (12) for the adopted physical properties and initial velocities of two colliding grains. A similar procedure is performed for calculating the restitution coefficient in the tangential direction. We use a time step of Δt=t_col/100 throughout this study, with t_col evaluated assuming an impact velocity of 25 m/s (t_col in (12) depends very weakly upon , as roughly (Shäfer et al., 1996)). The restitution coefficient in the default model is chosen to be very high (ϵ_n=0.98), such that the system is damped minimally. Although in one sense damping introduces time dependence at the contact scale, we find by varying the restitution coefficients from nearly 0 (complete damping) to nearly 1 (no damping) that they exert no significant influence on the system behavior in the slow-sliding regime of interest. For this reason we refer to the model as having no time dependence at the contact scale. The full details of the granular module of lammps are described in the lammps manual and several references (Brilliantov et al., 1996; Silbert et al., 2001; Zhang & Makse, 2005).

In addition to the default model, we have run simulations with a domain size twice the size of the default model, simulations with a grain and domain size 2 orders of magnitude smaller, simulations with grain-grain friction coefficients of 1.0 and 5.0 (default = 0.5), simulations with restitution coefficients ϵ_n of 0.003 to 0.82 (default = 0.98), and simulations with a quasi-exponential grain size distribution. The influence of most of these changes on the model results is rather modest, and we relegate detailed figures to the supporting information of this manuscript. An exception is the models with a different grain size distribution; these are described in section 5.2.6. A full accounting of the dimensionless parameters governing the model is provided in Appendix A. In principle, we wanted to prepare models that could isolate the influence of each parameter that we tested. However, because of the way we used the lammps random particle generator, in some cases there are slight variations in the total number of particles, which are reflected in different values of L_z (hereafter referred to as the gouge thickness H). Compared to the default model, for the simulations with different grain-grain friction coefficients H is larger by 10%; for the simulations with a grain and domain size 2 orders of magnitude smaller the ratio H/D_mean is larger by 7%, and in the simulations where L_x and L_y are 2 times larger, H is only 1.8 times larger (we continue to refer to this as the “2 times larger” model).

The velocity V in the RSF equations (1)–(3) is interpreted in laboratory experiments as the inelastic component of the relative tangential displacement rate between two parallel planes. This displacement rate is typically treated conceptually as occurring across a plane of zero thickness, but in fact it occurs across a zone whose thickness is generally unknown. In lab experiments, the relative displacement is measured between two points outside the zone of inelastic deformation, and the inelastic component of that displacement δ is determined by subtracting the estimated elastic displacement δ_el from the measured (total) displacement, that is,

(13)

where δ_lp is the measured “load point” displacement (in our simulations the displacement of the end of the spring not attached to the upper plate), τ the spring force divided by the nominal sample surface area (6 cm ×6 cm in our default model), and k the elastic stiffness of the combined testing apparatus plus sample between the measurement points. In our numerical simulations this stiffness is given by the effective stiffness of two springs in series,

(14)

where k_sp and k_H are the spring and gouge stiffness, respectively. To measure k_H, we performed several slide-hold-reslide simulations with a range of hold durations (Figure B1). The shear modulus can be estimated from the initially linear (assumed to be elastic) portion of the reslide following the longest holds in such simulations (e.g., Bhattacharya et al., 2017). From these tests, the shear modulus of the gouge layer is estimated to be in the range of G_H≈270to310  MPa, at a confining pressure of 5 MPa. This estimate is about 30–50% lower than previous experimental measurements on granular layers made from packing glass beads (Domenico, 1977; Makse et al., 1999; Yin, 1993) and granular simulations with properties similar to our model. However, those previous experiments and simulations were performed under specially designed preparation protocols, to produce a maximal packing fraction under a given confinement. We expect our simulation samples (that are generated under conditions similar to synthetic gouge experiments) to have a lower packing fraction and to exhibit a lower shear modulus. Although the appropriate value of G_H may vary modestly with the sliding history and packing properties of the gouge, we neglect this possibility here. For G_H from 270 to 310 MPa, the stiffness k_H varies from G_H/H=6.75×10⁹to7.75×10⁹ Pa/m, where H=0.04  m is the gouge thickness. To determine k_sp in Pa/m from the stiffness input in lammps in units of N/m, we divide by the sample surface area. For the default spring stiffness of 10¹⁴ N/m, k_sp∼3×10¹⁶ Pa/m ≫k_H, so k_eff∼k_H. This value of k_eff is so large that even large errors in G_H play no role in the Slip law fits to our simulated velocity steps (k_eff is essentially infinite).

Using (13) and (14) ensures that our analysis is consistent with both the conventional interpretation of equations (1)–(3) and standard laboratory protocols. For example, with k_sp essentially infinite and V_lp set to zero (a “hold”), the upper plate remains stationary, but due to granular rearrangements within the gouge, the inelastic displacement δ increases and V>0 as the stress relaxes.

4 Previous Studies of Granular Rheology Related to Rock Friction

The granular model has many dimensionless parameters, but most turn out to be unimportant in the region of parameter space of interest (A1). Within the physics literature, the most important is understood to be the Inertial number, defined as

(15)

where is the local shear rate (approximated as the slip speed divided by the gouge thickness in the second expression), P is the confining pressure (or normal stress, for the geometry of our simulations), and ρ and D_mean are the density and mean diameter of grains, respectively. The inertial number measures the ratio of the inertial forces of grains to the confining forces acting on those grains, such that small values ( ) correspond to the quasi-static state. A continuum model that has proven moderately successful in modeling steady-state granular friction is known as μ(I_n) rheology (Forterre & Pouliquen, 2008), where the local coefficient of friction depends only upon the local inertial number. However, in some regions of parameter space, the dimensionless pressure, defined as for the Hertzian contact law that we use, and as for a linear (F_n∝δ_ij) Hookean contact law (appropriate for 2-D simulations, with k_grain being the adopted grain-grain spring stiffness), also plays a role. Both versions of are proportional to the nominal elastic strain of grains subjected the applied load, given the adopted contact law (DeGiuli & Wyart, 2017; Salerno et al., 2018), and we only distinguish between them when necessary. For granular gouge with a quartz-like modulus (E∼50to70  GPa) and normal stresses from 2 to 50 MPa, varies from ∼10⁻³to10⁻²; the “rigid grain” (undeforming) limit is thought to be reached in the limit (DeGiuli & Wyart, 2017; de Coulomb et al., 2017).

The steady-state behavior of sheared granular layers has been studied extensively in the past two decades, using both simulations and experiments. Most numerical studies have explored values of I_n from roughly 10⁻⁵ to 10⁰, crossing the quasi-static to inertial transition. These studies generally find steady-state friction to be well fit by a power law of the form , with μ₀, b, and α being fitting parameters. When plotted versus or , friction is strongly velocity(rate) strengthening within the inertial regime, transitioning to weakly velocity strengthening and ultimately asymptoting to velocity neutral with decreasing I_n within the quasi-static regime (da Cruz et al., 2005; de Coulomb et al., 2017; Hatano, 2007; Kamrin & Koval, 2014). In contrast, some laboratory studies of sheared granular flow find velocity-weakening behavior within the quasi-static regime (Dijksman et al., 2011; Kuwano et al., 2013; Wortel et al., 2014), but potentially this could be due to time-dependent contact-scale processes not accounted for in the numerical simulations. However, in a theoretical study, DeGiuli and Wyart (2017) concluded that a sheared 2-D granular layer with a Hookean contact law changes behavior from velocity strengthening for to slightly velocity weakening at lower I_n, asymptoting to velocity neutral as I_n decreases further, provided .

Studies of granular gouge layers away from steady state are much less common and are mostly restricted to the geological literature. Using a model of a sheared granular fault gouge, Morgan (2004) observed both the direct and state evolution effects in velocity-stepping tests and the logarithmic-with-time healing of friction upon resliding in slide-hold-slide tests. In those simulations Morgan introduced a time-dependent grain-grain contact law, with . Likewise, Abe et al. (2002) implemented the Slip law version of state evolution to describe the time dependence of the grain-grain friction coefficient in slide-hold-slide simulations and again observed logarithmic healing of friction with time upon resliding. Because both of these studies introduced time dependence at the contact scale, it is difficult to isolate the purely geometrical contribution of granular flow to the transient frictional behavior they observed. Furthermore, neither study compared their results to laboratory experiments at the level of detail required, for example, to distinguish between competing state evolution laws. Hatano (2009) simulated velocity-stepping experiments, in three dimensions but using a linear (Hookean) contact law for grain-grain interactions, for a range of inertial numbers , and dimensionless pressures . He observed a critical slip distance that scaled linearly with the size of the velocity steps, behavior that is not reproduced by our simulations, and that is also inconsistent with laboratory rock friction experiments.

In the RSF framework, a steady-state velocity-weakening system and a system stiffness below a critical value are necessary conditions for stick-slip motion. Using a very soft spring for applying the sliding velocity (k_spring∼3×10⁻⁵k_grain, where k_grain is grain stiffness), Aharonov and Sparks (2004) performed DEM simulations of a two-dimensional confined sheared granular layer for and . They showed that the frictional behavior changes from stick slip to oscillatory motion to steady sliding as I_n increases. Similar behavior was later reproduced by Ferdowsi et al. (2013). Neither Aharonov and Sparks (2004) nor Ferdowsi et al. (2013) directly measured the steady-state friction coefficient as a function of velocity, so it is not clear if their systems were in the rate-weakening regime when stick-slip behavior emerged or whether in granular systems stick slip may occur despite the system being rate strengthening. One could imagine, for example, that with a sufficiently soft spring and a system small enough for only a small number of force chains to develop, collapse of a force chain might lead to sudden accelerations. The existence, origins, and controls of a transition from rate-weakening to rate-strengthening behavior in sheared granular layers are still a matter of much debate (Perrin et al., 2019; van Hecke, 2015). Recent experimental and numerical studies show that the variation of friction coefficient with shear rate and inertial number depends on the grain shape, surface roughness, and size distribution (Mair et al., 2002; Murphy et al., 2019, 2019; Salerno et al., 2018). In our preliminary results examining the influence of grain size distribution, we find that the behavior changes from velocity strengthening to approximately velocity neutral when the grain size distribution is changed from quasi-normal to quasi-exponential.

A continuum model for the flow of amorphous materials, recently applied to granular gouge, is known as Shear Transformation Zone (STZ) theory (Lemaître, 2002; Manning et al., 2007). In response to imposed velocity steps, STZ models exhibit both a direct velocity effect and an opposing state evolution effect, consistent with lab experiments and RSF (Daub & Carlson, 2008; Lieou et al., 2017). However, STZ models have yet to be compared to lab data at the level of, for example, establishing the basic result that the slip distance for stress (or state) evolution following an imposed velocity step is independent of the magnitude and sign of that step (Bhattacharya et al., 2015). Such tests matter because, as stated previously, simply documenting that a model has a direct and an evolution effect is insufficient justification for applying it to processes such as earthquake nucleation (Ampuero & Rubin, 2008). Furthermore, in the most recent versions of STZ (Lieou et al., 2017; Ma & Elbanna, 2018), variations in the state variable (“compactivity”) are assumed to be proportional to the gouge volume (thickness) change. However, both our granular simulations and laboratory friction experiments (to be discussed in section 5.2), and recent granular physics studies (Bililign et al., 2019; Puckett & Daniels, 2013), indicate that gouge thickness change is an inadequate description of state. Continuum approaches such as STZ theory may benefit from detailed studies of the granular physics of RSF of the sort described in this manuscript.

5 Results

5.1 Steady-State Friction

The results of granular simulations run to quasi-steady state at different normal stresses and driving velocities are shown in Figure 2. Because individual runs tend to be somewhat noisy, presumably due to the relatively small system size, each data point is averaged over seven different realizations (initial packings) of the granular fault gouge, and each of these realizations is averaged over a sliding distance of 5 times the mean grain diameter D_mean. Friction in this and all figures in this paper is defined as the ratio of shear to normal stress τ/σ, with τ and σ defined as the shear and normal force per unit area exerted by the gouge particles on the upper (driving) plate. This definition ensures that we are measuring the frictional strength of the gouge at the boundary with the upper plate, should that differ from the applied spring force (any mismatch leading to acceleration of the upper plate). In the absence of significant accelerations that are coherent when averaged over x−y planes, from force balance the shear stress as we have defined it is uniform throughout the gouge.

Figure 2

(a) The variation of steady-state friction coefficient with driving velocity at three different normal stresses. (b) The same data plotted as a function of inertial number (I_n). (c) The variation of steady-state gouge thickness at different driving velocities as a function of I_n, for the same three normal stresses. Error bars indicate 1 standard deviation of all friction measurements over a sliding distance of 5D_mean for each of the seven different realizations (initial grain arrangements) at each normal stress and V_lp. Most error bars in (c) are smaller than the symbol size. The dashed teal and brown lines in (c) show the temporal evolution (upper horizontal axis) of gouge thickness in the hold experiments shown in panel (d). (d) The evolution of gouge thickness with time during slide-hold experiments at V_i=2×10⁻¹ and 2×10⁻² m/s. Zero time in these plots marks the start of the hold (the halting of the upper driving plate). The teal and brown dots and arrows show the starting point and temporal progression of the curves that we plot in panel (c) (time progresses to the left in c). The confining pressure is σ_n=5  MPa.

The nominal friction coefficient in Figure 2a, ∼0.33, is low by laboratory standards. This low value is likely due to the use of spherical grains, as laboratory studies also show mean nominal friction coefficients in the range 0.25–0.45 for glass beads and for synthetic gouge layers produced from spherical grains (Anthony & Marone, 2005). Mair et al. (2002) also found that by changing grain shapes from smooth spherical to angular, the mean steady-state friction increases from ∼0.45 to ∼0.6. A recent computational study by Salerno et al. (2018) further shows that using nonspherical grains shifts the dynamic friction versus inertial number curves upward uniformly, increasing mean friction values from 0.25–0.35 for spheres to the 0.5–0.6 range for rounded-edge cubic grains. Note that for comparison to RSF we are primarily concerned with the variations of friction with slip rate and slip history. In the absence of thermal weakening mechanisms, numerical simulations of fault slip in an elastic solid depend only upon the time variation of friction and not its absolute value.

For our default model we find steady-state friction to vary essentially linearly with the logarithm of slip speed over the full range of parameters we have explored. Such behavior has been previously observed in solid-on-solid friction in many different materials (Baumberger et al., 1999; Berthoud et al., 1999; Dieterich, 1979; Karner & Marone, 1998; Ruina, 1983), as well as in experiments with spherical and nonspherical granular particles at low inertial numbers (Behringer et al., 2008; Hartley & Behringer, 2003) (although it is arguable that in experiments, time-dependent contact-scale processes may contribute to the observed logarithmic rate dependence;  Heslot et al., 1994; Nakatani, 2001). We find velocity-strengthening behavior over the range of parameters explored thus far, consistent with many experiments on gouge, although many other gouge experiments show nearly velocity-neutral behavior (Marone, 1998b; Marone et al., 1990). The value of |a−b| from the slope of our data, ∼0.0055, is slightly high by lab standards, but Marone et al. (1990) found values as high as 0.005 for laboratory gouge, and we emphasize that unlike standard RSF and STZ theory, this value is an output of the model and not a tunable parameter.

Note that in Figure 2a the friction coefficient increases slightly with decreasing normal stress. This is not a feature of standard RSF, but it is consistent with some laboratory data (e.g., Dieterich, 1972). If the data are plotted against the inertial number I_n rather than velocity (Figure 2b), there is a near collapse of all observations onto a single curve, as expected from previous work. Relative to previous numerical studies, we explore a somewhat lower range of I_n (roughly 10⁻⁷–10⁻², compared to 10⁻⁵–10⁰). While those previous studies found steady-state friction to have a power law dependence upon I_n, they are nonetheless consistent with ours in that for the overlapping range of I_n (∼10⁻⁵−10⁻²), they can be fit quite well by a logarithmic dependence of friction upon I_n, with a slope not much different than ours (Hatano, 2007). It is within the inertial regime of flow, for , that the steady-state friction versus curves in previous studies become strongly concave up and require a power law fit. Our steady-state results differ from previous simulations mostly in extending the range of I_n lower by ∼2 orders of magnitude, the lowest we can achieve in a few weeks of computation time. We find the logarithmic dependence to continue to those lower values, while the power law fits adopted by previous studies continue to flatten with decreasing I_n (for further discussion see supporting information section S1 and supporting information Figure  S1).

To estimate how our range of I_n compares to that accessed by typical laboratory gouge friction experiments, we note with reference to equation (15) that such experiments typically do not vary very far from our value of (ρ/P)^1/2. This means that if our adopted value of D_mean/H∼1/13 is appropriate, our simulations will have basically the same I_n as a lab experiment with the same V. The synthetic gouge experiments of Mair and Marone (1999), for example, spanned slip speeds of 0.3–3,000 μm/s, compared to our lowest V of 200 μm/s. Thus, typical low-velocity lab friction experiments can be expected to overlap the lowest values of I_n we explore and to extend to values of I_n several orders of magnitude lower still. At slip speeds within the upper half of our range, say 0.1 m/s, thermal weakening mechanisms are expected to dominate over classical RSF in rock friction experiments (e.g., Rice, 2006). To estimate I_n for the Mair and Marone (1999) experiments more precisely, we can use their P=25  MPa and initial value of D_mean/H≈1/30 (initial grain size 50–150 μm; gouge thickness 3 mm), to obtain 10⁻¹⁰<I_n<10⁻⁶. For experiments accompanied by grain comminution and strain localization over a thickness H_eff, I_n will vary to the extent that D_mean/H_eff varies from ∼1/30 (although the behavior at a given I_n could change for nonspherical particles, and if the grain size distribution becomes very large, then the appropriate choice of D_mean in the definition of I_n might need to be reexamined). Based upon experimental studies summarized by Rice (2006), shear bands in granular sands with a relatively narrow size distribution often satisfy D_mean/H_eff∼1/10−1/20.

The steady-state gouge thickness H in our simulations decreases with increasing normal stress but increases quasi-linearly with at a rate that is only weakly dependent on normal stress (Figure 2c). The logarithmic rate dependence of gouge thickness, with the gouge thickness change ΔH being ∼0.1D_mean per order of magnitude increase in driving velocity, also seems roughly consistent with laboratory observations (Beeler & Tullis, 1997; Marone & Kilgore, 1993; Rathbun & Marone, 2013). (We show in the next section that in our simulations 0.1D_mean∼0.05D_c, which enables a comparison with lab experiments where D_c is estimated but not D_mean.)

The temporal evolution of the gouge layer thickness in two slide-hold simulations is shown in semilog scale in Figure 2d. Both the friction coefficient (shown later in Figure 12a) and the gouge thickness show a relaxation with the logarithm of time. We compare the compaction rate of the gouge during the holds to the dilation rate as a function of inertial number in Figure 2c. The similar slopes of the thickness data from the steady sliding experiments (dots) and the holds (teal and brown lines) show that the reduction in gouge thickness that results from a tenfold increase in hold duration is comparable to the reduction from a tenfold decrease in inertial number (approximately slip speed). This suggests that the origin of the velocity dependence of steady-state gouge thickness may lie in the same slow relaxation process that operates during holds. Dieterich (1978) proposed a somewhat analogous equivalency between increased hold duration and decreased slip speed in laboratory experiments: That contact strength increased logarithmically with age, whether that age was defined as the duration of a hold or as the typical contact lifetime (contact dimension divided by the steady sliding speed).

5.2 Velocity-Step Simulations

The results of several granular velocity-step simulations, with load point velocity increases of 1–4 orders of magnitude, are shown in Figure 3a. “Slip” on the horizontal axis in this and all subsequent figures is the inelastic displacement as defined by equations (13) and (14). The solid curves show the measured friction relative to the future steady-state value. Immediately following the velocity increase, there is a stress increase, roughly proportional to the logarithm of the velocity jump, representing a direct velocity effect, followed by a quasi-exponential decay to the new steady-state value, representing a state evolution effect (the system is stiff enough that V over the stress decay is essentially identical to the load point velocity, so from equation (1) there is a linear relation between the change in friction and the change in log state following the friction peak). This friction decay occurs over a sliding distance of a few mean grain diameters.

Figure 3

(a) Results from step velocity increases with initial load point velocity V_i=2×10⁻⁴ m/s. The friction coefficient, plotted relative to its future steady-state value to emphasize the state evolution, is shown as a function of shear slip distance normalized by D_mean. Slip in this and later figures is defined to be zero at the time of the step. The curve for the four-order increase to V_f=2  m/s jumps discontinuously backward to a small negative slip value because equation (13) does not account for elastodynamic effects (see Appendix B). The gray curves are the friction signals rescaled as (the −0.05 is used just to make all signals visible on the same axis). The dashed lines show the prediction of the Slip law with b=0.0178, a=0.0247, and D_c=1.78D_mean (see text). (b) The solid lines show the variation of friction with normalized slip from panel (a). The dashed lines show the difference between the steady-state gouge thickness H_ss and the current thickness H, normalized by the mean grain diameter D_mean (the gouge dilates with slip). The results are averaged over seven different realizations of the same imposed loading conditions, with σ_n fixed at 5 MPa.

Given the increase in steady-state gouge thickness with slip speed/inertial number (Figure 2c), it seems reasonable to suggest that the direct velocity effect comes from sliding at the new (higher) slip speed but with the old (compacted) gouge thickness, while the state evolution effect is associated with the gradual approach to the new steady-state gouge thickness. A direct correspondence between state and gouge porosity has also been proposed in the context of both RSF (Segall & Rice, 1995; Sleep, 2006) and STZ theory (Lieou et al., 2017). However, although this view has some intuitive appeal, we show below that it is too simplistic; there is not a one-to-one relation between “state” and gouge thickness. (We also note here, in anticipation of results to be presented in section 5.2.2, that in simulations that use the same particle size distribution but a gouge thickness H 1.8 times larger, the gouge evolves to steady state over a slip distance roughly 1.8 times larger; that is, state evolution seems to be governed by a critical strain rather than by a critical slip distance. For convenience, we speak here of a critical slip distance. This does not alter our previous estimate of ΔH/D_c for a given log velocity change, where ΔH is the change in gouge thickness, because in our simulations both ΔH and D_c are proportional to H.)

The gray curves in Figure 3a show these friction changes normalized by the logarithm of the velocity jump, and are flipped for ease of visualization. That the gray curves all nearly overlap, that is, have approximately the same scaled amplitude and approach the new steady state over the same sliding distance, is entirely consistent with the Slip law description of state evolution with quasi-constant values of a, b, and D_c (Bhattacharya et al., 2015). Using a simplex method, we find the single set of (Slip law) RSF parameters that best matches these velocity jumps to be a∼0.025, b∼0.018, and D_c∼1.8D_mean. These values of a and b are on the high side but are within a factor of 2 of those commonly cited for rock and gouge, and we again emphasize that they are an output of the model and not an input. The dashed curves in Figure 3a show the Slip law predictions for these velocity steps, using these parameter values. The Slip law predicts the behavior of the granular model quite well, excluding the initial rounding that occurs over a slip distance of up to ∼D_mean in the simulations. For the 4-order velocity jump to 2 m/s there is some contribution to the measured shear stress from bulk inertia of the gouge; however, this contribution is expected to be small for slip distances larger than a modest fraction of D_mean and should not influence the Slip law fit to the data (Appendix B).

Figure 3b shows the variation of gouge thickness with slip distance (dashed lines) in comparison to the variation of friction coefficient, for the same velocity steps in panel a. The simulations show that the gouge layer approaches its future steady-state thickness H_ss over a slip distance comparable to the slip distance for the evolution of friction (the gouge dilates with slip, but we plot H_ss−H for easier comparison to the friction data). The good correlation between gouge thickness and friction (and hence log[state]) and the accepted parallels between state and gouge thickness (i.e., that the mushrooming of asperities that increases contact area also brings the surfaces closer together  Sleep, 1997) make it natural to ask whether variations in gouge thickness are a useful proxy for variations in state.

Figure 4a shows results for similar simulations with an initial steady-state load point velocity of 10⁻² m/s and velocity steps of up to +2 and −3 orders of magnitude. These show that friction evolves to its new steady state over a slip distance that is independent of the sign as well as the magnitude of the velocity step, again precisely the Slip-law description of state evolution. The variation of gouge thickness during these velocity steps is shown in Figure 4b, which indicates that the gouge thickness for velocity step increases evolves to its new steady state over a slip distance comparable to that for the evolution of stress, as in Figure 3b. In contrast, the gouge thickness during velocity step decreases evolves to its new steady state over a slip distance shorter than that observed for the friction coefficient in the same experiments, especially for the 2 and 3 orders of magnitude step downs. This is emphasized by the gray curves in Figure 4b, which show the thickness evolution for the step velocity decreases, flipped and rescaled to cover the same range as the corresponding 1- and 2-order step increases (the total thickness change is larger for the step increases). This asymmetry of the transient response to changes in driving velocity, in conjunction with the symmetric response of the friction coefficient, indicates that gouge thickness is an incomplete description of state. Other aspects of the granular structure, such as force fabric and structural anisotropy, must contribute to the state of the system.

Figure 4

The variation of (a) friction coefficient and (b) gouge thickness, in simulations with velocity steps up to +2 and −3 orders of magnitude. The initial driving velocity in all tests is V_i=10⁻² m/s. The simulation with V_f=10⁻⁵ m/s has yet to run to completion (the future steady-state values are estimates only) but is sufficient to demonstrate that the thickness initially varies much more rapidly than stress. The gray curves in (a) are the step down simulations, flipped to emphasize the stress symmetry between the step increases and decreases. The results in both panels are averaged over seven different realizations, with normal stress fixed at 5 MPa. (c) The variation of porosity in gouge experiments in response to ±1 order of magnitude increases and decreases in velocity from and back to the initial velocity of V=1 μm/s. The experiments were performed (as reported by Segall & Rice, 1995; Marone et al., 1990) on water saturated but drained (approximately constant pore pressure) layers of Ottawa sand. The gray curves in panels (b) and (c) are step down simulations (b) and the lab experiment (c), flipped and scaled to the same initial value as the corresponding step up, to emphasize the much more rapid response (with respect to slip) of porosity (thickness) to the velocity step decreases.

The prediction that gouge thickness evolves much more rapidly with slip in response to step velocity decreases than increases appears to be borne out by laboratory experiments (Figure 4c; see also Rathbun & Marone, 2013, Figures 6 and 7, and Mair & Marone, 1999, Figure 10a), although a more systematic comparison to existing lab data is certainly warranted. In fact, the asymmetric response of the gouge thickness in the simulations is very reminiscent of the Aging law prediction for friction, especially the modified form of the Aging law that Li and Rubin (2017) argued was more faithful to the underlying concept of contact “age” (their Figure 5a). We will return to this point during the discussion of slide-hold simulations.

Figure 5

The variation of normalized friction coefficient, , for velocity step ups of (a) 1 order, (b) 2 orders, and (c) 3 orders of magnitude, in systems with confining pressure σ_n=1, 5, and 25 MPa. With this normalization, rough estimates of a (the jump across the velocity step) and b (the amplitude of the decay following the peak) can be read directly from the vertical scale (the signal-to-noise ratio increases with the size of the velocity step). The initial driving velocity is V_i=10⁻³ m/s in all tests. The results are averaged over seven different realizations of the same imposed loading conditions.

Figure 6

(a) The variation of normalized friction coefficient with normalized slip for velocity step ups of 1–2 orders of magnitude, in the default system and the system with twice the domain size. (b) The variation of the gouge thickness normalized by the initial gouge thickness for the same velocity steps in (a). In both panels, the slip distance (x axis) is scaled by the ratio of the gouge thickness to the default gouge thickness H_default. V_i=10⁻² m/s and σ_n=5  MPa. The results are averaged over seven different realizations of the same imposed loading conditions.

Figure 7

(a, c, e) The variation of frictional resistance and normalized gouge thickness with slip distance in velocity step tests. Slip here is normalized by the nominal gouge thickness H₀, so the horizontal axis is the nominal shear strain. (a) The default model with step increases of 1 and 2 orders of magnitude. (c) The default model with step decreases of 1 and 2 orders of magnitude. (e) Step increases of 1 and 2 orders of magnitude in the system with twice the dimensions of the default model. The thickness change in these panels is normalized by H₀, so the vertical axis is the nominal dilatational strain, but the scale factor relating the normalized thickness and friction axes in each panel is the same as in Figure 4. (d, b, f) The variation of friction and normal to shear deformation rate with respect to normalized slip distance for the same experiments shown in the panels immediately above. The ratio between the dδ_n/dδ_s and friction scales (1.44) is the same in all panels. The dδ_n/dδ_s minima in panel (d) are at −0.25 and −1.38 (the latter off scale) for the 1- and 2-order step decreases, respectively. All experiments are performed at σ_n=5  MPa and V_i=10⁻² m/s; H₀ is taken to be the value of H_ss under these conditions. The results are averaged over seven different realizations of the same imposed loading conditions.

The single set of (Slip law) RSF parameters that best matches the velocity steps with V_i=10⁻² m/s is determined from the simplex method to be a=0.024, b=0.018, and D_c=1.7D_mean, very similar to the values determined previously for the step increases from V_i=2×10⁻⁴ m/s. Laboratory investigations of the velocity dependence of the RSF parameters show somewhat mixed results. For order of magnitude velocity steps on initially bare granite samples, Kilgore et al. (1993) found variations of a and b of no more than a few tens of percent for initial velocities ranging over 4 orders of magnitude. In contrast, similar experimental protocols conducted by Mair and Marone (1999) on synthetic fault gouge indicate that D_c increases systematically by up to 2 orders of magnitude and that (for sample slip distances exceeding ∼15 mm) a decreases systematically by a factor of 2–3, as the initial velocity increases over a range of 3 orders of magnitude. However, using similar starting materials, Bhattacharya et al. (2015) found that velocity step increases of 1 and 2 orders of magnitude from a single starting velocity, and step decreases of 1 and 2 orders of magnitude back to that same velocity, were fit extremely well by the Slip law with constant RSF parameters.

5.2.1 The Influence of Confining Pressure

In addition to velocity steps at a normal stress of σ_n=5  MPa and initial velocities V_i of 10⁻² and 2×10⁻⁴ m/s, we also conducted 1 to 3 orders of magnitude velocity increases at σ_n= 1, 5, and 25 MPa at V_i=10⁻³ m/s. The results, shown in Figure 5, indicate that the magnitude of direct and evolution effects varies slightly but not systematically with σ_n. We again search for the single sets of (Slip law) parameters that best match all the velocity jumps at each confining pressure, using the simplex method (Table 2). Except for D_c being modestly larger at the largest σ_n, and a and b being larger at the smallest σ_n, the parameters seem to be largely independent of confining pressure.

Table 2. The RSF Parameters Obtained for Velocity Steps at σ_n=1, 5, and 25 MPa and V_i=10⁻³ m/s

Normal stress	RSF parameters
σn	a	b	a−b	Dc/Dmean
1 MPa	0.0290	0.0226	0.0064	1.83
5 MPa	0.0202	0.0135	0.0067	1.92
25 MPa	0.0232	0.0145	0.0087	3.23

5.2.2 Critical Slip Distance or Critical Strain?

The critical slip distance in our default system velocity-step experiments is roughly 1.7 times the mean particle diameter D_mean (see Figures 3 and 4). This seems reasonable, given that in laboratory fault friction experiments the critical slip distance D_c is often interpreted as being close to an asperity size (Marone, 1998b; Dieterich, 1981). However, laboratory data are somewhat ambiguous with regard to whether a critical strain or a critical slip distance controls the approach to a new frictional equilibrium. Dieterich (1981) reported that the critical slip distance is largely independent of gouge thickness, an observation he interpreted as indicative of slip localization within the gouge (i.e., a critical strain over a layer thickness that was insensitive to gouge thickness). Marone and Kilgore (1993) reported that some gouges had a critical slip distance that increased quasi-linearly with gouge thickness (i.e., a critical strain), while others had a much weaker dependence upon thickness, possibly reflecting variable degrees of localization.

We have run step velocity increase simulations from V_i=10⁻² m/s using the model that has twice the dimensions of the default model (although 1.8 times the thickness), with all other grain and system properties being identical to the default model. A comparison to the default model is shown in Figure 6. We find that the critical slip distance following velocity steps is 1.9 times as long in simulations with 1.8 times the model thickness (RSF parameters: a=0.028, b=0.019, and D_c/D_mean=3.3, compared to a=0.024, b=0.018, and D_c/D_mean=1.7 for the default model at V_i=10⁻² m/s), suggesting that indeed it is a critical strain that governs the approach to the new steady state. As a result, rescaling the slip distance (x axis) by the ratio of the model dimensions shows that the frictional behavior for both systems almost collapses (with some noise) to a single curve. The critical strain, using γ_xz=∂u_x/∂z+∂u_z/∂x=∂u_x/∂z, is . In contrast, the gouge thickness curves, when normalized by their (future) steady-state values, do not completely collapse when plotted as a function of rescaled slip distance (Figure 6b). We obtained similar results (not shown here) for V_i=10⁻¹ and 10⁻³ m/s.

5.2.3 The Gouge Dilation Angle

Several authors have commented on the potentially important contribution of fault gouge dilatancy or compaction to the measured value of friction (Beeler & Tullis, 1997; Marone et al., 1990; Morrow & Byerlee, 1989; Morgan, 2004). Marone et al. (1990) proposed that the “apparent” friction, μ^A, defined as the ratio of the shear to normal stress τ/σ (what is measured in laboratory experiments and our numerical simulations), can be written as

(16)

where dδ_n/dδ_s is the instantaneous ratio of fault-normal displacement δ_n to slip δ_s (dilation taken to be positive and compaction negative here) and μ^f can be considered to be some hypothetical “intrinsic” friction that would be measured in the absence of fault-normal displacements.

Changes in dδ_n/dδ_s in lab experiments are often larger than changes in the observed friction μ^A. Because of this, Beeler and Tullis (1997) pointed out that if μ^f is thought to be given by equation (1), the direct effect parameter a would have to be negative; that is, at constant state, materials would have to weaken with increasing slip speed. As this violates standard interpretations of the source of the direct effect, they argued that μ^f should be interpreted not as resulting from the total energy dissipated in the fault zone but as only the energy dissipated in fault-parallel shear. They showed that with this definition of μ^f, the time-dependent plastic contribution to dδ_n/dδ_s should be neglected in equation (16).

For granular models we are not persuaded that it is useful to speak of an “intrinsic” friction that is distinct from the contribution of dilatancy to the measured μ^A. And, as a practical matter, it is not trivial to separate dδ_n/dδ_s as observed in laboratory experiments into time-dependent plastic and slip-dependent geometric components, as advocated by Beeler and Tullis (1997). Nonetheless, our measurements of dδ_n/dδ_s can be compared to both laboratory experiments and our measured μ^A. Figure 7 shows the evolution of friction, the gouge layer thickness, and dδ_n/dδ_s, for 1 and 2 orders of magnitude velocity step increases and decreases for our default model, as well as 1 and 2 orders of magnitude step increases for the model with dimensions twice as large. The scale factor between friction and thickness changes in panels a, c, and e is the same as in Figures 3, 4, and 6. As in those figures, there is a reasonably close correlation between the measured friction and gouge thickness for the step increases but not the step decreases. However, the correlation between the measured friction and dδ_n/dδ_s for the step increases, as well as for the step decreases once the system is close to steady state, is even more striking. Note the difference in scale; the variation in dδ_n/dδ_s is about 40–50% larger than the variation in μ^A. In steady sliding laboratory experiments on 2-D glass rods, Frye and Marone (2002) found a ratio closer to 1. Hazzard and Mair (2003) also found a ratio of ∼1 at steady state for both 2-D and 3-D granular simulations with Hertzian grain-grain interactions.

Our granular simulations show that upon a step increase in velocity, the maximum value of dδ_n/dδ_s exceeds the direct-effect friction change Δμ_direct by anywhere from a few tens of percent to a factor of about 2 (Figures 7b and 7f). The difference is larger in our simulated velocity-step decreases; because of the more rapid evolution of thickness with slip, dδ_n/dδ_s following the velocity step exceeds Δμ_direct by more than a factor of 5 for the 1-order step down and more than a factor of 10 for the 2-order step down (Figure 7d). These results are within the ballpark of laboratory values. In experiments on synthetic gouge in a triaxial shear apparatus, Marone et al. (1990, Figures 20 and 21) find that dδ_n/dδ_s exceeds Δμ_direct by a factor of 4–6, independent of the magnitude of the velocity step, for both step increases and the one step decrease shown. Using data from the same paper, however, and plotting thickness as a function of slip, Segall and Rice (1995) show an example (reproduced here as Figure 4c) for which dδ_n/dδ_s is significantly larger for the step down than the step up. Similarly, Mair and Marone (1999, Figure 10a) show thickness versus slip for a 1-order velocity step increase and decrease in a double-direct shear experiment on synthetic gouge where (dδ_n/dδ_s)/Δμ_direct is about 2.5 for the step increase but many times larger for the step decrease (for the step increase , where 0.005 is the value of a for σ_n=25 MPa, total slip 18–20 mm, and V=1 to 10 mm/s in their Figure 8a). Using a rotary shear apparatus, Beeler and Tullis (1997) present data from 1-order velocity step decreases where dδ_n/dδ_s exceeds Δμ_direct by a few tens of percent for initially intact granite that develops a gouge layer through wear and by a factor of about 2.5 for synthetic granite gouge. This is an area where a more thorough comparison between the granular gouge simulations and existing laboratory data is certainly warranted.

Figure 8

Results from constant-volume velocity-step experiments. (A) shows the variation of friction; (B) and (C) show the variation of shear and normal stress applied by gouge grains to the driving plate, as functions of slip distance. All experiments use the default model with an initial normal stress of 5 MPa and V_i=2×10⁻⁴m/s. Gray lines in (A) show the frictional behavior for the corresponding constant normal stress experiments. The results are averaged over seven different realizations of the same imposed sliding conditions.

As a final investigation of equation (16), we ran velocity-step simulations while enforcing a constant volume (gouge thickness) boundary condition (dδ_n/dδ_s=0, so μ^A=μ^f). For a step increase this entails a transient increase in normal stress, as the gouge, which dilates at constant normal stress, is prevented from doing so. Remarkably, the transient friction response in the constant-volume simulations is indistinguishable from that in the corresponding constant normal stress simulations (Figure 8). We are thus faced with the surprising observation that at constant normal stress there is a very close correlation between dδ_n/dδ_s and μ^A for most of the friction evolution after the step velocity increases in Figures 7b and 7f, seemingly consistent with the spirit of equation (16), while essentially identical friction evolution occurs in simulations in which dδ_n/dδ_s is forced to be  zero.

5.2.4 Is There Localization in the Granular Gouge Layer?

A plot of the particle velocity through the gouge, u_x(z), spatially averaged over x and y and temporally averaged over an upper plate displacement of 0.1D_mean, is shown in Figure 9a for a steady-state shearing simulation performed at the load point velocity V_lp=10⁻² m/s. The steady sliding velocity profile decays linearly away from the shearing plate and shows no sign of localization. Following an order of magnitude velocity-step increase, we further measure the velocity variation with distance from the driving plate during the first 0.001D_mean, 0.01D_mean, and 0.1D_mean shearing distance. The results are plotted in Figures 9b–9d and show no signs of strain localization immediately or shortly after the velocity step (in Figure 9b the shear wave generated by the velocity step at the upper pate has yet to reach the bottom plate; see Appendix B). Hatano (2015) suggested that the duration of the friction transient following his simulated velocity steps might correspond to the slip distance required for the gouge to approach its new steady-state velocity profile, but as this occurs over distances <0.1D_mean in Figure 9, compared to slip distances of several D_mean for the friction transient, this is clearly not the case in our simulations.

Figure 9

The velocity profile of the granular gouge in the default system. The driving velocity is initially V_i=10⁻² m/s, as in Figure 4. Panel (a) shows the velocity profile at steady sliding with velocity V_i, measured over a slip distance 0.1D_mean. Panels (b), (c), and (d) show the velocity profiles measured in the first 0.001D_mean, 0.01D_mean, and 0.1D_mean, respectively, following an order of magnitude step velocity increase. In (b), the shear wave generated by the velocity jump at the upper plate just 3×10⁻⁵ s earlier has traversed only about half the gouge thickness (see B1). The normal stress is fixed at 5 MPa. The indicated velocity is a spatial average over the x and y directions.

The absence of localization in our system is also consistent with the adopted dimensionless pressure ( for σ_n=5 MPa), which puts it near the stiff or rigid grain limit. The studies by de Coulomb et al. (2017) and Bouzid et al. (2015) show that in our range of dimensionless pressure and inertial numbers, systems do not show persistent localized deformation, although Aharonov and Sparks (2002) report periods of spontaneous transient slip localization in 2-D simulations with . In contrast, persistent patterns of localized deformation in the form of simple shear bands are expected in systems that operate in the soft grain regime ( ) (Amon et al., 2012; Darnige et al., 2011; de Coulomb et al., 2017; Le Bouil et al., 2014). In laboratory experiments on synthetic gouge (Sleep et al., 2000), and gouge formed by wear of initially intact rock (Beeler et al., 1996), slip appears to be localized, but this may be associated with processes such as grain breakage that are not included in our model (see Abe and Mair (2009) for a granular simulation that includes breakage at the grain scale, and Aghababaei et al. (2018) for atomistic simulations that include asperity breakage and wear at the atomic scale).

5.2.5 The Influences of Grain-Grain Friction Coefficient, Restitution Coefficient, and Grain Size

To explore the generality of our observations and which grain-scale properties may influence the results, we investigated the steady-state behavior, and the transient response to velocity steps, of systems with different grain-grain friction coefficients, grain-grain restitution coefficients, and (while keeping the ratio of gouge thickness to grain size fixed) grain size. Figure 10 shows the variation of the steady-state friction coefficient (a) and gouge thickness (b) with driving velocity for the default system and for systems with different grain properties. The variation of steady-state friction with driving velocity is somewhat sensitive to the details of the system, although frictional behavior remains velocity strengthening for all cases with mean slopes of . The variation of gouge thickness with velocity shows that the gouge layer remains logarithmically dilatant, with similar normalized dilation rates of roughly 0.01D_mean per decade, corresponding to normal strains of order 10⁻³ per decade, for all systems.

Figure 10

(a) The variation of steady-state friction coefficient with driving velocity in the default system at three different normal stresses and in systems with different values of the grain-grain friction coefficient, restitution coefficient, size (smaller by 100 times), ratio of steady-state gouge thickness H_ss to mean particle diameter D_mean (1.8 times larger), and grain size distribution (quasi-exponential). (b) The variation of H_ss/D_mean for the simulations in (a). For models that have a different number of grains per unit area (L_x×L_y) than the default model, the ratio H_ss/D_mean has been further normalized by the ratio of that number to the number of grains per unit area in the default model (a correction that is ≤10% for the models with the same L_x and L_y). This normalization is performed for the systems with quasi-exponential grain size distribution, with 1.8 times the H_ss/D_mean of the default model (2H) and with different restitution coefficients and grain-grain friction coefficients. Error bars indicate 1 standard deviation of all friction measurements over a sliding distance of 5D for each of seven different realizations (initial grain arrangements). Most error bars in (b) are smaller than the symbol size.

Note that increasing the grain-grain friction coefficient decreases the macroscopic friction slightly, consistent with previous studies (Silbert, 2010), presumably as a result of enhanced grain rolling. From dimensional analysis, decreasing the grain and system sizes by the same scale factor is not expected to lead to differences in macroscopic behavior, as this changes only the magnitude of the gravitational stress relative to the confining pressure, which is already extremely low (Appendix A). Comparing the default model to the system reduced in scale by a factor of 100 in Figure 10 shows that this is generally the case, to within the scatter of the  data.

The choice of restitution coefficient ϵ also has very little influence on frictional behavior. Figure 10a shows that values of ϵ_n ranging from nearly fully damped (0.003 and 0.01) to near-zero dampling (the default value of 0.98) show essentially the same value of μ_ss as a function of velocity. Previous numerical studies have also demonstrated that for inertial numbers , varying the grain-grain damping exerts almost no influence on the steady-state frictional behavior of the system (MiDi, 2004) (this is unlike the behavior at higher I_n, where increasing the damping during grain-grain collisions decreases the rate of velocity strengthening and dilation with increasing driving velocity and inertial number  (MiDi, 2004; Silbert et al., 2001)).

The influence of these grain-scale properties on the transient frictional response to velocity-step tests was also very modest. Although we have not formally fit the results to determine the RSF parameters a, b, and D_c, directly comparing the transient responses to those for the default model generally shows differences that are within or near the apparent noise level (supporting information Figures S3, S5, and  S6).

5.2.6 The Influence of Grain Size Distribution

Unlike the grain-scale properties of the previous section, we find that grain size distribution has a dramatic influence on the macroscopic behavior of the system. We have run simulations with a quasi-exponential grain size distribution, which better represents actual fault gouge (An & Sammis, 1994; Billi, 2005; Marone & Scholz, 1989; Sammis & King, 2007). For the quasi-exponential size distribution, we targeted generating a distribution with grain sizes ranging from 0.5 to 5 mm, with D_mean=1.5  mm, and distribution form , with distribution parameters and λ=2  mm. The resulting system, generated by a random particle generation algorithm in lammps, has D_min=0.5, D_max=4.9, and D_mean=1.5  mm. We reduced D_mean by half, relative to the default system, to ensure that the largest particle size was no larger than the 5-mm particles in the bounding rigid blocks (larger gouge particles led to a roughly 5 mm periodicity in friction during quasi-steady sliding). We also found that the exponential distribution led to apparently noisier (more variable) friction during steady sliding; on the assumption that a longer model dimension in the sliding direction would reduce the influence of individual force chains, the quasi-exponential system was given dimensions L_x=4L_y=6L_z=160D_mean (Figure 11). This reduced the apparent noise substantially. A few simulations of the same dimension using the quasi-normal grain size distribution verified that increasing L_x/L_z from 1.5 to 6.0 did not change the steady-state friction level, its dependence upon slip speed, the rate of change of gouge thickness with shear velocity, or the qualitative behavior of the system during velocity-stepping or slide-hold protocols.

Figure 11

(a) Visualization of the virtual rock gouge experiment with the quasi-exponential grain size distribution, with mean grain diameter D_mean=1.5  mm. Colors show the velocity of each grain in the x direction, averaged over an upper-plate sliding distance of D_mean. The driving velocity is V_lp=0.1  m/s. (b) The size distribution of grains in the gouge layer.

The variation, with driving velocity, of the steady-state friction coefficient and gouge thickness for the quasi-exponential grain size distribution are shown by the solid orange symbols in Figures 10a and 10b, respectively. Given the error bars in panel a, one could perhaps argue that the system is velocity neutral. However, because the gouge thickness, which has much smaller uncertainties, decreases as V_lp increases from 10⁻³ to 10⁻¹ m/s, and increases from 10⁻¹ to 1 m/s, we think that it is more likely that the system is steady-state velocity strengthening as the shear velocity increases from V_lp=10⁻¹ to 1 m/s, and nearly velocity neutral or slightly velocity weakening as V_lp increases from 10⁻³ to 10⁻¹ m/s (an association between steady-state velocity weakening and gouge thinning, and steady-state velocity strengthening and gouge thickening, underlies recent versions of STZ theory  Lieou et al., 2017). Therefore, the gouge thickness, and perhaps the friction coefficient, vary nonmonotonically with driving velocity. DeGiuli and Wyart (2017) previously observed a nonmonotonic variation of friction coefficient with shear velocity in 2-D granular simulations and in the range of and inertial numbers we have explored. The grain size distribution used in their model is not specified. The nonmonotonic variation of friction coefficient has also been observed in several experimental granular physics studies, including those by Dijksman et al. (2011), Wortel et al. (2014), and Wortel et al. (2016). However, it is not straightforward to separate the potential contributions of time-dependent contact-scale processes from purely granular rearrangements in those experiments. van der Elst et al. (2012) also observed a nonmonotonic variation of gouge thickness with shear rate in experiments using angular grain shapes, while experiments using spherical grains showed a monotonic increase of gouge thickness with shear rate. The friction coefficient and the influence of grain size distribution on the velocity dependence of gouge thickness were not explored in their study.

We also performed a limited number of velocity-step simulations using the quasi-exponential grain size distribution. The results are shown in the supporting information Figure S7. They include a subset of 1 and 2 orders of magnitude velocity step up or step down from initial driving velocities of 10⁻², 10⁻¹, and 1 m/s. Owing to the large model size, we averaged only three realizations of each set of conditions, and the results are much noisier than for our normal distribution simulations (although less noisy than the average of seven realizations of the exponential distribution using the default simulation size). All of these tests show a direct velocity effect and an opposing state evolution effect, with a∼0.0085 for the step with the highest signal/noise ratio (a 2-order step down; supporting information Figure S7c) and values not far from this for the others. This is within the range typically reported for laboratory experiments (e.g., Mair & Marone, 1999).

The variation of gouge thickness following step velocity changes between 0.1 and 1 m/s, where steady-state friction and gouge thickness increase with slip speed, is similar to the behavior of models with a quasi-normal grain size distribution, in that the thickness monotonically approaches its future steady-state value at a decaying rate. However, for steps between velocities in the range of 10⁻³ to 10⁻¹ m/s, where steady-state thickness (and perhaps friction) decreases with increasing slip speed, the transient thickness change becomes nonmonotonic. Following a velocity step decrease, for example, the gouge initially compacts, as for the quasi-normal grain size distribution, but then dilates by a greater amount to reach the new steady-state thickness. Where the signal-to-noise ratio is sufficient (e.g., supporting information Figures S7b and S7c, and to a lesser extent S7e and S7f), this transition from compaction to dilation seems to occur while the friction is monotonically (except for the noise) approaching its new steady state. The reverse behavior is seen for velocity step increases. We do not yet understand the origins of this behavior and see no dramatic changes in the particle velocity profiles over the course of the nonmonotonic thickness changes. The change in sign of δd_n/δd_s at these lower velocities, together with the monotonic nature of the (smoothed) friction (and hence state) transient, is inconsistent with the notion that state and porosity are linked in any simple way. Considering only the steady-state thickness changes (Figure 10b), the positive direct velocity effect (a stress increase for a velocity increase) is also inconsistent with the simple notion that the direct effect comes from sliding at the new velocity but the old porosity. However, this positive direct effect is consistent with the initial thickness change following a velocity step having the opposite sign than the steady-state thickness change (e.g., an initial thickness increase for a step velocity increase).

We ran one slide-hold simulation using the quasi-exponential grain size distribution, with initial sliding velocity V_i=0.1  m/s. It showed logarithmic-with-time stress relaxation and gouge compaction, with a compaction rate of about half that of the model with a quasi-normal grain size distribution, after normalizing by the different initial gouge thicknesses. Again considering only steady-state thickness changes in Figure 10b, this result seems inconsistent with the intuitive statement that the effect on gouge thickness of an order of magnitude increase in hold time is roughy comparable to the effect of an order of magnitude decrease in slip speed (Figure 2). And, as with the positive value of the direct velocity effect, the compaction during the hold seems qualitatively consistent with the initial compaction following a step velocity decrease for the quasi-exponential grain size distribution; however, the subsequent dilation following the velocity step remains unexplained.

5.3 Slide-Hold Simulations

The main emphasis of this paper has been granular simulations of velocity-step experiments, which have long been known to be well modeled by the RSF framework using the Slip law for state evolution (Bhattacharya et al., 2015; Ruina, 1983). We have shown that the granular simulations, like the Slip law, predict that following the initial direct velocity response, friction decays quasi-exponentially to its new steady state over a slip distance that is independent of the magnitude and sign of the velocity step. Moreover, with apparently no important free parameters, the granular model with our adopted quasi-normal grain size distribution produces a direct velocity effect and a subsequent state evolution effect with amplitudes that vary linearly with the logarithm of the velocity jump, with values of the RSF parameters a and b that are reasonably close to those determined empirically in the laboratory. Changing to our quasi-exponential grain size distribution changes only the magnitudes of a and b, while still leaving them close to lab values (and perhaps introducing enough velocity dependence to make the system transition from steady-state velocity weakening to velocity strengthening with increasing slip speed).

In this section, we present preliminary results from the default granular model using loading conditions intended to simulate slide-hold protocols. We focus on both the stress decay during the hold and the corresponding change in thickness of the gouge layer. Laboratory observations indicate that in response to an imposed load point hold, the stress decays in a manner consistent with the Slip law and not the Aging law, which exhibits too little decay due to time-dependent healing (Bhattacharya et al., 2017). Furthermore, during the hold, the gouge undergoes fault-normal compaction roughly as the logarithm of time. Although RSF classically makes no explicit prediction about fault-normal displacements, the conventional interpretation of log-time fault-normal compaction during holds is that it is consistent with the Aging law for state evolution. That is, compaction is viewed as going hand in hand with the plastic deformation of microscopic asperity contacts and log-time increase in true contact area under high local normal stresses (Berthoud et al., 1999; Sleep, 2006). This compaction is observed despite the fact that log-time healing as embodied by the Aging law for state evolution is ruled out by the stress data from the same slide-hold experiments.

We have thus far examined slide-hold simulations performed at two initial sliding velocities and σ_n=5 MPa. Figure 12a shows the variation of normalized friction with normalized hold time for these tests, with the initial velocities of V_i=2×10⁻² and 10⁻¹ m/s shown by the blue and black curves, respectively. For standard RSF (equations (1)–(3) with constant parameter values), these curves would plot on top of one another when normalized in this fashion, a result that follows from dimensional analysis. Although the stress decay for the black curve (V_i=2×10⁻¹ m/s) is not strictly log linear, a log linear fit to that curve would be similar to the curve for V_i=2×10⁻² m/s. The figure also includes the predictions of the Aging and Slip laws, shown by the dashed orange and green lines, respectively, using the RSF parameter values determined independently from Slip law fits to the numerical velocity-step tests. As described in the Computational Model section, for the RSF predictions, we use a shear modulus of 300 MPa, leading to a normalized stiffness of 425. For a velocity-strengthening system with such a large stiffness, increasing (decreasing) by a factor of 2 shifts the Slip law fit left (right) by a slightly larger factor, but does not change the slope at long hold times (Bhattacharya et al., 2017, Appendix c2). The comparison between the blue and dashed green curves shows good agreement between the granular model and the Slip law prediction, as for the laboratory experiments of Beeler et al. (1994) analyzed by Bhattacharya et al. (2017). And the Aging law underestimates the stress decay during the holds, for the same reason that it underestimates the stress decay during lab experiments. This initial result suggests that the granular model, like the empirical Slip law, may capture much of the phenomenology of laboratory slide-hold tests. Further testing of the granular model over a broader range of slide/reslide velocities and spring stiffnesses, for comparison to available lab data, is currently underway.

Figure 12

(a) The blue and black lines show the variation of friction coefficient, normalized by the RSF parameter b, as a function of normalized hold time, for granular slide-hold simulations with prior driving velocities V_i of 2×10⁻² (blue) and 2×10⁻¹ (black) m/s. The orange and green dashed lines show the predictions of the Slip and Aging laws, respectively, using the RSF parameters determined from the velocity-step tests in Figure 4. This panel also shows (in blue) results of a reslide at V=V_i=2×10⁻² m/s following a normalized hold time t_hold/(D_c/V_i) of 100, in comparison to the Aging and Slip law predictions. The peak friction upon reslide is indicated by μ_peak. The confining pressure in all simulations is 5 MPa, and the dimensionless stiffness . (b) Blue and black lines show the change in gouge thickness during the slide-hold granular simulations of panel (a). (solid dots) The change in gouge thickness during hold experiments on granite reported by Beeler et al. (1994), who used two different (low and high) machine stiffnesses. An estimated slip-weakening distance D_c≈3 μm is used to normalize results from the laboratory experiments. Both low and high stiffness laboratory experiments were performed at 25 MPa confining pressure.

In addition to seeming to match the stress decay during laboratory holds, the granular model qualitatively reproduces the observed reduction in gouge thickness with log hold time (Figure 12b). In the conventional RSF framework, because the stress data are well modeled by the Slip law with its lack of state evolution, the gouge would not be expected to compact. The paradox that it does so was also noted by Bhattacharya et al. (2017) in their analysis of the Beeler et al. (1994) slide-hold experiments. In contrast, and in agreement with laboratory experiments, our granular simulations show that log-time compaction during holds is present even though log-time healing as embodied by the Aging law is lacking. This behavior is reminiscent of the symmetric stress change/asymmetric thickness change in response to velocity-step tests in Figure 4 (much more rapid variation in thickness than stress, following a step velocity decrease), and is another indication that equating state and porosity (Lieou et al., 2017; Sleep, 2006) neglects some fundamental aspect of granular friction.

5.3.1 Slide-Hold-Slide Simulations

During both laboratory and simulated slide-hold-slide experiments, friction (shear stress) relaxes during the hold but upon the reslide overshoots its future steady-state value by an amount Δμ_peak, reflecting the “healing” (strengthening at a reference slip speed) of the gouge during the hold. As shown by Bhattacharya et al. (2017), neither the Aging law nor the Slip law can successfully model, with a single set of parameter values, both the stress relaxation and the subsequent Δμ_peak for each of the high and low stiffness slide-hold-slide laboratory experiments of Beeler et al. (1994). In particular, although the Slip law can match the stress decay during holds for both stiffnesses moderately well, the predicted Δμ_peak for the high stiffness setup is far too low to match the lab data (Figure 8a of Bhattacharya et al., 2017). The reason is that for the high-stiffness setup, the slip during the load point hold is too low for the Slip law to allow significant healing.

Figure 12a shows the predicted Δμ_peak for the granular simulation (blue), Aging law (orange), and Slip law (green), the latter two using the same values of a, b, and D_c used to model the holds. Note that the Slip law predicts Δμ_peak∼0, because almost no slip accumulates during the load point hold and that Δμ_peak from the granular simulation is much higher. This is the first sliding protocol we have modeled for which the stress history from the granular simulation differs qualitatively from that of the Slip law, and it differs (1) in the sliding protocol for which the Slip law most obviously fails to match lab data (the reslides following holds) and (2) in the proper sense to match the lab data better than the Slip  law.

5.4 Exploring the Microphysics of Granular Rate-State Friction

There is currently no well-accepted explanation for the empirical, but moderately successful, Slip law for describing the rate- and state-dependent frictional behavior of rock and gouge. The only heuristic explanation of which we are aware is that of Sleep (2006), who proposed that it results from the highly nonlinear stress-strain relation at contacting asperities (e.g., that the modestly smaller stress following a velocity step decrease results in an exponentially smaller strain rate, and a symmetric stress response to step increases and decreases when plotted against slip). In this paper we have presented a physical model that, despite lacking meaningful time dependence at the contact scale, reproduces the Slip law where that law matches experimental data well (velocity-step and slide-hold protocols) and may outperform the Slip law where that law does not work (the reslides following holds). We would therefore like to use the output of the granular model to understand the source of its lab-like (and RSF-like) behavior.

As a first step, we consider the source of the rate dependence of granular friction. We expect that the log-time densification and relaxation of stress during holds (and by extension the densification with decreasing slip speed during steady sliding) are due to a reduction of elastic potential energy associated with local grain rearrangements. These rearrangements generate seismic waves that perturb nearby grains which might themselves be near the threshold for hopping, at a rate that decays quasi-logarithmically with time, as the driving stress and the opportunities for continued compaction lessen. This picture of grains as always vibrating, being perturbed by neighbors, and occasionally overcoming activation energy barriers, is conceptually similar to the traditional atomistic-scale view that the logarithmic rate dependence in RSF (the term in equation (1)) arises from a thermally activated Arrhenius process (Chester, 1994; Lapusta et al., 2000; Rice et al., 2001). In that microscopic picture, the slip rate is , where the product of the Boltzmann constant, k_B, and the temperature, T, is a measure of the average kinetic energy (KE) of the atoms. The activation energy E has the form E=E₁−PΩ_A, where P is a representative pressure and Ω_A is the associated activation volume. In this equation, V₁ can be interpreted as an attempt frequency times a slip displacement per successful attempt. Such an interpretation reproduces the empirical logarithmic form of the direct velocity dependence of friction with

(17)

A histogram of the KE (E_k) of every grain in a steady-state granular simulation with V_lp=2×10⁻² m/s is plotted on log linear and log-log axes in Figures 13a and b, respectively. Assuming that this KE plays the role of k_BT in equation (17), we can use this measurement (mean value ∼2×10⁻⁵ J/grain) to estimate a. We take the product of pressure and activation volume to be given by the elastic strain energy of grain compression, leading to

(18)

where C is the average coordination number (number of contacts per grain), Δ_ij is how closely two grain centers approach one another under the contact force F_n, d() in the integral represents an infinitesimal change (not the grain diameter), and the last equality is derived using the nonlinear Hertzian contact law for F_n as a function of grain compression δ_ij (equation (8) with no damping, d_i=d_j, and F_n=Pd²; d here is grain diameter). If we use D_mean for d and C∼4 (a value obtained from our simulations), we find a∼0.022, close to the value determined independently from fitting our velocity-step tests.

Figure 13

(a, b) Histograms of the per grain KE during steady sliding at V_lp=2×10⁻² m/s in (a) log linear (x−y) axes and (b) log-log axes. (c) The variation of the mean per grain KE (E_k) for steady sliding simulations at a range of shearing velocities and confining pressures, as well as for a system with twice the size of the default model, compared to the estimate (solid line) of per grain KE assuming homogeneous shear between the driving plates. (d) The per grain KE, normalized by PΩ_A from equation (18) (our estimated RSF a), for the same steady sliding simulations as in (c), here expressed as a function of the inertial number. The dashed line corresponds to I_n=10⁻³, which traditionally is considered the limit above which inertial effects become nonnegligible. The upward pointing triangles in (c) and (d) show the “fluctuating granular KE” (δE_k) as defined by Ogawa et al. (1980) and Ogawa (1978). In the low inertial number regime of interest, the difference between KE and δE_k is insignificant.

There is certainly slop associated with this estimate, including whether the activation volume is more appropriately thought of as a single grain or a few grains that rearrange collectively (as in STZ theory), and whether it is the total normal displacement or the incremental displacement from the background state that determines the activation volume (similar questions pertain to the classical RSF estimate of a, e.g., whether the activation volume corresponds to a single atom or a unit cell). Nonetheless, we find the order of magnitude agreement to be encouraging. But this agreement is insufficient; if the granular KE is to play the role of temperature, it must be insensitive to both the sliding speed and the confining pressure, and it is not apparent that this need be the case. Empirically, however, we find that the mean value of granular KE at any particular P changes only modestly over several orders of magnitude variation in V_lp, at the low driving speeds of interest (Figure 13c). For comparison, the solid line on the same plot (of slope 2) shows the KE that would result from a layer of uniformly sheared grains as a function of V_lp. For P=5  MPa the quasi-constant granular KE intersects this trend at V_lp∼2  m/s, the inertial number I_n∼3×10⁻³, and the system is traditionally considered to leave the regime of quasi-static flow (Forterre & Pouliquen, 2008). Furthermore, if we normalize the per grain KE by the estimate of PΩ_A from equation (18), as in (17), our proposed estimates of a collapse for all confining pressures onto a single curve in the quasi-static regime (Figure 13d). The prediction is thus that a changes very slowly for a range of shearing velocities and pressures in the quasi-static regime, consistent with both our granular simulations and many laboratory rock and gouge friction experiments.

In Figure 13 we used mean grain kinetic energy as a measure of the effective temperature T_eff of the granular gouge. A number of more rigorous thermodynamics- and statistical mechanics-based relationships have been proposed for measuring T_eff in granular materials (e.g., the rate of change of energy with entropy); this remains an area of active research (Bi et al., 2015; Blumenfeld & Edwards, 2009; Ono et al., 2002; Puckett & Daniels, 2013). Ono et al. (2002) showed that for zero-temperature foam, seven of these definitions are internally consistent in that they yield the same variation of T_eff with shear rate. Further experimental investigations showed that two of these measures of T_eff become approximately constant at low shear rates (Corwin et al., 2005; Song et al., 2005). This is similar to our finding in Figure 13e, although these other measures are even more constant than our granular KE at low I_n. Such measurements are necessary to confirm whether different measures of temperature converge toward the same behavior, if they also agree with the variation of kinetic energy, and become nearly constant within the quasi-static regime. Such measurements could elaborate the cause of near constancy of the granular temperature—which to this date remains unknown—by making analogies to the behavior of other glassy materials (like foam) as they approach the glass transition.

6 Conclusions

In this work, we explored the frictional behavior of a granular gouge layer with no time-dependent plasticity at the grain-grain contact scale. We imposed velocity steps over a range of driving velocities and normal stresses that are relevant to earthquake nucleation and laboratory rock friction experiments. We further performed a limited number of slide-hold granular simulations. The system is mechanically stiff enough that, following a step change in driving velocity, the inelastic sliding velocity is essentially constant and variations in the friction coefficient are proportional to variations in log state. We found that the behavior of the granular model appears very similar to the Slip law version of the RSF equations, under conditions where the Slip law agrees well with laboratory data, that is, velocity-step and slide-hold tests. In particular, we observed that (i) following velocity steps that vary by several orders of magnitude, friction approaches its future steady-state value over the same sliding distance (or strain, if gouge thickness is varied), (ii) the frictional response of the system to velocity-step increases and decreases is symmetric, (iii) the amplitude of frictional evolution following velocity steps scales with , and (iv) the ranges of the RSF parameters a (0.020–0.029) and b (0.014–0.023) are not very different from those typically found in laboratory rock and gouge friction experiments. In addition, the slide-hold granular simulations appear to be well described by the Slip law, using parameters derived from fits to the velocity steps, as is the case (or nearly the case) for laboratory friction data. Finally, preliminary slide-hold-slide simulations indicate that the peak stress upon the reslide exceeds the prediction of the Slip law, using the same parameters that fit the hold well, as is also the case with lab data (Bhattacharya et al., 2017).

Future work should include investigating whether the granular model can reproduce observations of the friction peak upon the reslide in slide-hold-slide experiments (often referred to as “frictional healing”) (Bhattacharya et al., 2017; Karner & Marone, 1998; Marone & Saffer, 2015), over a broader range of normal stresses, driving velocities, and system stiffnesses that we have explored thus far; the recent observation that the slip-weakening distance following the reslide increases systematically with log hold time (Bhattacharya et al., 2017), and the friction and thickness changes observed in normal-stress stepping tests (Kilgore et al., 2017). These will be important tests for the granular model, as none of the current empirical constitutive relations for the behavior of rock interfaces reproduce these observations acceptably.

The conventional understanding that state evolution in RSF results from contact plasticity suggests that state and gouge thickness are closely related. However, we found that even though gouge thickness seems to be a useful proxy for variations in state following step velocity increases, the gouge thickness evolves over much shorter slip distances than does friction following step decreases. Qualitatively, the asymmetric response of gouge thickness to velocity step increases and decreases appears similar to the asymmetric response of friction (i.e., log state) predicted by the Aging law. Related behavior is seen during load point holds, where the friction coefficient appears to decay as predicted by the Slip law, implying very little state evolution, while the gouge layer compacts as log time, reminiscent of the (log) state increase predicted by the Aging law. The asymmetric response of the gouge thickness to changes in driving velocity, in conjunction with the symmetric response of the friction coefficient, indicates that gouge thickness is at best an incomplete description of state. The log-time compaction of the gouge during holds in which the friction decay is well described by a law that predicts very little state evolution suggests the same. Aspects of the granular structure other than porosity, such as force fabric and structural anisotropy, must also contribute to the state of the system (Lechenault et al., 2006; Puckett & Daniels, 2013).

Both the asymmetric response of the gouge thickness to velocity step increases and decreases, and the log-time compaction during load point holds, are predictions of the granular model that seem consistent with laboratory rock and gouge friction experiments. Models of coupled fault gouge dilatancy/pore pressure diffusion (e.g., Segall et al., 2010) are likely to be most consistent with existing lab experiments if porosity is tied to the Aging law for state evolution when state is increasing (Vθ/D_c<1 in equations (2) and (3)) and the Slip law when state is decreasing (Vθ/D_c>1), even while the frictional strength is more accurately modeled by the Slip law under both conditions.

We explored a range of parameters and material properties that could have influenced our observations. We found that grain-grain friction coefficient, restitution coefficient, and grain size had only minor effects on system behavior. Using a system with roughly twice the thickness of the default model, we found that the critical slip distance scales with gouge thickness and can instead be expressed as a critical strain (of about 13%, when defined as D_c/H). We also examined the influence of changing the grain size distribution from a quasi-normal to a quasi-exponential distribution. This reduced the value of a to about 0.008, near the low end of the range typically cited for rock and gouge. More significantly, we found that changing from a quasi-normal to quasi-exponential grain size distribution changed the steady-state friction from velocity strengthening to something closer to velocity neutral. Although within the noise of the simulations the quasi-exponential system could be argued to be strictly velocity-neutral, the close association between the observed velocity dependence of friction and the clearly nonmonotonic steady-state gouge thickness leads us to favor the interpretation that the steady-state friction transitions from velocity-weakening to velocity-strengthening with increasing slip speed. A nonmonotonic dependence of steady-state friction on driving velocity has not often been observed in numerical simulations of frictional granular systems (da Cruz et al., 2005; Kamrin & Koval, 2014; Koval et al., 2009; MiDi, 2004), but within our adopted range of dimensionless pressures and inertial numbers, it is consistent with recent theoretical predictions (DeGiuli & Wyart, 2017). The effect of grain size distribution was not explored by DeGiuli and Wyart (2017), however. In the velocity-strengthening regime, where the quasi-exponential gouge layer dilates with increasing slip speed, following a velocity step the layer approaches its new steady-state thickness monotonically, just as does the velocity-strengthening gouge with the quasi-normal size distribution. In the velocity-weakening regime, however, the gouge thickness for the quasi-exponential system varies nonmonotonically following a velocity step, for example first compacting following a step decrease before dilating by a larger amount with continued slip. This initial response seems consistent with a positive direct velocity effect and is consistent with the observed compaction during holds for the quasi-exponential system, but the nonmonotonic evolution of thickness with slip is yet another indication that there is not a simple relation between gouge porosity and state, and we do not understand its cause.

By making an analogy between granular rearrangements in a potential energy landscape and a thermally activated Arrhenius process, we estimated the magnitude of direct velocity effect (the RSF parameter a) in our model. For this purpose, we used the mean kinetic energy of grains as a measure of granular temperature, and assumed that this was equivalent to the thermodynamic temperature in a thermally activated process. We found a value of a close to that obtained independently from fitting our velocity-step tests. Furthermore, this value was found to be independent of confining stress and nearly independent of slip speed. This nearly constant value of a is consistent with our simulation results and with much lab  data.

The successful adoption here of the granular temperature may motivate its future implementation as a state variable for granular RSF. In standard thermodynamics, involving thermal materials, energy is the conserved property. However, the granular materials in our simulations, and in many others in the physics literature, are athermal, in the sense that the actual temperature plays no role. Recent progress in the granular physics community points toward a revised version of granular temperature, called keramicity and defined in the stress ensemble, where instead of energy the conserved quantity is the force-moment tensor of the granular packing (Bi et al., 2015). Ideally, one would like to devise a state variable that would obey the laws of thermodynamics for granular systems and be path and protocol independent. It would be interesting to investigate whether a state variable defined in the stress ensemble could be used for effectively describing the rate- and state-dependent frictional behavior of rocks.

While our observations here focused on rock gouge and the frictional behavior of fault rocks, they could be potentially relevant for transient frictional behavior and hysteresis of a broad range of disordered Earth materials, such as soils on hillslopes (Ferdowsi et al., 2018; Handwerger et al., 2016), fluvial sediments (Houssais et al., 2015; Johnson, 2016; Masteller et al., 2019), and subglacial till (Rathbun et al., 2008).

Acknowledgments

BF acknowledges support from the Department of Geosciences, Princeton University, in form of a Harry H. Hess postdoctoral fellowship, and from NSF award EAR-1547286 and the US Geological Survey (USGS), Department of the Interior, award G19AP00048, both to AMR. BF performed some slide-hold-slide tests during his PhD research with the granular physics model he developed in his PhD. Some elements of the current model, which is distinct from the model in his PhD, were developed during BF's stay at the University of Pennsylvania, partially supported by the 2016 Southern California Earthquake Center (SCEC) award 16059 to David L. Goldsby. A study on the granular origins of rate- and state-dependent friction for fault gouge was also proposed in that same award to David L. Goldsby. BF benefited from conversations with Chris Marone, Pathikrit Bhattacharya, Anders Damsgaard, Nicholas M. Beeler, Heather Savage, Emily Brodsky, Andrea J. Liu, Corey O'Hern, Troy Shinbrot, Norman Sleep, Rob Skarbek, Paul Segall, Karen E. Daniels, Shmuel Rubinstein, Rob Viesca, Melodie French, Julia Morgan, Jean M. Carlson, Ahmed Elbanna, Andreas Kronenberg, David Sparks, and Hiroko Kitajima. AR benefited from a subset of those. BF would also like to acknowledge support he has received from D. J. Jerolmack and D. L. Goldsby, and the insightful discussions he had with D. J. Jerolmack, D. L. Goldsby, C. A. Thom, and Carlos P. Ortiz on this topic during 2015-2016. Parallel programs were run on computers provided by the Princeton Institute for Computational Science and Engineering (PICSciE). The 3-D visualizations of the model were performed using the open-source visualization software “The Persistence of Vision Raytracer” POV-Ray (http://www.povray.org). Most of the data analysis were carried out using the open-source Python library, NumPy (https://numpy.org). The 2-D plots were made with the Python library Matplotlib (www.matplotlib.org). The computer codes for LAMMPS simulations of this paper with the information about the version of LAMMPS used for the simulations are available on the Dryad digital repository at https://doi.org/10.5061/dryad.2z34tmphk. We thank two anonymous reviewers whose suggestions helped to improve and clarify this manuscript. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Government.

Appendix A: Model Dimensionless Parameters

The adopted DEM model has many dimensionless parameters, each of which could potentially affect the system behavior. However, only a few of these seem to be significant. Here we give a full accounting of these parameters, along with a qualitative assessment of their relevance to the observed RSF parameters in the slow shearing regime of interest.

Neglecting parameters associated with the adopted grain size distribution, Table A1 lists 15 independent dimensional parameters using three dimensions (writing Pa as kg m⁻¹ s⁻²), implying that 12 dimensionless parameters govern the system. Some of the parameters in Table A1 can be considered as equivalent to an equal number of different parameters; for example, the grain normal and shear contact stiffnesses are derived from the elastic shear and Young's moduli (G and E), and the normal and shear restitution coefficients are derived from the normal and shear damping coefficients γ_n and γ_t. Additional parameters depend upon those listed, for example, the grain mass m and the bulk density ρ_H and shear modulus G_H, as well as the measured friction coefficient.

Table A1. DEM Parameters for Steady Sliding Simulations

Symbol	Parameter	Units
D	Median grain diameter	m
H	Nominal gouge thickness	m
Lx	Domain length in slip direction	m
Ly	Domain length in slip-perpendicular direction	m
kn	Grain normal contact stiffness	Pa
kt	Grain shear contact stiffness	Pa
ϵn	Grain normal restitution coefficient	—
ϵt	Grain shear restitution coefficient	—
μg	Grain-grain friction coefficient	—
ρ	Grain density	kg m^−3
Vlp	Driving velocity	m s^−1
σn (P)	Applied normal stress	Pa
ksp	Driving spring stiffness	Pa m^−1
g	Gravitational acceleration	m s^−2
Δt	Numerical time step	s

Before listing dimensionless parameters, we introduce some relevant time scales:

1. t_γ, time scale for bulk strain of 1: H/V_lp.
2. t_w, time scale for elastic shear wave propagation across layer: .
3. t_i, inertial time scale for an initially stationary a grain to move a distance D, given an applied force PD²:
4. t_col, collision time (obtained by solving equation (12) in the text).

Table A2 lists a reasonable set of choices for the 12 dimensionless parameters. Three involve ratios of lengths. It has been proposed that the ratio H/D determines the ability of the gouge to localize deformation, with no localization for values (Tsai & Gollub, 2005). We see no localization in the velocity profiles for our default model with H/D∼13.3, or in the model with H/D=24 and L_x/H and L_y/H unchanged. We also see no change in the RSF parameters between the two simulations, provided we speak of a critical strain rather than a critical slip distance D_c (Figure 6). The ratios L_x/H and L_y/H are not expected to be significant as long as they are sufficiently large; if force chains typically form at ∼45°, then L_x/H should at a minimum exceed 1. We see no significant difference between L_x/H=1.5 and L_x/H=5 for both normal and exponential grain size distributions, other than the expected result that simulations with L_x/H=5 exhibit less variability during steady sliding.

Table A2. DEM Governing Parameters

H/D;  Lx/H;  Ly/H

ϵn;  ϵt;  μg

kn/kt;  ksp/(G/H);  ρgH/P

Δt/tc

ti/tγ (Inertial number In, equal to

(P/E)^2/3 (dimensionless pressure)

We have varied ϵ_n over nearly the full range of 0 to 1. Consistent with previous results for steady-state friction (MiDi, 2004), we find negligible influence of the restitution coefficients on the RSF parameters in the low-I_n regime of interest (supporting information Figure S3). We have always assumed that the damping coefficients satisfy γ_t=0.5γ_n, the default for lammps, from which the restitution coefficients are derived, but given the existing results, we expect no significant change for γ_t≠0.5γ_n. We also see no significant changes in the RSF parameters when doubling μ_g from the default value of 0.5 to 1.0 (supporting information Figure S6). This is consistent with previous studies that show little influence of μ_g on steady-state friction (MiDi, 2004).

The ratio k_n/k_t is fixed by the elastic moduli of the grains and is not a free parameter. The ratio k_sp/(G/H) controls the elastic deformation of the driving spring relative to that of the gouge layer (the more relevant bulk gouge shear modulus G_H depends upon G and the granular packing). We made k_sp extremely large, to make the effective stiffness of the system as large as possible; this ensures that sliding velocity following a step change in V_lp is constant, such that changes in friction correspond directly to changes in log(state), facilitating a “by eye” comparison of the measured friction transient to different state evolution laws. Significantly reducing k_sp/(G/H) will change the loading history of the gouge layer for a given V_lp history, but traditionally, the RSF parameters are assumed to be independent of loading history. The ratio ρgH/P determines the relative magnitude of the gravitational stress at the base of the gouge layer to the applied stress; in our simulations it is so low (10⁻⁶ to 10⁻⁸) that we expect it to be negligible, although it may lead to some grain sorting during the packing of the gouge layer prior to imposing the confining pressure. In the future it would make sense to dispense with gravity during the sliding and most of the packing phases.

For numerical accuracy we employ Δt/t_col=0.01, small enough that it does not influence the simulations.

The inertial number I_n, equal to t_i/t_γ, is a well-established control parameter for granular systems, but from the figures in this paper it does not affect the RSF parameters much. This is consistent with many laboratory rock and gouge friction experiments.

The dimensionless pressure (P/E)^2/3 is equal to the grain strain (grain compression at a contact divided by the initial grain radius) under the Hertzian contact law. For P from 1 to 50 MPa, this ratio varies from 0.7×10⁻³ to 10⁻², near to but perhaps not within the “rigid grain” limit (DeGiuli & Wyart, 2017). We find that the RSF parameters vary only modestly, and not necessarily consistently, over this interval (Table 2).

This information can be used to extrapolate beyond the simulations already run. For example, we use a relatively large D_mean of 3 mm, but from Table A2, if we reduce the grain size and all model dimensions by the same factor (say 2 orders of magnitude) and keep V_lp and all other parameters the same, we change nothing other than to increase k_sp/(G_H/H) and decrease ρgH/P by the same 2 orders of magnitude. These ratios were already so large and so small that we expect to see no significant changes to the model output, consistent with Figure 9.

In summary, despite the large number of dimensionless parameters in Table A2, remarkably few of these are free parameters available for tuning the values of the RSF coefficients. Their influence might be largest

on the value of (a−b), as this depends upon the difference between two numbers of comparable magnitude. Significantly, the sign of (a−b) seems sensitive to the grain size distribution. This is a parameter that, along with grain shape and perhaps others, is not referenced in Table A1.

Appendix B: The Inertial Contribution to the Measured Shear Stress

Equation (13) in the main text assumes that the elastic component of the gouge deformation occurs quasi-statically and uniformly across the gouge, such that for constant load point and sliding velocities the shear stress increases linearly with time and load point displacement. However, for a linearly elastic system, following a sudden change of upper plate velocity ΔV_pl, a shear wave traverses the layer that, until the arrival of the reflected wave from the stationary lower plate, imposes an instantaneous shear stress change at the base of the upper plate given by

(B1)

where is the elastic shear wave speed, with G_H being the shear modulus and ρ_H the density of the gouge layer (Rice, 1993). Dividing this by the normal stress gives an estimate of the inertial contribution to the “apparent” direct velocity effect,

(B2)

For the example of the 4-order velocity increase to V_f=2  m/s in Figure 3a, we see an instantaneous, not linear-with-time, apparent Δμ_inertial of about 0.19. Plugging this value into the left side of (B2), and on the right 5 MPa for σ_n and a typical porosity of 0.45 to estimate ρ_H, we calculate G_H=165  MPa. This is just over half the 270–310 MPa we estimated from the reloading (at much lower slip speeds) of the gouge following long holds (Figure B1), but it is certainly possible that at the large stresses associated with the 4-order velocity jump, some inelastic deformation is occurring.

Figure B1

(a) The variation of friction coefficient with load point displacement for the reslide portion of several slide-hold-reslide simulations (obtained from the default model with V_lp=2×10⁻² m/s and σ_n=5  MPa). In (b), the friction coefficient at the end of the hold is subtracted from the signals, so the initial slopes of the reslide curves can be more easily compared. From the asymptotic slope at zero displacement (using the longest hold over a load point displacement of 0.008D_mean), we estimate an elastic shear modulus of ∼270–310 MPa.

Note that Figure 9b shows a snapshot in which the shear wave front following a 1-order velocity jump (to 0.1 m/s) has yet to traverse the gouge layer. The postjump displacement of the upper plate is 10⁻³D_mean, or 3×10⁻⁶ m, in (at V_lp=0.1  m/s) 3×10⁻⁵ s, and the shear wave has progressed approximately 0.02 m, implying a propagation velocity of ∼670 m/s. Setting this equal to and estimating ρ_H as above, we can derive a third independent estimate of G_H: 610 MPa, about twice the estimate from Figure B1. As we have not investigated in detail the nature of the shear wave propagation across the gouge layer, we continue to use our midrange estimate of G_H∼300  MPa, consistent with our nearly quasi-static reloading simulations (V_lp=2×10⁻² m/s), with previous experimental, numerical, and theoretical estimates for granular systems under comparable conditions (Domenico, 1977; Makse et al., 1999; Yin, 1993), and with standard methods of estimating G in laboratory rock and gouge friction experiments (e.g., Bhattacharya et al., 2017). As we note in the main text, 300 MPa is large enough that variations in G_H of a factor of 2 do not change the Slip law fits to our simulated velocity steps, and do not change the slope of the slip law prediction at large hold times in Figure 12.

It is clear that for any value of G_H in the vicinity of 300 MPa, for V_f=2  m/s inertia contributes significantly to the apparent Δμ for early times. However, note that once the steady-state velocity profile in the gouge is reached, the contribution from bulk inertia to the measured Δμ at later times is zero. The approach to that steady-state velocity profile is likely a complex process involving multiple reflections from the bounding rigid plates. We can make an estimate of Δμ_inertial in this case from the inertial force per area A required to change the velocity profile from one steady state to another over a time Δt:

(B3)

where is the spatially averaged acceleration of the gouge particles and ΔV_pl/2 comes from assuming the steady-state velocity profile to vary linearly across the gouge layer. Note that Δμ_inertial is proportional to 1/Δt as well as to ΔV_pl. For example, peak friction for V_f=2  m/s in Figure 3 is reached at 6×10⁻⁴ s. If this is the time at which the steady-state velocity profile is reached (roughly 5 times the two-way shear wave travel time across the gouge as estimated from Figure 9B), the inertial contribution to the apparent Δμ would average only 0.018 up to that point (11% of the plotted peak value), would likely be lower at that slip distance, and would be zero at greater distances. If the steady-state velocity profile was not reached until a slip distance of D_mean, the contribution to the measured friction up to that point would average more than a factor of 2 smaller. In light of these results, we conclude that inertia can contribute modestly (or zero) to the measured friction in the vicinity of the friction peak for V_f=2  m/s in Figure 3, but that it provides a negligible contribution to the overall Slip law fit to that friction curve.

For the 10 times smaller jump to V_f=0.2  m/s, the inertial contribution to the measured friction would be 10 times smaller from equation (B2), as well as from (B3) assuming that the same time Δt is required to reach the new steady-state velocity profile. For this velocity step the peak friction does not occur until a slip distance of ∼0.4D_mean, at which point the steady-state velocity profile is almost certainly established (see Figures 9c and 9d for velocity profiles at slip distances 40 and 4 times smaller, for V_f=0.1  m/s), and the contribution from bulk inertia would be zero (if not, from B3 the average contribution up to that point would be 0.0019, which at the scale of Figure 3a is completely negligible). We conclude that bulk inertia plays no discernible role in our simulated velocity steps where the larger (initial or final) slip speed is 0.2 m/s or smaller, and that even at 2 m/s inertia will only affect the friction curves significantly for slip distances smaller than some tenths of D_mean.