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Keywords:

  • granular media;
  • rock physics;
  • fault gouge

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[1] Two-dimensional numerical simulations of shear in a gravity-free layer of circular grains were conducted to illuminate the basic mechanics of shear of granular layers (such as layers of fault gouge). Our simulated granular layers exhibit either stable (steady state) or unstable (stick slip) motion. The transition from steady to stick-slip sliding depends on loading velocity and applied confining stress in a way similar to a simple model of a block on a frictional surface. We investigate the conditions which lead to naturally occurring stick-slip behavior and study in detail the systems behavior prior to and during slip events. Matching our numerical results to a spring block model, the system of grains was found to have bulk static and dynamic coefficients of friction that differ by about 0.1. This differing static and dynamic friction emerged spontaneously, from the collective behavior of grains, and was not prescribed a priori via a frictional rule between grain contacts. Results show that the micromechanics of contact forces is responsible for stick-slip behavior: During the “stuck” phase, and in preparation for slip, more and more grain contacts which carry low forces slide, resulting in accelerating internal stress release. When enough of the low-force contacts frictionally slide, the granular layer weakens and losses rigidity, leading to motion of contacts that carry larger forces and large-scale slip. Our results may have implications to the understanding of the stability of gouge layers and are thus related to the underlying physics of earthquakes.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[2] The origin of friction has been a topic of much study since Leonardo De Vinci's first work in the fifteenth century. Friction is a measure of the resistance experienced by surfaces to sliding past each other. This resistance comes about from chemical and mechanical forces such as adhesion and interlocking between surfaces, and may be relieved by various means: Plastic and elastic deformation, ploughing, riding, and brittle fracture (for a review, see Scholz [2002, and references therein]). Even though this picture of friction is complex enough, natural faults are even more complicated than a system with a single sliding surface. Faults with significant displacement are often filled with a layer of broken up pieces of rock termed “fault gouge”. Slip on a gouge-filled fault involves frictional sliding, rolling, breaking and rearrangement of the grains in the gouge. There are two basic modes of frictional sliding between surfaces: stable sliding at a nearly constant velocity, and unstable “stick-slip” sliding, in which the surface remains locked for a long period of time, and then surges forward rapidly. Experimental systems that include a simulated gouge layer have been observed to switch from a velocity strengthening (favoring stable sliding) regime to a velocity weakening (allowing unstable sliding) regime as the gouge matures through continued comminution [e.g., Dieterich, 1981; Biegel et al., 1989; Marone, 1998; Mair and Marone, 1999]. Earthquakes are manifestations of unstable motion on natural faults. It is important to understand the sliding stability of gouge layers, since it may control the mechanical properties of fault zones, and, as a result, the mechanisms of earthquake nucleation and propagation.

[3] Despite much interest in the dynamics of granular media, it has eluded complete understanding. There is, at present, no consistent theoretical framework that describes the behavior of a large collection of grains [Jaeger et al., 1996; Savage, 1998] under all relevant conditions. Even the simple questions of the distribution of velocities in a shearing granular layer, or the boundary conditions to be used in any proposed theory, are still open. Fortunately, a large amount of recent work in this area has begun the development of a basic understanding of stress transmission, heterogeneity and deformation in granular layers (e.g., Edwards and Wilkinson [1982], Radjai et al. [1996], Miller et al. [1996], Ouaguenouni and Roux [1997], Radjai et al. [1998], Cates et al. [1998], Veje et al. [1998], Howell et al. [1999], and Aharonov and Sparks [2002], to list a few). A promising avenue of research is discrete numerical simulations, [e.g., Cundall and Strack, 1979; Donze et al., 1994; Scott, 1996; Aharonov and Sparks, 1999; Mora and Place, 1998; Morgan and Boettcher, 1999]. In these models, each grain is a discrete entity that interacts via shear and normal forces with its neighbors. Complex behavior emerges naturally from the collective behavior of many grains. This paper uses such a model to investigate in depth the stability of sliding in a simulated granular layer and the behavior during the different phases of unstable sliding.

[4] Although unstable (stick-slip) sliding has been studied theoretically in the context of elastic blocks sliding past each other along a rough interface [Dieterich, 1978, 1979; Ruina, 1983], it has been much less explored in the more complicated case of a granular layer filling the gap between the two plates: Thompson and Grest [1991] used a two-dimensional numerical discrete element model to study a transition from continuous to stick-slip motion at a critical pulling speed. In their numerical system the gravitational body force of the grains was a significant fraction of the exterior confining stress, and the stick-slip transition was attributed to a gravity-driven density transition. A set of illuminating laboratory experiments were conducted by Nasuno et al. [1997, 1998], in which a horizontal plate lying atop a layer of glass beads was pulled horizontally by a spring. As loading velocity was decreased below a critical value, the velocity of the plate went from nearly constant to highly oscillatory, to finally a discontinuous stick-slip motion. The stick-slip behavior was shown to be described by a model in which the granular layer had different static and dynamic coefficients of friction [Nasuno et al., 1998]. In addition, recent laboratory experiments explored the effect grain shape had on sliding stability, demonstrating that spherical grains are especially prone to unstable sliding [Mair et al., 2002; Frye and Marone, 2002].

[5] Our goal in this study is to further explore the origin of stick-slip behavior during shear of granular media. We present two-dimensional numerical experiments of shear in granular layers, under conditions similar to those of Thompson and Grest [1991] and Nasuno et al. [1998], but in contrast to previous models our model lacks gravity forces. Our simulations are relevant to either horizontal granular layers that are buried deeply relative to their thickness, or to vertical granular layers, such as a gouge zone in a strike-slip fault. We report a transition from continuous to stick-slip motion related to the configurational stability of the granular layer, and in which gravity plays no role.

[6] In order to introduce terms and concepts that will be used later, we first review a simple analytic model for sliding on a single interface in section 2. Then, the setup of our numerical models is described in section 3. Section 4 reports our simulation results and the conditions under which continuous or discontinuous (stick-slip) motions in the layer are mapped out. We show that some aspects of the granular model (stability, slip pulses, characteristic timescales) are, to first order, well described by the simple spring-block model of sliding along a single frictional interface, and discuss the implications of these similarities. In section 5 we compare modeling results to the simple analytical model presented in section 2, and find that the model provides a good description to our numerical results, assuming the layer has different static and dynamic friction coefficients. Section 6 investigates the difference between the apparent frictional resistance of the granular layer and the friction prescribed for single grain interactions. We study the relation between characteristics of force distribution on the grain-scale and the large-scale motion and find that the changing statistical characteristics of forces on grain contacts determine the emerging frictional behavior and are thus the source of the unstable sliding.

2. Spring-Block Model of Frictional Sliding

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[7] In this section we review a simple mathematical model of frictional sliding [cf. Scholz, 2002]. This model of a single frictional interface will be useful in understanding the numerical results that will be presented later. A rigid block, at position x(t), is pulled across a frictional surface by a spring. The force exerted by the spring is

  • equation image

where Δx is the displacement of the spring from its equilibrium length. The driving velocity of the spring is Vsp. At time t = 0, the block is at rest at x = 0, and the spring is stretched to the static friction limit,

  • equation image

where μs is the coefficient of static friction and N is the normal force exerted by the block on the surface. Further stretching of the spring overcomes the frictional force, and the block begins to move.

[8] Once the block is in motion, the frictional force is assumed to immediately decrease to a lower dynamic value, μd. During the ensuing motion, the force balance on the block is given by

  • equation image

where mb is the mass of the block. If the motion of the block is unstable, consisting of short-duration events of large slip, then during this slip period, the change in the position of the driven end of the spring is small compared to the change in position of the block. The equation of motion can then be approximated as a simple harmonic oscillator:

  • equation image

where Δμ = μs − μd. The solution for equation (4) has the form

  • equation image

A “slip” event will last until the block velocity first vanishes, at which point the static friction again pertains. The block will remain stuck until the shear force rebuilds to the critical value given by equation (2).

[9] There are two characteristic timescales in this system, “recurrence time” and “slip time”, and their relative sizes determine the type of motion that the block will have. The duration of a slip event (“slip time”) is given by

  • equation image

At the end of a slip event, the force on the spring is reduced to

  • equation image

Assuming the block remains at rest, the recurrence time, Tr, is the time required to rebuild the spring force back up to the static friction force equation (2):

  • equation image

Stick-slip motion occurs when Tslip < Tr (e.g., high normal stress on the interface, or very slow loading). In the limit of very short recurrence time (e.g., when the dynamic and static frictions are nearly equal) Tslip > Tr, and the motion of the block will become continuous, sliding at the loading velocity Vsp. Setting Tslip = Tr, the normal stress required to bring about this transition from stick-slip to continuous motion is given by

  • equation image

The transitional normal stress increases linearly with the loading velocity. If the normal stress on the frictional interface is larger than the right hand side of equation (9), then the motion of the block will be unstable. If the normal stress is further increased, relative to NT, the instability is enhanced, i.e. longer recurrence times and larger slip events.

[10] In a gouge-filled fault zone, sliding occurs on many surfaces, and the individual grains may roll, reducing the amount of frictional sliding. However, the transition described by equation (9) will still occur. To investigate the behavior in the more complex system, we turn to numerical simulations.

3. Discrete Numerical Model of a Granular Layer

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

3.1. Discrete Element Method

[11] We use a two-dimensional discrete element method [cf. Cundall and Strack, 1979] to simulate the behavior of a simplified layer of grains. This method treats individual grains as inelastic disks which undergo linear and rotational accelerations due to grain interactions and external forces. The mathematical model and numerical techniques used are very similar to those described in our two previous papers [Aharonov and Sparks, 1999, 2002]. The separation of the centers of two disks, i and j, is described by the vector rij. When the distance between the centers is less than the sum of the radii, Ri + Rj (i.e., the grains overlap), an interaction force is exerted on each grain at the point of contact. The grains are noncohesive, so this force vanishes when the overlap reaches zero. The contact force has shear, Fs, and normal, Fn, components. The normal component consists of a linear elastic repulsive force and a damping force dependent on the relative grain velocities, equation imageij:

  • equation image

Here equation image = (rij · equation image, rij · equation image)/rij is the unit vector parallel to the contact; kn is the normal spring stiffness, mij is the harmonic mean of the two grain masses, and γ is a damping coefficient. The spring stiffness is taken to vary inversely with grain size,

  • equation image

where equation image is the average grain diameter in the system. This variable stiffness is introduced so that grains of all sizes obey a stress-strain relationship with uniform stiffness.

[12] Shear forces on contacts are determined using an elastic/friction law [Cundall and Strack, 1979]

  • equation image

where equation image = (rij · equation image, −rij · equation image)/rij is the unit vector tangent to the contact, μ is the surface friction coefficient, ks is the shear stiffness, and Δs is the shear displacement since the formation of the contact. Equation (12) implies that relative motion of the grains in the tangential direction at the contact point is initially opposed by an linear elastic force, which increases with sliding until it reaches a maximum value of μFijn. The two spring constants, kn and ks, are chosen so that the overlap of the grains and the elastic sliding distance will be small compared to the grain size.

[13] Energy is dissipated through two mechanisms. The last term in equation (10) introduces a viscous damping that ensures that collisions between grains are inelastic. In the simulations presented here, most of the dissipated energy is lost during frictional sliding between grains. The coefficient of surface friction used in this study was held fixed at μ = 0.5.

[14] We nondimensionalize this system to facilitate computation and to the highlight the important interplay of physical processes. Distance is scaled by the average grain diameter, equation image, stress by Young's modulus, E, and mass by the mass of a grain with the average diameter, equation image. We also introduce characteristic scales for force, F0, spring stiffness, k0, and time, t0:

  • equation image

The choice for the timescale was guided by the vibrational period of a purely elastic interaction between two characteristic grains. Velocities are scaled by equation image/t0, approximately equivalent to the acoustic wave speed in a grain. The resulting force equations, with dimensionless primed variables, are

  • equation image
  • equation image

The dimensionless damping parameter is Γ = γequation image/Eequation image2. The simulations presented here are heavily damped, with Γ = 1; this corresponds to a restitution coefficient of about 0.3 for a collision of two grains of diameter equation image (see Aharonov and Sparks [1999] for details); ks′ is the ratio ks/kn. The value of this stiffness ratio is taken to be 0.5, within the range of values that give reasonable agreement with particle collision experiments [Schafer et al., 1996].

[15] We limit the applied nondimensional confining stresses to less than 0.01, so that the most highly stressed contacts will have overlaps of only 1–2% of a grain diameter. Assuming quartz grains, with a Young's modulus of 60 GPa, our simulations were conducted at confining stresses between 18 and 1800 MPa.

[16] The contact forces on each grain are summed to give the total linear and rotational accelerations of the grains. There is no gravity applied to this system, so there are no body forces. The accelerations are integrated through time to calculate grain position, and linear and rotational velocity, using a “leapfrog”-like Verlet algorithm [Allan and Tildesley, 1987]. Stability and accuracy requires a time step small enough to resolve the evolution of forces during a collision of grains. In the simulations we present, the grain velocities are always a small fraction (<10−2) of the acoustic wave speed, so that motion of a grain during a time step is very small. The exact choice of time step does not affect the outcome of the runs, but many time steps (typically 105–107) are required to do significant rearrangement of the grains. As a result, dimensionless loading velocities were kept above 10−4, and the total shear strain achieved in the slowest cases was only about 2.

3.2. Model of a Granular Layer

[17] Simulations were performed in roughly square systems with about 550 (24 × 24) grains. For simplicity, the grains were all of a similar size. Since a uniform grain size will introduce ordering effects, the grain diameters were drawn from a Gaussian distribution with a standard deviation of 0.5 equation image, clipped at plus/minus one standard deviation. For convenience we define “horizontal” and “vertical” directions, although there is no gravity vector to orient the system. The top and bottom boundaries were composed of grains of different sizes glued together along their centerlines to form rigid rough walls. The system is periodic in the horizontal direction, analogous to a laboratory rotary shear apparatus [Mandl et al., 1977; Savage and Sayed, 1984; Hanes and Inman, 1985; Beeler et al., 1996]. The systems were initiated as tall loosely packed boxes, which were compacted vertically by gradually applying normal stress to the top walls. A typical compressed system of grains is shown in Figure 1. Because the system is periodic in the horizontal direction, vertical stresses applied to the top and bottom walls create a confining pressure, N, that is isotropic if averaged over the whole system. The bottom wall was held fixed, while a horizontal force, Q(t), was applied to the top wall, introducing right lateral shear stress. This force mimics that of a compliant spring (Figure 1), increasing as one end of the spring is pulled, and decreasing as the wall moves:

  • equation image

Vsp is the constant loading velocity on the spring and Vwall is the horizontal velocity of the grains of the top wall. The first term on the right-hand side emulates the buildup of tectonic shear stresses on a natural fault or the applied shear stresses in a laboratory friction experiment. In the models presented here, ksp was taken to be 3 × 10−5k0. As in the sliding block model (equation (9)), the transition between stable and unstable sliding will be controlled by a parameter that combines ksp, Vsp and N. Because there are limits to the computationally accessible ranges of driving velocities and confining stresses, ksp was chosen so that the transition to unstable sliding could be investigated efficiently.

image

Figure 1. Configuration of packed circular grains in a 24 × 24 horizontally periodic granular layer. Shaded circles are grains, and black lines show the orientations of contact normal forces, with the thickness of the lines proportional to the total force on the contact. The top and bottom walls are composed of semicircular grains of different radii, glued together along a line. The top wall is connected to a spring with stiffness ksp, which is pulled at its other end at a constant velocity Vsp. A constant normal stress N is applied to the horizontal boundaries.

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4. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[18] We ran a series of numerical experiments at a range of loading velocities, and normal stresses. To facilitate comparison, each experiment discussed here was run with the same initial packing of grains. Two different regimes of dynamical behavior can be defined by the variations in the horizontal velocity of the top wall, V, (as by Nasuno et al. [1997]): “continuous” motion, where V fluctuates about the loading velocity Vsp; and “stick-slip” motion, in which V is near zero for extended periods of time and reaches very large values during short bursts. The transition between these two behaviors is gradual, so some experiments in which V vanishes momentarily are classified as “oscillatory” [Nasuno et al., 1997]. Figure 2 shows the time series of frictional resistance (measured shear stress on the boundary, normalized by the applied normal stress N) and normalized wall velocity (V/Vsp) for experiments that exhibit these different behaviors. The rest of this section will focus on the characteristics of the stick-slip runs, since these runs exhibit a very interesting and rich behavior emerging from a relatively simple system, and are relevant to the study of earthquakes.

image

Figure 2. (top) Velocity of the top wall, V, and (bottom) shear force measured on the spring versus time for three different numerical experiments, representative of the stick-slip regimes (Vsp = 10−3, N = 10−3), oscillatory motion (Vsp = 10−3, N = 3 × 10−4) and continuous motion (Vsp = 3 × 10−3, N = 3 × 10−4).

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[19] During a stick-slip run, there are prolonged periods when the plate does not move and stress builds up on the wall and the spring, followed by short bursts of motion (“slip events”), where the plate surges forward at velocities one to three orders of magnitude larger than Vsp (Figure 2, top left), and a variable fraction of the stress on the system is relieved. Thus the spring force versus time curve is characterized by a saw-tooth pattern (Figure 2, bottom left). Because each slip event rearranges the grains in the system, the time between slip events (i.e., recurrence time) and the displacement during events can be highly variable. The average displacement and recurrence time increases as the system moves farther into the stick-slip regime. Because of the increasing recurrence time, the number of slip events during the experiments, particularly those away from the transition region, was small. In the runs conducted, the slip displacement can vary from 1 to 10 grain diameters and is typically 4–5.

[20] Figure 3 shows two different measures of the granular packing during a typical stick-slip experiment: Porosity, ϕ, which varies as the wall separation changes; and coordination number, Z, a system average of the number of contacts per grain. Both of these quantities vary strongly during a slip event, with in general higher porosity and lower coordination during the peak of slip. A correlation between shear motion and dilation has also been observed experimentally [e.g., Marone et al., 1990; Geminard et al., 1999] and noted already by Reynolds [1885]. At the end of the slip event, ϕ and Z may take on values higher or lower than their preslip ones, depending on the new configuration of grains. There does not appear to be a critical porosity that allows slip to begin, but instead onset of rapid dilation and slip (i.e., wall displacement) are simultaneous, making it difficult to establish a cause and effect relationship between the two processes.

image

Figure 3. (a) Spring force versus time in typical stick-slip run. (b) Coordination number and porosity versus time during the same run. Coordination number fluctuates more widely than porosity and generally is observed to reduce slowly during stick, while porosity during stick shows only minor changes. Slip is generally accompanied by a large fluctuation both in porosity and coordination numbers, accompanied by coordination number reduction and porosity increase. After slip porosity and coordination number can take on larger or smaller values than before the slip.

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[21] Some “creep” of the wall occurs during “stick” periods, where the top wall moves at a small velocity, <1% of the loading velocity. Creep typically displaces the top wall by less than 0.05 grain diameters, equivalent to a strain of about 10−3. The creep is accompanied by a slow dilation of the layer by as much as 0.01 grain diameters (an increase in porosity of 0.001). Although some of this creeping is due to elastic distortion of the system, which increases as the spring force is increased, a substantial part of the creep is due to irreversible grain rearrangements, which begin to take place when strain exceeds 10−4, as Aharonov and Sparks [1999] showed. Grain rearrangements during the stick period occur by frictional sliding along grain contacts, resulting in an internal release of stress. Figure 4 shows the sum of the shear stresses on all sliding contacts prior to a single representative slip event, plotted as function of time. The observed internal stress release, which is a combined measure of the number of sliding contacts and the stresses on these contacts, accelerates as the slip event approaches. Benioff strain, or the total internal strain [Bufe and Varnes, 1993] on a fault zone, has also been observed to accelerate in a characteristic fashion prior to many major earthquakes [Zöller and Hainzl, 2002; Ben-Zion and Lyakhovsky, 2002], and prior to slip events in granular models [Mora and Place, 2002]. If a linear relation can be assumed between stress and strain then internal stress release would be proportional to Benioff strain. Benioff strain provides a much sought after precursor for major earthquakes, though its usefulness in prediction is not clear yet. Figure 4 also shows that internal stress release, S(t), in our model is well fit by a power law

  • equation image

with m = 0.38; m in our model varies between slip events with a mean value around 0.35, in agreement with values obtained from Benioff strain release in earthquake sequences [Ben-Zion and Lyakhovsky, 2002].

image

Figure 4. Circles are measured internal stress release (sum of shear stress on all sliding contacts) prior to and during the onset of slip. Solid line is wall velocity, V, and dashed line is equation (17) using m = 0.38.

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[22] Following the termination of a slip event, the system usually experiences decaying elastic vibrations, with accompanying decaying oscillation of wall velocity. The fundamental elastic period of the system should be given by Tsys = 2πequation image, where M is the mass per unit area of a vertical column of grains, l is the height of layer, and G is the shear modulus of the whole system. The shear modulus was measured for these systems by observing the relaxation of a very small applied uniform shear [Durian, 1997; Aharonov and Sparks, 1999] and was found to be about 0.15, i.e., about 9 MPa in dimensional form. Using this value, the predicted nondimensional period of elastic oscillations is 390, while the observed oscillation periods range from about 370 to 450. Experiments with larger systems confirm that the oscillation period increases with system size, l.

5. Spring-Block Model Applied to Granular Layers

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[23] The simple spring-block model (SBM) described in section 2 agrees well with several aspects of our numerical models of granular shear zones, even though the SBM only incorporates sliding along a single frictional interface. Using the classification of the top wall motion described in section 4, we can build a phase diagram delineating the different types of motion as a function of two important parameters, confining stress N and loading velocity Vsp (Figure 5). A system that is sliding continuously goes into stick-slip motion if either the loading velocity is decreased or the confining stress is increased, as also seen in earlier simulations [Thompson and Grest, 1991] and experiments [Nasuno et al., 1997].

image

Figure 5. Phase diagram showing the mode of shear deformation in the numerical model of granular layers, as a function of imposed normal stress and loading velocity. The different symbols distinguish the stick-slip and continuous regime and a transitional regime between the two. The line shows the predicted boundary between the stick-slip and continuous regimes from the spring block model: The line on which Tslip = Tr, for an assumed value of Δμeff = 0.1. As conditions move farther from the boundary into the stick-slip regime, slip events become larger and less frequent. As conditions move farther into the continuous regime, the wall velocity becomes more constant.

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[24] The transition between the stick-slip regime and the continuous regime is well fit by a straight line in log N − log Vsp space, as predicted by the SBM (equation (9)). The intercept of this line in the SBM depends on the mass of the block and on Δμ, the difference between the static and dynamic coefficients of friction on the interface. The block mass in SBM corresponds, in the numerical system, to the mass of that part of a vertical column of grains that moves during an event. This varies between slip events, but on average is about 12: the average mass of grains is 1.0, the average layer thickness is about 24, and about half of the layer moves during a typical slip event. In most events, the slip is either localized in the middle of the layer, or uniformly distributed, so that the average velocity of all grains is about half of the wall velocity. The velocity profile is similar to that discussed by Aharonov and Sparks [2002] as the “solid-like” mode of deformation.

[25] In our numerical formulation, we implement a frictional force such that there is only a single value of dynamic friction (chosen here to be 0.5), and Δμ = 0 on the grain contacts. However, the existence of stick-slip motion in the layer implies that the system as a whole may be modeled as having an effective static friction that is larger than the dynamic friction that applies during slip, and thus Δμeff > 0, where the subscript eff refers to effective. It has been noted from other simulations that sheared granular systems can display effective coefficients of friction (shear to normal force ratios) that are significantly different from either the intrinsic static or dynamic friction, attributed to rolling and grain shape [Mora and Place, 1998; Morgan and Boettcher, 1999; Morgan, 1999]. A value of Δμeff ∼ 0.1 yields a line on the phase diagram that provides a good fit for the division between the numerical stick-slip and continuous regimes (Figure 5).

[26] Although the peaks in spring force are highly variable, if we average the maximum shear to normal stress ratio over many events in different experiments, we can estimate the effective static friction coefficient of this system of grains to be about 0.35. We then estimate that the effective dynamic friction coefficient is between 0.2 and 0.3. The average value of the effective friction from the simulations in the continuous motion regime is about 0.29, but may fluctuate from this value by as much as 0.10. All of these values are significantly less than the prescribed surface friction value of 0.5, and are within the range of reported values observed in other numerical studies using circular grains [Morgan, 1999; Mora and Place, 1999], and two-dimensional experiments in rods [Frye and Marone, 2002].

[27] The simple spring-block model also describes other aspects of our numerical experiments fairly well. The predicted dimensionless slip time from equation (6) is in good agreement with the length of slip pulses in the numerical experiments (Figure 6a), using the assumption that the slipping mass includes half of the total weight of the grains in the layer, as well as the mass of the upper block. To check the predicted dependence of slip time on system mass, we conducted a few experiments in which the density of the grains in the top wall was greatly increased, so that they had 154 times their normal mass. This simulates the effect of a large stiff plate attached to the top wall. The increased predicted slip time is again a good match to our simulations (Figure 6b). For both values of the sliding mass, slip times may be slightly longer in experiments with smaller loading velocities. This may indicate that a larger number of grains move in these experiments.

image

Figure 6. Several scaled and superimposed velocity versus time pulses from slip events in numerical experiments with (a) a normal top wall, and slipping mass of 12 grains per column, and (b) a weighted top wall, and slipping mass of 166 grains per column. In each plot, the solid curves are velocity pulses for experiments with Vsp = 10−3, and the dashed curves are for Vsp = 10−4. The bold line is the theoretical velocity pulse, from equation (5), using Δμeff = 0.1, for each slipping mass.

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[28] The slip velocities shown in Figure 6 have been normalized by their maximum velocity for comparison, but actually vary greatly between events (see Figure 2, left). This is usually observed in SBM systems with some finite heterogeneity and complexity [e.g., Burridge and Knopoff, 1967] and in granular systems with heterogeneous grains [Nasuno et al., 1998]. The shape of the scaled velocity pulse from our model is qualitatively similar to the sine wave predicted by the spring-block model (equation (5)), although the pulses in the granular models are more complex, as seen in comparing theory (bold line) to simulation results in Figure 6. The most noticable deviation from the predicted pulse shape is high frequency variations, which are related to grain collisions that occur in the interior and along the rough wall. These collisions result in decelerations that reduce the slip velocity and the overall stress drop. Another difference is the gradual acceleration of the top wall at the beginning of a slip event, as opposed to the discontinuous initiation of slip assumed in the spring-block model. We identify this acceleration with a “commitment to slip” that appears in the system prior to slip, and accelerating internal stress release as seen in Figure 4.

[29] The recurrence times between slip events in a simulation are much more variable than the slip times because the peak stress and the stress drop are dependent on the particular configuration of grains. From Figure 2 it can be seen that time intervals between significant slip events in a given run can vary by as much as a factor of 5, and we also observe that in general the recurrence time is related to the magnitude of stress drop, as in real earthquakes. The average recurrence time agrees quite well with the value predicted by the spring-block model (equation (8)) using a value for Δμeff of 0.075 to 0.1 (Figure 7). Our numerically constrained range in loading velocity, and large variability in recurrence times (shown by error bars representing standard deviations of recurrence times in Figure 7), allows to investigate first-order relations between recurrence times and load velocity, so we conclude that TrV−α, where α = 1 ± 0.05, in agreement with the slider block models, experiments and measurements from real earthquakes [Karner and Marone, 2000; Beeler et al., 2001].

image

Figure 7. Mean recurrence times in six different stick-slip simulations plotted against normal stress divided by loading velocity and spring stiffness. Dashed and dotted lines are theoretical predictions, following equation (8), using Δμeff = 0.1 and Δμeff = 0.075, respectively. Error bars are calculated from standard deviations of recurrence times during runs.

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6. Source of Apparent Static and Dynamic Friction Values

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[30] Since our model subscribes a constant value of surface friction, the effective static and dynamic frictions implied by the fit to the stick-slip SBM must arise from some micromechanical reorganization processes, in which packing geometry affects the strength of the layer. In this section we will discuss the reason for variable strength and the progressive loss of strength that precedes the onset of slip.

[31] The majority of contacts between grains carry forces smaller than the mean force, shown by the histograms in Figure 8. However, the distribution of contact forces are characterized by two subsets, that are different for the smaller (smaller than the mean) forces and larger forces. The probability of finding a force within a particular range is nearly independent of the magnitude of the force, for forces less than the mean. For forces greater than the mean, the probability density function, P(f), decreases exponentially with the magnitude of the force. P(f) ∼ exp(−βf), where β ≈ 1.49. This type of bimodal distribution, with a change in behavior around the mean force, is found both in simulations and in experiments under a variety of conditions, both static and flowing [e.g., Radjai et al., 1996; Liu et al., 1995; Aharonov and Sparks, 2002].

image

Figure 8. Histograms of contact forces, P(f), where f is force on a contact scaled by the mean force in the system, and P(f) is the probability density function. Three contact force distributions are shown, one measured just prior to slip, the second measured during slip, and the third during stick. All distributions show that forces with f < 1 obey a different distribution than forces with f > 1. The dashed line is an exponential function with slope = −1.49.

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[32] The two subsets of the grain contacts in this system show a distinct behavior in a stick-slip run: forces larger than the mean (f > 1) usually belong to force chains, which transmit large forces from one boundary to another. Contacts that carry forces smaller than the mean (f < 1) can be better described as interchain contacts, which transmit smaller forces between chains. In Figure 9 we look separately at sliding of contacts from the two populations: Contacts carrying f > 1 and f < 1 forces, termed “strong” and “weak” contacts, respectively. In preparation for large-scale slip the weak contacts are observed to start sliding with the number of weak contacts slipping increasing rapidly prior and during the initiation of slip. Sliding of the weak contacts is also responsible for accelerating internal stress release (Figure 4), since as demonstrated in Figure 9, at most 0.1% of the strong contacts are sliding.

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Figure 9. (top) Spring behavior during a stick-slip run. (bottom) Fraction of sliding contacts during the same run measured from two populations of contacts: Contacts that carry forces larger (strong) and those that carry forces smaller (weak) then the mean. Observe that mostly weak contacts slide and the fraction of sliding contacts accelerates prior to a slip event, even though the wall of the system is stuck. Less than a percent of strong contacts start sliding as slip approaches, and even during slip their fraction remains low. The dashed lines fits exponential acceleration in fraction of sliding contacts.

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[33] The two subsets of forces, the slipping weak forces and the more or less stationary strong forces, have also different mean orientations. Figure 10 shows a series of rose diagrams (i.e., polar histograms) of contact orientations at several times during a stick-slip run. The subset of large contact forces (defined here to be those with a force at least 1.25 times the average contact force) is plotted separately from the smallest contact forces (with a force less than 0.75 times the average contact force). The large-force contacts, which can be identified with force chains, are highly anisotropic, peaked strongly around the direction of maximum principle compression σ1, at ∼135°, as seen in the lower row of rose diagrams in Figure 10. This is true even throughout slip events, even though individual force chains are being rapidly formed and destroyed by shear. However, the small force contacts (interchain forces) have a different preferred orientation that changes significantly through the stick-slip cycle. During the buildup of shear stress in the stick phase, the orientation of interchain contacts becomes increasingly anisotropic, but in a direction perpendicular to the force chains, i.e., peaked around 45°, parallel to the minimum compressive direction σ3, as seen in the upper row of rose diagrams. This preferred alignment has been seen in other numerical models of systems being deformed in continuous shear [Radjai et al., 1999; Aharonov and Sparks, 2002]. As internal slip accelerates two conjugate directions develop in the weak set, with orientation around 75° and 15°, in agreement with expectations from Reidel shear directions [Scholz, 2002]. Conjugate sets can be seen in both the weak and strong contacts during most of the cycle, but are most pronounced in weak contacts before slip. During the peak velocity, i.e., large-scale slip, the orientation of the small forces is randomized. The horizontal preferred orientation that sometimes appears arises from collisional forces between rapidly moving grains. The effective frictional resistance of the layer (calculated from the shear resistance force measured on the wall minus the accelerations of the system) decreases steadily throughout the slip event.

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Figure 10. Polar histogram of contact directions (rose diagrams) during a stick-slip cycle. The top row shows rose diagrams of contacts carrying forces smaller than the mean. The bottom row shows rose diagrams of contacts carrying forces larger than the mean. Letters indicate the time rose diagrams were calculated during the stick-slip cycle.

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[34] The orientation of weak forces plays a crucial role in controlling granular layer strength. Weak contacts, when oriented at 45°, perpendicular to stress chains at 135°, act to connect and brace different force chains, and to prevent rolling of grains within the chains. This is the configuration of the system during a stick period: although many of the weak contacts are oriented randomly, those that are oriented in the σ3 direction act as a “confining stress network”. Once enough weak contacts are sliding, as demonstrated in Figures 4 and 9, the support for the stress chains is lost. The system then transitions to a weaker mode, characterized by a lower effective dynamic friction. A cartoon of this concept is given in Figure 11, showing the two different configurations: Figure 11a is a strong configuration, during stick phase. This configuration has the higher static friction. Figure 11b is the contact distribution during large-scale slip, after much internal slip caused redistribution of weak contacts. This configuration lacks the organized support structure in the perpendicular direction to stress chains. It is characterized by a lower resistance to slip (low dynamic friction), due to the ability of stress chains to rotate and buckle. Thus the network of weak contacts plays a major role in determining the system strength and stability.

image

Figure 11. Cartoon illustrating how grain contacts in a granular system change during the stick-slip cycle. Black lines represent contacts carrying strong forces, which are always oriented in the maximum compressive direction, σ1. Grey lines represent weak contacts. (a) During the stick phase, weak contacts are mostly oriented in the minimum compressive directions σ3 ∼ 45°. (b) During and prior to slip the weak contacts lose this preferred orientation and are randomly distributed, although the stress chains remain at the maximum compressive direction, σ1 ∼ 135°.

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[35] The sequence of images in Figure 12 shows how sliding on weak contacts leads to buckling of stress chains and rolling of grains, followed by large-scale rearrangements. A mechanism of failure via buckling and rolling is in agreement with previous observations [Cundall and Strack, 1983; Morgan and Boettcher, 1999]. Grain rolling, instead of sliding, may also provide an explanation for the low apparent friction we and others have measured both in granular models [Scott, 1996; Mora and Place, 1998, 1999] and in experiments performed with round grains [Mair et al., 2002] and rods [Frye and Marone, 2002].

image

Figure 12. Closeup of a region of slipping contacts at the onset of slip. (top left) Wall velocity. (a–e) progressive snapshots. Grains are drawn as faint gray circles. Dark lines represent forces on grain contacts, with thickness increasing with force. Slipping contacts are black, while stationary contacts are gray. White lines within grains are angle indicators, to be used for observing grain rotation. Figures 12a and 12b show slipping of weak force contacts surrounding a stress-supporting chain. Figure 12c shows buckling of this chain by slip on a strong force (heavy line) contact, and rolling of stress-supporting grains. Figures 12d and 12e demonstrate that this buckling allowed major stress rearrangement to occur in the area and reformation of new stress-supporting paths.

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7. Stick Slip in Actual Gouge Zones

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[36] The simplification of using two-dimensional circular disks in place of three-dimensional angular grains makes the problem tractable, but does affect the behavior of the system. One documented effect of round grains is a reduction in the effective friction due to enhanced rolling [Mora and Place, 1998, 1999; Morgan and Boettcher, 1999]. Laboratory experiments that compare circular rods with spherical beads suggest that the effective friction also increases with dimensionality [Frye and Marone, 2002]. It was also recently discovered that stick slip is more easily induced in spherical grains, than in broken and angular shaped ones [Mair et al., 2002]. This shape effect may be due to larger spatial heterogeneity of stress, and the large role of stress chains in spherical grains, although this suggestion is merely a speculation, as nearly no works address the role of angularity.

[37] An additional factor that may influence the resistance to shear is the grain-size distribution. Though our numerical runs show little sensitivity to moderate variations in grain size distributions, a system with significantly wider range of grain radii, when the smallest grains are smaller than 1/6 of the largest grains, is expected to exhibit a different effective friction coefficient [Morgan, 1999]. Such a wide grain size distribution allows small grains to fill in holes between large grains, increasing the coordination number and thus changing the mechanics slightly [Bideau and Troadec, 1984].

[38] Since the effective friction would depend on the particular shape, size distribution and surface characteristics of the grains, different systems under the same applied conditions could exhibit different sliding behaviors. The experimentally observed transition of a gouge system from stable sliding to unstable stick-slip motion with increased displacement [Marone, 1998] may indicate that the change in grain size distribution due to comminution either increases the difference between the bulk static and dynamic coefficients of friction, or decreases the mass of the system involved in the sliding. The latter may occur due to a transition from distributed deformation to localized boundary shears, which is frequently observed in gouge experiments [e.g., Mair and Marone, 1999]. Friction in distributed deformation may be strongly affected by geometry and configuration of grains. However, if sliding occurs on a localized surface, chemical healing between surfaces and time-dependent surface friction (for review see [Scholz, 2002]) possibly overtakes the geometrical and configurational effects on friction.

[39] Our results suggest that apparent friction of a granular layer is a combination of geometrical affects of grain contact rearrangements and the actual form of grain surface friction, which most likely depends on the sliding rate and age of the contact. In the simulations presented here we held the coefficient of surface friction fixed at a single value (0.5) to investigate how time-dependent shear behavior arises even from simple forms of the frictional resistance, just due to geometrical rearrangements of contacts. The relative affects and interplay between complex forms of surface friction (e.g., rate- and state-dependent friction [Ruina, 1983]), and rearrangements of grain configurations, is an important direction for future study. The time and length scales controlling the sliding process will be determined by this interplay.

8. Summary

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[40] In this paper we have investigated the stability of shear motion between two plates, where the gap is filled with circular grains, and the forces and rearrangements of the grains control the frictional behavior of the system. The purpose of this simplified system is to shed light on the processes involved in stick-slip deformation in granular gouge layers, and to investigate what happens prior to and during slip events. We have demonstrated that a deformation of a simulated gouge layer has similarities to the sliding of a single block on a frictional surface. The transition from continuous sliding to stick-slip motion that is predicted by the spring-block theory matches the observations of our simulations implying the existence of different effective static and dynamic coefficients of friction, even though the frictional sliding laws imposed on each grain have only a single-valued friction. The phase diagram derived from the simulations (Figure 5) therefore may be used as a rough guide for choosing experimental conditions (confining stress, machine stiffness, loading velocity) to produce a desired mode of deformation (continuous sliding or stick slip).

[41] During stick-slip motion especially interesting behavior emerges: The granular layer strength changes according to grain configurations. Strain-dependent strength of the system arises due to the existence of a bimodel distribution of grain contacts: one set of contacts, oriented in the maximum compressive stress direction, supports most of the load. The other set of contacts, with preferred orientation during the stick period in a direction parallel to the minimum stress direction, braces the stress-supporting contacts against buckling, and determines the strength of the system. When this set of contacts starts to slide, precursors to large-scale slip are seen, including accelerating stress release and growth in the numbers of sliding contacts, reminiscent of acceleration of deformation prior to many large earthquakes [Ben-Zion and Lyakhovsky, 2002]. Such accelerating deformation was also found in granular dynamic simulations by Mora and Place [2002] and experimental deformation of granular layers [Nasuno et al., 1998].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References

[42] This work was funded by NSF grant EAR-9804855 and ISF grant 3794. We wish to thank C. Scholz for helpful discussions. E.A. would especially like to thank G. Ziv for preparing Figure 9 and for very stimulating discussions. E.A. is the incumbent of the Anna and Maurice Boukstein Career Development Chair.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spring-Block Model of Frictional Sliding
  5. 3. Discrete Numerical Model of a Granular Layer
  6. 4. Results
  7. 5. Spring-Block Model Applied to Granular Layers
  8. 6. Source of Apparent Static and Dynamic Friction Values
  9. 7. Stick Slip in Actual Gouge Zones
  10. 8. Summary
  11. Acknowledgments
  12. References