The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

4.05 The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle C Marone and DM Saffer, The Pennsylvania State University, Univ...

2MB Sizes 0 Downloads 3 Views

Recommend Documents

Instability of buildings during seismic response
The issue of gravity induced instability during response to severe seismic excitation is examined. It is contended that

Relationships between along-fault heterogeneous normal stress and fault slip patterns during the seismic cycle: Insights from a strike-slip fault laboratory model
•The seismic cycle on a strike-slip fault is simulated using a new experimental model.•The along fault normal stress loa

The effect of chalone on the cell cycle in the epidermis during wound healing
A cut was made on the ear conch of mouse and an extract containing epidermal chalone was injected subcutaneously 2 days

Laboratory observations of transient frictional slip in rock–analog materials at co-seismic slip rates and rapid changes in normal stress
Knowledge of frictional (shear) resistance and its dependency on slip distance, slip velocity, normal stress, and surfac

Stick–slip instability of decelerative sliding
Considering different friction laws the stability of decelerative sliding motions of a driven mechanical system is inves

Crustal dynamics and active fault mechanics during subduction erosion. Application of frictional wedge analysis on to the North Chilean Forearc
The forearc region of the non-accreting South American Plate margin in northern Chile is characterised by subduction ero

Strain-rate effect on frictional strength and the slip nucleation process
Kato, N., Yamamoto, K., Yamamoto, H. and Hirasawa, T., 1992. Strain-rate effect on frictional strength and the slip nucl

4.05 The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle C Marone and DM Saffer, The Pennsylvania State University, University Park, PA, USA ã 2015 Elsevier B.V. All rights reserved.

4.05.1 Introduction 4.05.1.1 Application to Faulting 4.05.1.2 Shear Failure, Frictional Healing, and Friction Constitutive Laws 4.05.1.3 Repetitive Frictional Failure and the Upper Stability Transition from Stable to Unstable Sliding 4.05.2 Laboratory Experiments 4.05.2.1 Transducers, Control Electronics, and Data 4.05.2.2 Materials 4.05.2.3 Velocity and Frictional Force Measurements 4.05.3 Laboratory Data 4.05.3.1 Experimental Boundary Conditions and Range of Major Parameters 4.05.3.2 SHS Tests, Creep, and Static Friction 4.05.3.3 Frictional Healing, Creep Consolidation, and Dilation 4.05.3.4 Effect of Slip Velocity on Frictional Healing 4.05.3.5 Creep, Precursory Slip, and Relocalization of Strain Following Healing 4.05.4 Analysis and Discussion 4.05.4.1 Forward Modeling of Frictional Healing 4.05.4.2 Origin of the Loading Rate Effect on Healing 4.05.4.3 Recovery of Rate State Friction Parameters from SHS Tests 4.05.4.4 Microphysical Explanation of Frictional Healing and Rate Dependence 4.05.5 Conclusions Acknowledgments References

4.05.1

Introduction

Stick-slip frictional motion is a common characteristic of a wide variety of mechanical systems, ranging from wood flooring to automobile brakes, polymer films, and plate tectonic faults (Persson, 1998; Scholz, 2002). Most studies of stick-slip friction focus on the dynamic motion during slip because of its importance in generating vibrations, heat, wear, and, in some cases, seismic radiation. However, the processes that govern friction recovery (healing) and the nucleation of instabilities are also fundamental to understanding earthquakes and the seismic cycle. For example, many types of observations exhibit strong signatures of frictional healing, including (1) the depth frequency distribution of seismicity at subduction megathrusts (Figure 1) and (2) the structural evidence of repeated failure – in the form of the deformed margin wedge and faults therein. Geologic and geophysical data record both repeated seismic and aseismic slip on faults and the way faults regain frictional strength after failure. Subduction zones host the world’s largest earthquakes, and it is well known that large earthquakes occur on preexisting, large faults that have experienced multiple cycles of seismic failure (e.g., Scholz, 2002). Moreover, in evaluating the distribution of seismic and aseismic deformation along Earth’s major plate bounding faults, including subduction zones, a key unresolved question centers on processes and rock properties that control the upper transition, at a depth

Treatise on Geophysics, Second Edition

111 113 114 115 115 115 116 116 116 116 118 118 121 121 123 126 127 132 135 135 135 135

of 5–10 km, from aseismic to seismic slip (Figure 1) (e.g., Beeler, 2007; Hyndman et al., 1997; Ikari et al., 2007; Marone and Scholz, 1988; Moore and Saffer, 2001; Saffer and Marone, 2003). To host repeated earthquakes, these structures must exhibit both frictional healing and frictional and elastic properties that allow instabilities to nucleate. In this sense, the processes of stick-slip frictional failure and frictional strength recovery are intimately related. Both the rate dependence of friction that affects sliding stability and nucleation and the recovery of frictional strength are governed by processes that can be described by constitutive equations applicable to the complete cycle of stick-slip failure (e.g., Dieterich, 1992, 1994; Gu et al., 1984; Rice and Ruina, 1983; Tse and Rice, 1986). Much progress has been made in developing friction constitutive laws to describe variations in the frictional strength of faults with slip rate, slip history, contact time, normal stress, and temperature (Angevine et al., 1982; Beeler, 2007; Beeler et al., 1994, 2008; Behnsen and Faulkner, 2012; Blanpied and Tullis, 1986; Boettcher and Marone, 2004; Bos and Spiers, 2000, 2002; Carpenter et al., 2011; Chester, 1994; Chester and Higgs, 1992; den Hartog and Spiers, 2013; den Hartog et al., 2012a,b; Dieterich, 1978, 1979, 1981; Dieterich and Kilgore, 1994, 1996a,b; Fredrich and Evans, 1992; Frye and Marone, 2002; Goldsby and Tullis, 2011; Hong and Marone, 2005; Ikari et al., 2011; Kanagawa et al., 2000; Karner and Marone, 1998; Karner and Marone, 2001; King and Marone, 2012; Linker and Dieterich, 1992;

http://dx.doi.org/10.1016/B978-0-444-53802-4.00092-0

111

112

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Offshore Nicoya, Costa Rica Microseismicity 1999–2001 0

Depth (km below seafloor)

Frontal prism

Slope sediments

Newman et al. ( 2002)

Margin wedge ic

10

Aseism

genic eismo

S 20

30

0

t

n crus

s ocea

Igneou

VE = ∼1.5X ∼20 km

20 40 60 80 100 120

# of events

Figure 1 Cross section showing basic structural features of the subduction zone megathrust at Costa Rica. The downgoing slab includes igneous ocean crust overlain by marine sediments; major faults include the de´collement (subduction plate boundary) and a series of faults in the upper plate margin wedge. Left panel shows histogram of seismicity from Costa Rica (Newman et al., 2002). Dashed line shows the upper stability transition from aseismic to seismogenic fault behavior, defined as the depth below which 90% of the seismicity occurs.

Marone, 1998a,b; Marone and Saffer, 2007; Marone et al., 2009; Muhuri et al., 2003; Niemeijer and Spiers, 2005, 2006, 2007; Niemeijer et al., 2008, 2010; Ohnaka et al., 1997; Olsen et al., 1998; Paterson, 1995; Perrin et al., 1995; Persson, 1998; Persson, 1998; Reinen et al., 1994; Renard et al., 2012; Rice, 1983; Rice, 1993; Richardson and Marone, 1999; Rubinstein et al., 2004; Rubin, 2008; Ruina, 1983; Saffer and Marone, 2003; Saffer et al., 2001; Scholz, 1998; Segall and Rice, 1995; Shimamoto and Logan, 1981a,b; Tenthorey and Cox, 2006; Tenthorey et al., 2003; Tesei et al., 2012; Tullis, 1988; Yasuhara et al., 2003, 2005; Zheng and Elsworth, 2012, 2013). In particular, the rate dependence of friction and its link to potential unstable sliding have been explored extensively through laboratory experiments, for both natural and synthetic fault materials (e.g., Brown et al., 2003; Carpenter et al., 2011, 2012; Ikari et al., 2007, 2009, 2011; Saffer and Marone, 2003). In contrast, frictional healing has received relatively less attention. Notwithstanding the considerable progress made in laboratory experiments to illuminate the mechanics of frictional aging (e.g., Dieterich and Kilgore, 1994, 1996b), and in particular recent work at the atomic scale (e.g., Li et al., 2011), we have only a rudimentary understanding of the underlying processes and rates of frictional healing in fault zones. For example, in the context of the simplest friction law, involving only static and kinetic friction, the question arises: how is static friction reset between slip events? For some materials, an understanding is beginning to emerge based on plastic deformation and adhesion at highly stressed contact points (BenDavid and Fineberg, 2011; Ben-David et al., 2010a,b; Bhushan et al., 1995; Dieterich and Kilgore, 1994, 1996b; He et al., 1999; Johnson, 1996, 1997; Li et al., 2011; McLaskey et al., 2012). However, for fault rocks and granular materials, the answer is less clear. Is fault healing a temporally continuous process, or does it operate only below a threshold slip velocity? Do depth variations in frictional healing processes control the updip limit of the seismogenic zone (e.g., Figure 1)? How do the properties and compositions of mineral surfaces, pore fluids, and fault gouge affect frictional healing? What role, if any, does healing play in determining the nature of earthquake rupture propagation (e.g., Liu, 2013; Marone, 1998b; Marone et al., 1995; Obara, 2002; Peng and Gomberg;, 2010; Peng

et al., 2005; Rubin and Ampuero, 2005; Shelly et al., 2007; Tadokoro and Ando, 2002; Vidale et al., 1994; Zheng and Rice, 1998)? These are complex issues involving a large number of coupled processes, but a better grasp of the mechanics of frictional healing is an important step toward understanding the seismic cycle and the spatiotemporal character of the seismogenic zone. We focus here on the nature of repetitive stick-slip instability, frictional healing, and the nature of the stability transition from seismic to aseismic deformation in tectonic fault zones, through a review of laboratory experimental results coupled with numerical simulations. We draw in particular from direct shear experiments on simulated fault gouge at normal stresses appropriate for the upper part of the seismogenic zone and for shearing velocities appropriate for earthquake nucleation, in the range of tens to hundreds of micron/s (Figure 2). This chapter is in part a review and in part a presentation of new results. The review is extensive but not fully comprehensive, due to time and space constraints, and because we draw more heavily on our own work than is probably fully justified. The primary purposes of this chapter are (1) to explore the nature of frictional restrengthening and fault healing, with particular emphasis on laboratory experiments focused on the effect of loading rate on frictional healing, (2) to evaluate the role of consolidation and lithification for effecting changes in elastic properties of subduction zone and, in general, fault zone materials, and (3) to investigate the possible relationships between elastic stiffness, frictional healing, and frictional stability. The latter issue in particular has received relatively little attention, compared to other possibilities, as an explanation for the updip limit of the seismogenic zone. We are able to more rigorously consider this explanation using new friction data and measurements of elastic parameters from recent field studies. In particular, the expected changes in frictional behavior and elastic stiffness inferred from laboratory experiments on relevant materials are consistent with the observed transition from stable, aseismic deformation to unstable, seismic slip at depths of 5–10 km (Figures 1 and 3). In the context of Coulomb failure including cohesion and friction (or internal friction), time-dependent strengthening can occur via increased friction or cohesion. Existing works

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

113

0.8 2.5 Velocity-step tests

0.6

Slide-hold-slide tests

0.5 2.0 0.4 0.3 Layer thickness

0.2

Layer thickness (mm)

Shear stress/normal stress

0.7

1.5

0.1 m074

0.0

0

5 10 15 Load point displacement (mm)

20

Figure 2 Friction and layer thickness data from a representative experiment. Strain hardening occurs upon initial shearing and is terminated by removing and reapplying shear load. Velocity step tests are used to assess the progression toward a steady-state friction behavior. Three sets of slide-hold-slide (SHS) tests are shown. Layer thickness data show rapid compaction during initial strain hardening and a progressive decrease in compaction rate with continued shear. The roughly linear rate of thinning reached after 10 mm of slip corresponds to geometric thinning of the layer. Vertical line at zero displacement shows compaction associated with the application of normal load (from 0 to 25 MPa).

address these issues to some extent (Karner et al., 1997; Marone et al., 1992; Muhuri et al., 2003), and additional work is needed. We focus here on the normal stress-dependent part of the shear strength, which is the frictional part.

Effective K r = 50 m 2

4.05.1.1

Effective K r = 10 m K > Kc stable

Depth (km)

4

Stability transition 6 K < Kc unstable

Kc 8

10

0

200

400

600

800

K (MPa m-1) Figure 3 Plot showing K and Kc as a function of depth along the plate interface for a subduction zone example. Rate dependence of friction (b  a) is defined by fitting the data of Ikari et al. (2007); Dc is fixed at 30 mm. Effective normal stress is computed assuming pore pressure is 85% of the lithostatic stress (e.g., Davis et al., 1983). The effective stiffness is computed for a circular slip patch of radius 10 and 50 m, respectively, embedded in an elastic half-space. For this range of parameters, a transition from stable to unstable slip would occur where K < Kc, at a depth of 4–6 km.

Application to Faulting

Seismogenic fault zones are expected to be spatiotemporally complex regions, hosting a range of mineral types, fracture geometries, and fluids as well as a spectrum of slip velocities and strain rates ranging from dynamic to interseismic (Beroza and Ide, 2011; Ide et al., 2007; Figure 1). Field observations show that faults contain granular wear materials, referred to as fault gouge, and that these zones can be up to hundreds of meters wide (e.g., Chester and Chester, 1998; Li and Vidale, 2001; Li et al., 1998; Montgomery and Jones, 1992; Unsworth et al., 1997). Ideally, this range of complexity would be included in laboratory experiments designed to study fault zone rheology. However, our current understanding of friction falls well short of that necessary to incorporate the full range of expected complexity within natural fault zones. One problem is that of the sensitivity of slip stability to small variations in frictional properties (Rice and Ruina, 1983). The stability of frictional sliding is often determined by subtle variations of friction representing only a small fraction of the total frictional force, and this appears to apply to a wide range of materials (e.g., Baumberger et al., 1994; Beeler, 2007; Collettini et al., 2011; Dieterich and Kilgore, 1994; McLaskey et al., 2012; Persson, 1998; Scholz, 2002). Previous work shows that all of the following variables have significant effects on the stability of fault slip: particle size distribution of gouge, particle comminution during shear, shear localization and fabric development, clay content and mineralogy, roughness of the bounding surfaces, net slip, porosity, fluid chemistry, temperature, effective normal stress, and absolute slip velocity (e.g., Beeler et al., 1996; Boulton et al., 2012; den Hartog et al., 2012a,b; Di Toro et al., 2006,

114

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

2011; Dieterich and Kilgore, 1994; Marone, 1998a; Niemeijer and Spiers, 2005, 2006; Niemeijer et al., 2011; Noda and Shimamoto, 2009; Saffer et al., 2001; Tesei et al., 2012). Tackling all of these in a single study has proven challenging. Instead, laboratory investigations have generally focused on isolating the effects of a few variables.

4.05.1.2 Shear Failure, Frictional Healing, and Friction Constitutive Laws Measurements of frictional aging have been available for over 200 years, since at least the time of Coulomb and his predecessors Amontons, Desaguliers, and da Vinci. Coulomb (1785) measured the time dependence of static friction, primarily of lubricated woods and also of metals, and derived a power-law aging relation F ¼ A þ mT n

[1]

where F is friction force, A represents cohesive or adhesive effects, T is time of contact, and m and n are empirical constants (see Dowson, 1979, for a summary). Coulomb’s value of n was 0.2, which for his range of available contact times is consistent with recent works in which aging is generally written in a log form: F ¼ A þ m ln(T ). Coulomb refined his aging relation to account for short- and long-time saturation effects and suggested alternative forms to eqn [1]. In addition, he made a connection between aging and his second empirical friction law, which has been used widely as a brittle failure criteria in studies of faulting (see, e.g., Belardinelli et al., 1999; Baumberger and Caroli, 2006, and references therein): F ¼ A þ mP

[2]

where P is normal force and m is a coefficient of friction. Coulomb noted that the aging relation [1] provided an explanation for the difference between static friction and kinetic friction in eqn [2] and proposed a physical mechanism involving interpenetration of fibrous surfaces for time-dependent static friction. The distinction between static friction and kinetic friction defines the simplest friction law. However, as Coulomb and many subsequent workers have noted, it involves a nonphysical singularity upon the initiation of sliding. Rabinowicz (1951, 1956, 1958) resolved this problem and inspired the next generation of friction laws by showing that friction evolves over a characteristic sliding distance Dc, which he related to physical characteristics of asperity contacts. Rabinowicz’s insights became formulated as a slip-weakening friction law (e.g., Ida, 1972; Palmer and Rice, 1973). However, slip-weakening friction only complicates matters regarding frictional healing. That is, now in addition to asking how static friction is reset following failure, we may ask how the characteristic sliding distance is reset. Is Dc a property of the static or the dynamic contact, and is it therefore expected to vary with either time or sliding velocity? Moreover, once we admit time dependence of static friction and velocity dependence of kinetic friction, how, if at all, do these effects relate to resetting of static frictional strength following failure or to the onset or arrest of unstable slip? The rate and state friction (RSF) laws were developed to address these issues (Dieterich, 1972, 1978, 1979; Rice and

Ruina, 1983; Ruina, 1980, 1983). In the context of the RSF laws, time-dependent frictional healing and slip-dependent frictional healing are manifestations of the same processes that lead to velocity-dependent kinetic friction and memory effects. There are two distinct views of healing (see, e.g., Bayart et al., 2006; Beeler et al., 1994; Dieterich and Kilgore, 1994; Karner and Marone, 1998, 2001; Linker and Dieterich, 1992; Marone, 1998a; Ruina, 1983). In one, which we refer to as the Dieterich law, healing is fundamentally a time-dependent process involving growth of asperity contact area (e.g., Dieterich and Kilgore, 1994, 1996a,b) or increase in contact junction strength (Li et al., 2011; Tullis, 1996). In the alternative view, referred to as the Ruina law or slip law, any evolution of the physical surface, including healing, requires slip (e.g., Ruina, 1980, 1983) and the healing rate is zero for true stationary contact. In both views of friction state evolution, frictional contacts have an average age determined by the inverse of sliding velocity divided by Dc, and macroscopic frictional strength increases with contact junction age. However, whereas timedependent healing is apparently straightforward – if one imagines contact growth and strengthening via elastoplastic processes – the mechanisms that might cause slip-dependent healing are less evident. In the Ruina law interpretation, slip leads to both healing and weakening (as discussed more fully in the succeeding text). For multicontact solid surfaces, slip-dependent healing could arise from ploughing or shearenhanced indentation, and in granular materials, it might arise from shear-enhanced consolidation. However, existing experimental evidence is limited and equivocal, with some results supporting the Dieterich law interpretation and others supporting a Ruina law interpretation (Bayart et al., 2006; Beeler et al., 1994; He et al., 1999; Karner and Marone, 1998; Li et al., 2011; Linker and Dieterich, 1992; Losert et al., 2000; Nakatani and Mochizuki, 1996; Richardson and Marone, 1999). We address this issue in the succeeding text with new data and analysis. In previous studies of frictional healing, the effect of loading velocity was not widely appreciated (for exceptions, see Beeler et al., 1994; Johnson, 1981; Kato et al., 1992; Marone, 1998b). Measurements of frictional aging were focused primarily on time-dependent static friction, and they were carried out either in slide-hold-slide (SHS) tests or by bringing two surfaces in contact and waiting for a predetermined time before applying a shear load (e.g., Brockley and Davis, 1968; Coulomb, 1785; Dieterich, 1972). Details of the shearing rate were considered unimportant and not generally reported. However, experimental results show that aging and static friction vary with loading velocity, such that the static friction increases systematically with loading rate (Baumberger et al., 1999; Marone, 1998b). Moreover, in SHS tests, healing varies with shear stress (Berthoud et al., 1999; Karner and Marone, 1998, 2001; Nakatani and Mochizuki, 1996) and degree of granular consolidation (Bos and Spiers, 2002; Niemeijer and Spiers, 2006; Niemeijer et al., 2008, 2010; Richardson and Marone, 1999). Thus, measurements reported without such details are of limited value. Conversely, each of these variables can be usefully exploited as laboratory control parameters to study frictional healing, as was done by Beeler et al. (1994).

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

4.05.1.3 Repetitive Frictional Failure and the Upper Stability Transition from Stable to Unstable Sliding A second prerequisite for repeated stick-slip failure (in addition to frictional healing) is that the fault exhibits rate-weakening friction – that is, friction decreases with increased sliding velocity. In its simplest form, the conditions for nucleation of unstable slip can be described by the force balance in a 1-D elastic system (e.g., Gu et al., 1984): K < Kc ¼

ðb  aÞs0 Dc

[3]

where K is the effective elastic stiffness of the loading system, (b  a) is the friction rate parameter, and s0 is effective normal stress. The RHS of eqn [3] defines the ‘critical stiffness,’ Kc, such that unstable slip is expected if K < Kc (e.g., Gu et al., 1984; Ranjith and Rice, 1999). A great deal of work has focused on defining the frictional rate dependence of natural fault materials and synthetic analogs, with the goal of testing hypotheses proposed to explain the upper transition from shallow aseismic creep to seismogenic, stick-slip behavior along active tectonic faults (e.g., Boulton et al., 2012; Carpenter et al., 2012; Collettini et al., 2009; den Hartog et al., 2012a,b; Ikari et al., 2007, 2009, 2011; Marone and Scholz, 1988; Moore and Saffer, 2001; Saffer and Marone, 2003). Because phyllosilicates are abundant in many fault zones (e.g., Deng and Underwood, 2001; Schleicher et al., 2006; Vrolijk, 1990; Vrolijk and Van der Pluijm, 1999), and commonly exhibit rate-strengthening behavior, these studies have been sufficient to rule out a range of hypothesized compositional controls on the upper stability transition (e.g., Saffer and Marone, 2003). Recent studies have shown that thermally activated deformation mechanisms and/or the onset of quartz deposition, associated with clay transformation and pressure solution, may drive a transition to velocity-weakening behavior at 150  C, coinciding approximately with the upper stability transition in subduction zones (e.g., den Hartog et al., 2013; Ikari et al, 2007). The other elastic and frictional properties in eqn [3] have been less extensively explored but play an equally important role in governing frictional stability. In Figure 3, we provide one example illustrating these combined effects of friction rate dependence, the characteristic slip distance Dc, and loading system stiffness, K. We consider a subduction zone example, parameterized with laboratory-derived friction data from clay and clay–quartz mixtures analogous to natural fault gouge (Ikari et al., 2007), with the effective stiffness of the loading system defined by elastic constants measured for subduction zone sediments over a wide range of stresses (Gettemy and Tobin, 2003; Knuth et al., 2013). In this example, we predict a transition to unstable slip, based on the point at which the local stiffness K around a growing dislocation (e.g., Gu et al., 1984; Roy and Marone, 1996) drops below Kc. This occurs at a depth of 4–6 km – comparable to the depth of the observed upper aseismic–seismic transition (cf. Figure 1) (e.g., Byrne et al., 1988; Hyndman et al., 1997; Moore and Saffer, 2001). The results in Figure 3 show that concomitant changes in elastic stiffness, rate weakening, and Dc associated with consolidation and lithification may play a central role in determining the updip limit of the seismogenic zone (Marone and

115

Scholz, 1988; Saffer and Marone, 2003; Scholz, 1988a,b). Fundamentally, the processes within tectonic faults that drive changes in friction constitutive properties (e.g., rate dependence or Dc) with depth, such as lithification, cementation, mineral transformation, consolidation, or shear localization, are also likely to affect the loading system stiffness and fault healing behavior (cf. Figure 3) (e.g., Ikari et al., 2007; Moore and Saffer, 2001). Thus, three interlinked factors are necessary for effecting the transition from stable aseismic deformation, at shallow depths, to the seismic cycle of repeated frictional failure: (1) changes in frictional properties such that the critical stiffness Kc increases with depth in such a way that the criterion for instability is met (eqn [3]; Figure 3), (2) material stiffening brought about by consolidation of fault gouge and surrounding material, along with reduction of plastic strain, so that elastic stresses can be supported, and (3) the rate of frictional healing that must be sufficient to allow fault strength to increase at a rate that is higher than the rate of stress relaxation by creep and plastic deformation.

4.05.2

Laboratory Experiments

We introduce a limited set of new laboratory data collected in experiments designed to address frictional healing in a broad context. Direct shear experiments have played a key role in studies of fault zone friction. Here, we briefly describe this approach. The double-direct shear geometry (inset to Figure 2) is optimized for a biaxial load frame. This geometry eliminates the need for roller bearings, lubricated surfaces, or thrust bearings, as required for (single) direct shear (Dieterich, 1972; Hoskins et al., 1968). In the double-direct shear geometry, shear is accomplished using a three-block configuration in which a central bock is driven between two stationary side blocks (Dieterich, 1981; Ruina, 1983). The side blocks are shorter than the center block along the slip direction; thus, nominal contact area remains constant with slip.

4.05.2.1

Transducers, Control Electronics, and Data

The servo control system can be operated in either force or displacement feedback. Force measurements are typically made using custom-built, beryllium-copper load cells with active, full-bridge strain gauge networks for temperature compensation. The load cells and processing electronics are typically capable of resolving 1 N (a stress of 0.1 kPa on friction surfaces with nominal contact dimensions of 10 cm  10 cm) over a range of forces up to 1 MN, and the servo control system is capable of maintaining stresses constant to this precision. Displacements are measured with direct current displacement transducers (DCDTs) or linear variable differential transformers (LVDTs) and the primary control transducers are referenced to the loading rams, via fixed brackets (e.g., Karner and Marone, 1998). Thus, in displacement servo control, the position of a load point (the ram face) is controlled. Additional DCDTs are used to measure slip directly across the frictional surfaces in some cases. These are attached via brackets screwed directly to steel forcing blocks or via a spring-loaded clamp for granite. The clamp spans the length of the center block and thus average motion is measured in that case. When steel forcing

116

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

blocks are used, brackets are attached at a single screw point. Typically, the steel is assumed to be rigid in comparison with the gouge layer, which seems reasonable for quasistatic motions. Although DCDTs have unlimited resolution in theory, in practice, resolution is set by the maximum displacement range, processing electronics, and noise. These factors typically limit measurement precision to 0.1 mm in most configurations. Servo control amplifiers may be implemented in analog electronics or via computer control. Digital systems typically involve two analog-to-digital conversions: one to convert the signal from the analog transducer, to be used by the comparator and amplifier, and one to convert the digital output system to analog, to be used by the servo valve (for hydraulics) or stepper motor. Analog systems are typically faster but less flexible than digital systems. For complex load histories, and to control load point velocity, a digital signal and ramp generator may be used. Although the digital signal is typically converted to analog via a 16 or 24 bit converter, it is important that digital step size remains smaller than the resolution of interest. For friction experiments such as described here, load point velocities are implemented as digital ramps of 0.1 mm step size, and data are recorded with either 16 or 24 bit analog-todigital converters and a nominal sampling interval of 10 kHz, averaged to recording rates ranging from 1 Hz to 1 kHz depending on loading velocity.

4.05.2.2

Materials

We focus on laboratory experimental examples using granular quartz available under the product name F-110 from the U.S. Silica. F-110 is a nearly pure quartz sand (99.8% SiO2) with minor amounts of Fe2O3, Al2O3 (<0.1% each), and other oxides. Particles are rounded to subangular and the size distribution is as follows: 8% 53–75 mm, 25% 75–106 mm, 44% 106–150 mm, 18% 150–212 mm, and 4% 150–212 mm, with <1% beyond the upper and lower limits. For the experiments we describe here, gouge layers of uniform initial thickness were prepared with a leveling jig. The layers were handtamped to achieve uniform initial packing. Gouge layers were sheared between two types of surfaces: Westerly granite with roughness produced by sandblasting and steel with corrugations machined perpendicular to the sliding direction. Granite surfaces had rms roughness of 80–100 mm (see Marone and Cox, 1994, for additional details on roughness characteristics for similar surfaces), and steel surfaces had a sawtooth roughness with amplitude of 0.9 mm and a spacing of 1 mm. In both cases, roughness was sufficient to inhibit slip at the boundary. The dimensions of the forcing blocks were as follows: side blocks 10 cm  10 cm  4 cm and center block 15 cm  10 cm  8 cm (Figure 4). Small, lubricated cooper shims were used beneath each side block to accommodate lateral motions. The center block has double thickness to account for the extra shear load it carries in the double-direct shear geometry and to ensure that strains are equal in each of the forcing blocks.

4.05.2.3

Velocity and Frictional Force Measurements

The equation governing position x(t) of the central forcing block in the experimental configuration is

  M€ x ¼ K V lp t  x  F

[4]

where M is the mass of the center block and gouge, K is the apparatus stiffness, Vlp is the load point velocity, and F is the frictional force. In a reference frame fixed to the side blocks, x is the fault position and V ¼ dx/dt is the fault slip velocity. For the conditions of the present experiments, inertia can be neglected and friction force is thus F ¼ K(xl  x), where xl and x represent the load point and fault position, respectively, relative to rest (e.g., inset to Figure 4). Since normal force N is held constant in the experiments, the nondimensional friction force can be written as  F K ¼ xlp  x N N

[5]

The frictional contact area is constant in the experiments; thus, F/N ¼ t/s (where t is shear stress and s is normal stress), and it is convenient to plot the ratio t/s as a function of load point displacement (Figure 2). Because the conditions of present interest center around small perturbations from steady sliding, F is nearly always equal to mN, where m is the coefficient of friction; thus, the force balance for perturbations to sliding reads   dðF=N Þ dm ¼ ¼ k V lp  V dt dt

[6]

where k is the stiffness divided by normal stress (k equals 1  103 mm1 for the experiments reported here). Parts of this development are contrary to simple notions about frictional strength, in which motion is zero until a threshold force is reached (e.g., static friction), but consistent with modern views based on the rate- and state-dependent laws in which macroscopic frictional force obeys a flow law that arises from microscopic thermally activated processes (e.g., Baumberger and Gauthier, 1996; Baumberger et al., 1999; Berthoud et al., 1999; Rice et al., 2001).

4.05.3

Laboratory Data

4.05.3.1 Experimental Boundary Conditions and Range of Major Parameters Experiments are carried out at constant normal stress and a range of shearing rates under room temperature and humidity. Shearing rate was imposed as load point velocity and ranged from 1 to 300 mm s1. For the range of layer thicknesses (1–3 mm), the average shear strain rate ranged from 5  104 to 2  101 s1. At the initiation of each experiment, normal stress was raised to 25 MPa and maintained constant during shear (Figure 2). Dilation and compaction normal to the layers were measured continuously and these values are divided by two to obtain thickness variations for each layer. Initial shearing resulted in significant compaction and strain hardening (Figure 2). Previous works have shown that the effect of strain hardening is reduced by cycling shear load to zero and back to failure (Marone et al., 1990; Shimamoto and Logan, 1981a), and this approach is used commonly. The results shown in Figure 2 are consistent with previous works on quartz, which show that frictional strength evolves systematically with accumulated shear strain due to comminution

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

117

Load point

Coefficient of friction

0.77

Fault surface

0.75

0.73

Slide

0.77

ms

Hold 0

Friction force/normal force =m

Slide

0.71

Dm

500

1000

Time (s)

0.75 Hold

Dmc

0.73

Load point Fault surface 1044 s hold Vs/r = 10 mm s–1

0.71

m080

Reload

(a)

1000

Reload

Time (s)

Slip velocity ( mm s–1) 0

500

0.1 1.0 10.0

Hold Load point Fault surface

0

21

Slip on fault surface (mm)

(b)

21.2

21.2

21.1

21

Reload Hold

21 (c)

21.1 Displacement (mm)

21.1

21.2

Load point displacement (mm)

Figure 4 Details of a single SHS test. Upper left inset illustrates experimental geometry and displacement measurements. Upper right inset shows temporal history. Vl is zero during the hold and 10 mm s1 before and after the hold. (a) Nondimensional friction force, F/N ¼ m, is plotted versus load point displacement (dotted line) and slip measured at the fault surface (solid line). (b) Temporal history of the load point and fault position. Note that V reduces by over 2 orders of magnitude but remains finite during the hold period. (c) Relative motion between the load point and fault. From eqn [7], the two positions track one another when friction force is constant. △m and △mc are the amount of frictional healing and creep relaxation, respectively.

118

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

and shear localization (Frye and Marone, 2002; Karner and Marone, 1998; Mair and Marone, 1999; Marone, 1998a; Marone and Scholz, 1989; Richardson and Marone, 1999; Saffer and Marone, 2003). In any laboratory testing configuration involving direct or pure shear of confined layers, the layers undergo geometric thinning until shear becomes fully localized on boundary parallel shear bands (Rathbun and Marone, 2010, 2013; Scott et al., 1994). Initial compaction and quasilinear geometric thinning are clear in the data in Figure 2. Also, small amounts of quartz are lost from the front and rear edges of the layers if they are not jacketed or otherwise confined (Dieterich, 1981). Thus, the nominal contact area may decrease slightly with slip. Because of this, some investigators use lubricated plates at the front and back of the layers to eliminate loss along these edges (Mair and Marone, 1999). However, postexperiment inspection of the layers (e.g., Haines et al., 2009) indicates that gouge loss is limited to the outer edge of the perimeter (to a depth of about 1 layer thickness), and thus, the change in nominal contact dimension is small. Moreover, minor gouge loss occurs over large displacements relative to those of interest for frictional healing and second-order rate- and state-dependent friction effects. Accordingly, no artifacts of this effect have been reported.

4.05.3.2

SHS Tests, Creep, and Static Friction

SHS tests are commonly used to measure frictional healing and time-dependent aging under an applied shear stress (e.g., Beeler et al., 1994; Dieterich, 1972; Marone, 1998a). The tests measure stressed aging, as distinct from so-called unstressed aging, for which shear stress is zero during the waiting period (e.g., Baumberger et al., 1999; Berthoud and Baumberger, 1998; Berthoud et al., 1999; Bocquet et al., 1998; Ge´minard et al., 1999; Karner and Marone, 1998; Karner et al., 1997; Muhuri et al., 2003). In SHS tests, the hold period begins when Vl is set to zero and ends when loading resumes (Figure 4). Holds initiate during steady-state sliding, and thus, from eqn [6], V ¼ Vl at the initiation of a hold. We vary Vl, but keep the initial sliding velocity equal to the reload velocity. Thus, it is convenient to define a ‘slide-reload’ velocity Vs/r, which equals Vl both before and after a hold (e.g., Beeler et al., 1994). Frictional creep begins immediately upon initiation of a hold, but because of finite apparatus stiffness, fault slip velocity does not reduce to zero. Figure 4 shows detail of the creep and friction coefficient as a function of independent measurements of fault slip and load point displacement during a 1044 s hold. Note that during the hold period, sliding velocity V at the fault surface drops from 10 mm s1 to well below 0.1 mm s1 but remains finite (Figure 4(b)). During the hold period, friction decays logarithmically with time (upper inset to Figure 4). The parameter △mc denotes the amount of creep relaxation during the hold. When reloading begins, fault slip is nearly zero initially, but slip initiates well before the peak in friction. This is consistent with previous works showing precursory slip prior to stick-slip failure (e.g., Scholz et al., 1972) and with the observation that fault slip rate does not reduce to zero (Figure 4). Our data indicate that frictional shear is

continuous, albeit very slow, during hold periods conducted under shear load. The peak friction value is generally taken to define the coefficient of static friction in a SHS test (Marone, 1998a). However, the fault surface is never truly static (e.g., Rabinowicz, 1956). From eqn [6], it is seen that peak friction merely represents the point at which fault slip velocity reattains its initial value (i.e., prior to the hold). Yet, for historical purposes and because it represents a critical yield strength, we measure peak friction and refer to it as the static friction. Upon reloading, as slip continues after the friction peak, V overshoots Vl for a brief period (Figure 4(c)) due to finite apparatus stiffness. This velocity overshoot corresponds to the oscillation in friction after the peak (Figure 4(a)). It is important to note, however, that the maximum velocity during overshoot is only slightly above the loading velocity, 30–40 mm s1, compared to the loading velocity of 10 mm s1. Thus, the slight oscillation in friction force (which is also shown clearly in Figure 5(a) and subsequent figures) does not result in an inertial effect.

4.05.3.3 Dilation

Frictional Healing, Creep Consolidation, and

Figure 5 shows data for two hold times and two loading velocities. Holds were carried out in pairs to assess measurement precision and experimental reproducibility. For a given velocity (Vs/r), the degree of both healing and relaxation increases with hold time, and for a given hold time, both △m and △mc increase with velocity. These data illustrate a principal result of RSF. That is, frictional healing and relaxation vary systematically with both hold time and loading velocity (Kato et al., 1992; Beeler et al., 1994; Marone, 1998a,b). Figure 5 also includes layer thickness measurements, shown as relative changes in porosity, where △f ¼ △h/h and h is layer thickness. Consolidation occurs during creep relaxation and dilation takes place when loading resumes (Figure 5(c)). Creep compaction mimics the time-dependent decay of friction, proceeding logarithmically in time (Beeler and Tullis, 1997; Marone, 1998b). Upon reloading, dilation occurs in two distinct phases. An initial rapid dilation occurs over approximately the same displacement scale as attainment of the friction peak (Figure 5). This is followed by a second slower phase of dilation, which occurs as friction regains its steady-sliding value (Figure 5(c)). The duration of the second phase of dilation is longer for lower Vl, implying that it is driven by displacement rather than time (Figure 5(c)). For example, compare the 100 s holds at 3 and 1 mm s1 in Figure 5(c). Like the friction data, consolidation and dilation associated with healing show a systematic relationship with both hold time and Vs/r. The initial dilation that occurs upon reloading influences frictional strength via the work necessary to expand the sample against the normal stress (e.g., Bishop, 1954; Edmond and Paterson, 1972; Marone, 1991; Marone et al., 1990), and recent works have accounted for this effect when evaluating frictional healing (e.g., Bos and Spiers, 2002; Frye and Marone, 2002; Karner and Marone, 2001; Niemeijer et al, 2008). The effect of loading velocity on healing can be seen in a given experiment or by comparing separate experiments at different Vl (Figure 6). Figure 6 shows data for a series of

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Vs/r = 3 mm s–1

Vs/r = 1 mm s–1

ms

0.65

Dm

0.64 Hold

0.63

H 10 10

Dmc 10 100

100 s

0.62

24.0

R

3 mm s–1

Coefficient of friction

0.65

m044

26.0

1 mm s–1

Dm

0.64

0.63 Hold Reload

0.62

1.0 m m

Porosity change, ´10–3

100

ms

mmin (b)

100

25.0 Load point displacement (mm)

(a)

0.66

10

mmin

1.5

1.0

Df

Layer thickness change

Coefficient of friction

0.66

119

H

0.5

R

2000

2500

(c)

3000

3500

Time (s)

Figure 5 Healing characteristics for two loading velocities in a single experiment. (a) △m is the relative change in static friction and △mc defines relaxation creep. Hold times are given below data. (b) Temporal characteristics of relaxation and reapproach to steady sliding. Note consistency of steady-state friction level for a given Vl. Friction exhibits velocity weakening (sliding friction is lower for higher Vl). (c) Gouge compacts during hold periods and dilates when loading resumes. Changes in layer thickness are shown as porosity change △f ¼ △h/h, where h is layer thickness (scale bar gives △h). Thickness data have been corrected for geometric thinning (e.g., see Figure 2) by removing a linear trend.

SHS tests from two separate experiments at different velocities. Precision of the experimental data and reproducibility are demonstrated. For a given hold time and velocity, healing and relaxation are nearly identical in repeat tests but differ consistently as a function of loading velocity. The degree of healing increases with Vl and, for a given hold time, creep relaxation is greater for higher Vl. The displacement required

to regain steady slip also increases with hold time (e.g., Figure 6(b)). Consistent frictional behavior is observed for a given displacement range; however, details of healing and relaxation evolve with net slip. For example, Richardson and Marone (1999) showed that for a given hold time and loading velocity, healing increases with net shear displacement, while creep

120

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

0.75

Vs/r = 10 μm s-1

1 μm s-1

1 μm s-1

Coefficient of friction

0.74 0.73 0.72 0.71 0.70 10

10 10 30

0.69

30

Hold time (s)

30 100

100 100 m051

0.68 18 Load point displacement (mm)

17 (a)

19

Vs/r = 30 μm s-1

0.76

10 20 μm s-1 μm s-1

Coefficient of friction

0.75 0.74 0.73 0.72 0.71

3 3 10 10

0.70

30

30

300 300

Hold time (s) 0.69 15 (b)

(dcdt offset)

100 100 1000

16

1000

17 Load point displacement (mm)

m074

18

19

Figure 6 Healing characteristics for separate experiments and a range of hold times. (a) Three repeats at each hold time demonstrate experimental reproducibility. Healing and relaxation creep increase systematically with hold time. Note velocity-weakening frictional behavior as demonstrated by the response to a velocity step before and after the holds. (b) Data for loading at 30 mm s1. For a given hold time, healing and relaxation are systematically greater than for Vs/r ¼ 10 mm s1. Note that the ordinate range is the same in both panels.

relaxation and compaction decrease. These trends are evident in the raw data (Figure 7). Figure 7 shows six sets of SHS with hold times ranging from 3 to 1000 s. Comparing the last pair of holds in each set, which are 1000 s, shows that △m is clearly larger and △mc is smaller after a displacement of 30 mm than at 15 mm. The effect is also evident in the aging rate b ¼ △m/ △log10(th), creep rate, and consolidation, which show systematic changes as a function of shear displacement (Figure 8). Figure 8 shows that for a 1000 s hold, healing increases by a factor of 1.5 as shear displacement increases from 13 to 28 mm. Existing data on granular quartz rich gouge show

that healing parameters change nonlinearly with displacement, with most of the effect occurring prior to a shear of 20 mm (e.g., Richardson and Marone, 1999). In an SHS test, creep relaxation occurs until strength meets or exceeds the applied stress. Thus, the displacement effect on healing and △mc is consistent with the notion that the rate of time-dependent restrengthening (e.g., at contact junctions) increases as a function of net slip. This would result in less creep relaxation and greater static frictional strength. Previous works included calibration experiments so that the effects of shear displacement could be removed (Richardson

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

121

Slide-hold-slide tests, Vs/r = 10 μm s-1 0.70

Layer thickness (mm)

Coefficient of friction

1.6

0.65

Friction

Layer thickness

0.60

m107 15

20 25 Load point displacement (mm)

30

1.5

Figure 7 Six identical sets of healing tests showing the effect of cumulative displacement. The first set begins at a displacement of 13 mm and ends at 15 mm. In each set, holds of 3, 10, 30, 100, 300, and 1000 s are implemented in successive pairs. Note that for a given hold time, frictional relaxation and creep compaction decrease with slip, while healing and static friction increase with slip. The origin of the displacement effect is poorly understood but likely related to particle fracture and shear localization. Data from Richardson and Marone (1999).

and Marone, 1999). However, here, we consider data from only a narrow displacement range (e.g., Frye and Marone, 2002). Thus, in the detailed comparisons to follow, we use only data from net shear displacements of 15–19 mm.

presumably due to minor experiment-to-experiment differences in load history or initial conditions, as suggested by previous work on solid granite surfaces and granite gouge (Beeler et al., 1994; Dieterich, 1972).

4.05.3.4

4.05.3.5 Creep, Precursory Slip, and Relocalization of Strain Following Healing

Effect of Slip Velocity on Frictional Healing

Shearing velocity has an important effect on frictional healing. We explored a range of shear slip velocities to investigate the loading rate effect on healing. Figure 9 shows data from five experiments with loading rates from 1 to 100 mm s1. Within the scatter, healing increases linearly with the logarithm of hold time and velocity. The degree of healing increases by about the same amount for a tenfold increase in hold time as for a tenfold increase in loading velocity (Figure 9). Corresponding measurements of creep and compaction show similar trends to the healing data, with both increasing linearly with log hold time (Figure 10). Consolidation data are shown from two experiments, and others show similar trends. Creep during quasistatic holds tends to saturate for higher velocities and hold times. A clear Vl effect on △mc is evident for holds of less than about 300 s (Figure 10). These data show the importance of controlling and reporting loading velocity when carrying out measurements of static friction and aging. The effect of higher loading rates was investigated for Vl up to 300 mm s1. These data indicate that the velocity effect on healing extends to at least that level (Figure 11(a)). Data scatter is significantly lower for a single experiment than for multiple experiments (compare Figures 10 and 11),

The slips that occur during creep relaxation and reloading are important aspects of frictional healing. Figure 12 shows measurements of these parameters. The inset shows details of an example slip history taken from the experiment shown in Figure 6(a) (the example is the second 100 s hold shown there). We define three quantities to characterize slip associated with healing: dh is the slip that occurs during a hold, from the beginning to point 1; dp is the precursory slip during reloading, from points 1 to 2; and ds is a measure of the residual slip necessary to regain steady-sliding friction, from points 1 to 3 (Figure 12). Creep slip dh increases roughly linearly with log hold time for a given velocity. These data mimic frictional creep (Figure 10(a)), which is not surprising given that the slip measurements are derived from measurements of load point displacement and friction force. Independent measurements of fault slip (e.g., Figure 4) show the same relations. For short hold times, dh increases with loading velocity. For holds greater than about 300 s, the velocity effect on dh is negligible (Figure 12(a)). During reloading, precursory slip prior to peak friction ranges from 5 to 20 mm and decreases with increasing loading

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Frictional healing, Δm

122

0.03

0.02

0.01 Vs/r = 10 μm s-1

(a)

Frictional creep, Δmc

0

0.03

0.02

0.01

(b)

0 Shear displacement Creep compaction, μm

13 mm 19 mm 4

28 mm

2

m107

(c)

0 100

101

102 Hold time (s)

103

Figure 8 Systematic variations of (a) healing, (b) creep, and (c) compaction as a function of shear displacement. The first, third, and last sets of SHS tests from Figure 7 are shown; labels correspond to the initiation displacement. For a given hold time, frictional relaxation and compaction decrease with displacement, while healing increases with displacement. Data from Richardson and Marone (1999).

velocity (Figure 12(b)). For shorter hold times, there is a significant velocity effect, such that for a given hold time, more precursory slip occurs at lower velocity. The effect is nonlinear as a function of log-th, and the reduction in dp saturates at about 5 mm before increasing. Clearly, additional data are needed to resolve this issue. Healing also has longer-term effects on friction and reestablishment of steady sliding (e.g., Figure 6). The residual slip necessary to reestablish steady sliding seems to scale with hold time. We quantify these effects with the parameters ds. Ideally,

residual slip would be defined as the slip necessary to regain steady-state friction, for example, the slip at which friction– displacement curves become flat again as in Figure 6. However, this is made difficult by minor variations in the base friction level as a function of accumulated slip and subtle differences in the postyield region of the friction–displacement curves. Thus, we chose to measure the slip at the first friction minimum following the peak (e.g., point 3 in Figure 12 inset). Our measurements indicate that ds increases systematically as a function of hold time and loading velocity for shorter hold

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

1 μm s-1 0.04

123

Vs/r

3 μm s-1 10 μm s-1 30 μm s-1 100 μm s-1

Frictional healing, Dm

0.03

0.02

0.01

m074+ 0 0 10

101

102 Hold time (s)

103

104

Figure 9 Data from multiple experiments showing the effect of loading velocity on frictional healing. Healing △m and, to a lesser extent, healing rate b increase systematically with loading velocity. The data indicate that ‘static’ friction and aging depend on both hold time and loading velocity and that the magnitude of the velocity effect is about equal to that of the effect of th; for example, a factor of 10 change in th produces about the same change in static friction as a factor of 10 increase in Vs/r.

times, below about 100 s, but that it tends to saturate for longer th. Part of this effect is likely related to the way we define of ds and perhaps other measures of residual slip would be useful.

4.05.4

Analysis and Discussion

The slip rate and state variable friction laws were originally developed to model velocity step tests (Dieterich, 1979; Ruina, 1983); however, the RSF laws can also model frictional healing and relaxation. Healing data illuminate different friction characteristics than velocity step tests, because healing generally involves large changes in slip rate and large perturbations from steady-state sliding. There are several advantages of modeling data from different types of tests, including the opportunity to evaluate the internal consistency of constitutive parameters and the RSF laws. For the present discussion, we restrict attention to the RSF laws for constant normal stress and direct interested readers to other works that address the role of variable normal stress (Boettcher and Marone, 2004; Hong and Marone, 2005; Kilgore et al., 2012; Linker and Dieterich, 1992; Richardson and Marone, 1999). We evaluate two forms of the RSF laws. In each case, the coefficient of friction is written in terms of its dependence on slip velocity V and a state variable y that represents physical characteristics of the shearing region (e.g., contact area in solid friction or porosity in granular media):

    v vo y i mðyi , vÞ ¼ mo þ a ln þ b ln Dci vo

[7]

where mo is a constant representing base-level friction at a reference velocity Vo, the parameters a and b are nondimensional empirical constants, and Dc is a characteristic slip distance and the subscripts for y and Dc represent separate evolution processes. Previous works have discussed generalization of eqn [7] for shearing within a region of finite thickness, as in the present experiments, rather than on a single surface (e.g., Marone and Kilgore, 1993; Marone et al., 2009; Sleep et al., 2000). In that case, V is replaced by a strain rate, and eqn [7] is written as     e_ e_ o y m ¼ mo þ a ln þ b ln [8] eo e_ o where e_ ¼ V o =ho (assuming spatially uniform strain), Dc is understood as a critical strain eo ¼ Dc/ho, and ho is the total layer thickness. This form has the advantage of explicitly accounting for the effect of strain rate localization (see Rathbun and Marone, 2010; Sleep et al., 2000). That is, when strain rate is localized in a shear band of thickness h, the macroscopic and experimentally observed (apparent) value of the critical slip distance Dapp is related to Dc for uniform strain by Dapp ¼

hDc ho

[9]

124

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Vs/r 0.04

1 μm s-1 3 μm s-1

Frictional creep, Δmc

10 μm s-1 30 μm s-1 0.03

100 μm s-1

0.02

0.01

m074+ (a)

0 5

Porosity change (´ 10-3)

4

3

2

1 m080, m094 0 100

101

(b)

102 Hold time (s)

103

104

Figure 10 Details of the systematic effect of loading velocity on creep relaxation and compaction. (a) Creep increases with both hold time and loading velocity. The data indicate a tendency of creep relaxation to saturate at longer hold times and higher loading rates. (b) Compaction data are shown for two of the five loading velocities (others show similar trends). Consolidation during holds depends on both hold time and loading velocity in a way that mimics frictional creep.

as discussed by Ruina (1980) and shown experimentally by Marone and Kilgore (1993). To model strain rate localization and to illuminate the relationship between the RSF laws and other rheologies, eqn [8] can be rewritten as a flow law (King and Marone, 2012; Rice et al., 2001)  b=a hm  m i eo o e_ ¼ e_ o [10] exp a e_ o y This formulation makes a clear connection between the RSF laws, thermally activated creep processes, strain localization,

and strain delocalization during healing (e.g., Beeler, 2007; Berthoud et al., 1999; Heslot et al., 1994; King and Marone, 2012; Rice et al., 2001; Ruina, 1980; Sleep et al., 2000). For example, if m is understood as a creep stress s, then eqn [10] is readily rearranged as a logarithmic flow law s  S lnðe_ =_eo Þ, where S  a is a constant equal to kT/scO (kT is energy in temperature units, sc is contact indentation strength, and O is activation volume for the thermally activated process), or a power-law creep relation s  A_en , if the exponent n is small, where A is a constant.

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

125

50

Vs/r 100 μm s-1 300 μm s-1

d h 1 μm s-1

40

d h 10 μm s-1 30

d h ( μm )

Frictional healing, Dm

0.03

0.02

20

0.01 10

m084 m051

0 100

1

10

(a)

2

10 Hold time (s)

103

(a)

0 25

0.05 20

Vs/r 10 μm s-1

15

d p ( μm )

Frictional healing, Dm

0.04

1 μm s-1

0.03

10

0.02 5

0.01 (b)

2

102

103

104

μ

105

Figure 11 Healing data obtained from single experiments, for comparison of the data collections shown in Figure 9. (a) The velocity effect on healing extends to at least 300 mm s1. (b) Detailed comparison of two loading rates and hold times. The roughly log-linear relation between frictional healing and hold time seems to extend to 104 s.

Friction state evolution has been described by two forms:   dy Vy ¼1 Dieterich law : [11] dt Dc   dy Vy Vy ln ¼ [12] Ruina law : dt Dc Dc We focus here on the simpler forms of the RSF laws using macroscopic velocity, rather than strain rate. An important question is that of whether the first-order aspects of friction can be modeled with the RSF laws and, if so, whether the data distinguish between the two evolution laws [11] and [12]. There are two ways to address this question. One involves fitting the time series of a given SHS test (e.g., such as in Figure 5 or 6). The other (e.g., Beeler et al., 1994) is to fit the aggregate healing and relaxation data for a series of tests (e.g., such as given in Figures 10 and 11). We begin with the latter approach and address the former in the context of an analysis

3

1

60

Hold time (s)

dh 0

50

100

Slip (μm)

50

d s ( μm )

(b)

101

m051

0 70

m051

0 100

dp

Hold begins

40

30

m051

(c)

20 100

10

1

2

10

3

10

4

10

105

Hold time (s)

Figure 12 Details of slip characteristics during SHS tests. Inset shows representative friction-slip data set; hold begins at zero slip. dh is the slip that occurs during the hold, up to point 1. dp is the slip that occurs prior to peak friction, from points 1 to 2, and is referred to as precursory slip. ds is a measure of the slip necessary to reach the original value of sliding friction, from points 1 to 3. (A) The slip during dh increases systematically with log hold time and loading velocity. (B) Precursory slip decreases with hold time for low velocities. Note that dp is relatively independent of hold time and that it may be slightly lower for higher loading velocity. (C) ds increases nonlinearly with log hold time and, within the scatter, tends to be larger for higher loading velocity.

126

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Figure 13 show healing and relaxation as a function of hold time for two velocities. The simulations mimic several aspects of the experimental data, including a velocity effect on healing and a region within which the healing parameter △m increases linearly with the log of hold time (Figure 13(a)). For a given hold time and loading rate, the Dieterich law predicts greater healing and lesser relaxation than the Ruina law (Figure 13). The Dieterich law also predicts that frictional creep relaxation saturates for long hold times. Note that the rate of change (△mc/△th) drops significantly between about 10 and 100 s (Figure 13(b)). The simulations show a short-time cutoff effect at small hold times, which is consistent with experimental observations in the sense that, within measurement resolution, no healing is observed for hold times below about 1s for

of the consistency of friction parameters obtained from different tests conducted in the same experiment.

Forward Modeling of Frictional Healing

Figure 13 shows numerical simulations of healing for each friction state evolution law. Forward models are calculated using typical values of the constitutive parameters for F-110 and the laboratory value of apparatus stiffness for the tests described earlier. The coupled equations [7] and [11] or [7] and [12] are solved using a fifth-order Runge–Kutta method. The inset provides details of the slip history for a single case, along with definitions of the healing parameters (these are the same as measured in the laboratory). The main panels in

0.02 Hold begins

m − mo

0.01

Slide-hold-slide simulation 10 s hold Vs/r = 10 μm s−1

μ

0

Hold ends

100

Dieterich law

0

10

8

b = 0.009, b − a = 0.001 Dc = 5 μm, k = 1 ´ 10−3 μm−1

Slip/Dc

Healing, Dm

10

10

mc −0.01

12

Vs/r Ruina Dieterich (μm s−1) law law

6

0.05

4 m¢ = 2.302... log10th

(a)

Δm¢ = Δm/b

4.05.4.1

2

(c)

0

0

0.03

0.02 2

0.01

1

(b)

0

Δm/b

Relaxation, Δmc

= mo − mmin

3

10−2

(d)

10−1

100

101

102

Hold time (s)

103

104

105

10−2 10−1

100

101

102

103

104

105

0

Nondimensional hold time, t¢= (Vs/r th)/Dc

Figure 13 Numerical simulations of SHS tests showing healing and relaxation predicted by the RSF laws and elastic interaction. Inset to (a) shows details of a single simulation for which steady sliding is prescribed prior to a hold that begins at a normalized slip of 0 (the friction level is arbitrary and chosen as 0). The RSF laws predict that friction decreases during the hold due to creep and elastic interaction. Each point in the main panels shows results from a simulation. Four cases are shown corresponding to two velocities for each of the friction laws. Panels (a) and (b) show variations in the coefficient of friction. The same results are shown in nondimensional form in panels (c) and (d). Both laws show that healing and relaxation scale with loading rate and hold time, consistent with the experimental results. Simulations were carried out using the constitutive parameters given in the figure.

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle driving velocities of 10 mm s1. We note that data for the different velocities collapse when plotted versus nondimensional hold time t0 (Figure 13). The panels on the left side in Figure 13 show healing and relaxation data as changes in the coefficient of friction. The same simulation data are plotted in a normalized form in the right-hand panels and indicate that the distinction between time and velocity effects is accounted for when Vs/r is normalized by an effective velocity Ve ¼ Dc/th, to produce a nondimensional hold time t0 (Figure 13). Both laws predict that, for a given set of constitutive parameters, healing (Figure 13 (c)) and relaxation (Figure 13(d)) scale with the product of loading rate and hold time. The nondimensional form also shows that for the Dieterich law, the asymptotic healing rate b is equal to b ln(th) for long hold times (Figure 13(c)), as noted by previous workers (Beeler et al., 1994). This is expected, because for large th, V becomes very small and from eqn [11], y_  1; thus, y(th)  th. This can be seen by taking m ¼ mo during sliding prior to a hold; thus, healing is given by

1 ´ 106 3 ´ 105

Dieterich law

10

4.05.4.2

Origin of the Loading Rate Effect on Healing

Numerical simulations were carried out to investigate the origin of the loading rate effect on healing. Figure 14 shows results of simulations carried out for a range of hold times (102 to 104 s) and slide/reload velocities Vs/r (10– 103 mm s1) using the parameters given in Figure 14(b). Figure 14(a) and 14(b) shows the complete time series computed for SHS tests, with data shown from initiation of the SHS to the peak value of friction. Figure 14(c) and 14(d) shows measurements of the healing slip parameters (see Figure 12) from a suite of simulations run using the values of th and Vs/r shown. In Figure 14(a) and 14(b), friction values are normalized by the state evolution parameter b and slip is normalized by Dc

Vs/r

3

dp

10 μm s−1 100 μm s−1

1 ´ 105

8

1000 μm s−1

3 ´ 104 1 ´ 104

6

2

3000

4

dp/Dc

( m − mo)/b

△m(th) ¼ ms(th)  mo, and substituting into [7] with Vo ¼ Vs/r (recall that peak friction is defined by the point at which V ¼ Vs/r) yields △m(th)  b ln(th).

3 ´ 106

12

1000 300 100 30 10 3

2 1

0 −2

Dieterich law

1 Ruina law

(Vs/r th)

(a)

127

(c)

dh

0 12

Ruina law

10

3 ´ 106 k = 1 ´ 10 −3 μm −1 b = 0.009, b − a = 0.001 Dc = 5 μm, k¢ = 5

6

4

3 ´ 104

4

3000

2 0

1

−2

(b)

0

Ruina law

3 ´ 105

d h/ D c

8

(m − mo)/b

5

3 Dieterich law

2

300 30 3

1 (d)

1

2

3 Slip/Dc

4

=d

5

6

0

10−2

10−1

100

101

102

103

104

105

Hold time (s)

Figure 14 Characteristics of frictional slip during SHS simulations. Holds begin at zero slip and end when friction begins to increase (see inset to Figure 13(a)). The simulation is stopped when peak friction is reached. Panels (a) and (b) show results for the Dieterich and Ruina laws, respectively, plotted in nondimensional form. Curves are labeled by the product of loading rate and hold time and all simulations were carried out using the parameters given in panel (b). For a given velocity and hold time, the Dieterich law predicts less creep and slip during holds than the Ruina law. The parameters dh and dp (see Figure 12) indicate slip at the end of the hold and at peak friction, respectively. Panels (c) and (d) show the systematic changes in dp and dh as a function of velocity and hold time.

128

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

(Figure 14). Note that curves are labeled with the product of th and Vs/r. We also document the effect of loading rate and hold time on frictional creep and slip (Figure 14(c) and 14(d)). The slip parameters dh and dp are computed for a range of Vs/r values and hold times. The simulations indicate significant differences between the two state evolution laws (Figure 14). For a given hold time and loading velocity, the Dieterich law predicts significantly less creep slip and greater healing than does the Ruina law (Figure 14(a) and 14(b)). For the Ruina law, creep slip during the hold period exceeds 5Dc for the longest hold times. These simulations illuminate the origin of short-time, non-log-linear healing behavior (e.g., Figure 13(a)), in the sense that creep relaxation and healing are nearly zero for values of the product th Vs/r below about 10. We note that in an SHS test, creep occurs until strength meets or exceeds the applied stress. Thus, the prediction of greater creep slip for the Ruina law is consistent with the relation between slip and friction evolution (eqn [12]). That is, for the Ruina law, the rate of friction evolution goes to zero as slip velocity approaches zero. In contrast, the rate of friction state evolution becomes linear with time as V goes to zero for the Dieterich law, and thus, (1) creep relaxation is consistently smaller for the Dieterich law than for the Ruina law, and (2) creep relaxation becomes independent of loading velocity for the Dieterich law, for long hold times, but varies systematically with velocity for the Ruina law (Figure 14(d)). The precursory slip during the reload following a hold period is quite similar for each law at short hold times. For hold times shorter than roughly 10 s, slip during the reload decreases with increasing hold time and slip velocity (Figure 14(c)). Smaller values of dp indicate a stiffer, more brittle response, which is consistent with the expectation that frictional contact junctions become stronger with increasing hold time. Similarly, the effect of Vs/r on dp indicates a stiffer, more brittle response for higher loading velocities (Figure 14 (c)). This is consistent with the friction direct effect, which produces a logarithmic increase in friction with velocity at constant state. For longer hold times and higher loading velocities, the Ruina law predicts a stiffer, more brittle response than the Dieterich law (Figure 14(c)). This is also evident in Figure 14(a) as the extended slip prior to reaching peak friction for the Dieterich law. The effect of the Vs/r on dp saturates for hold times longer than roughly 100 s for both laws, consistent with our laboratory results (Figure 12). For a given hold time and loading velocity, the Dieterich law predicts larger precursory slip than the Ruina law, consistent with the stronger dependence of the rate of state evolution on slip velocity (eqn [11]). Clearly, additional work is needed to better constrain these parameters. Figure 15 shows a comparison of friction versus slip and the corresponding friction versus slip velocity (so-called phase plane plot: cf. Beeler et al., 1994; Rice, 1983) for two sets of simulations. We compare the responses for 100 s holds at two velocities for the Dieterich law and the Ruina law using the same friction parameters a ¼ 0.008, b ¼ 0.009, Dc ¼ 5 mm, and k ¼ 1  103 mm1. For the phase plane plots (Figure 15(b) and 15(d)), slip velocity is normalized by the initial velocity prior to the hold, so that Vo ¼ 1 or 10 mm s1. Note that friction decays to a lower velocity for Vs/r ¼ 10 mm s1 than for

1 mm s1 (Figure 15). This may seem counterintuitive: Why would slip velocity become lower if the initial velocity is higher? But this is an important part of the effect of velocity on healing rate. The rate of state evolution depends on slip velocity. Higher slip velocity results in greater slip, and together, they cause faster state evolution – that is, larger negative values. We note that differences in deceleration history during an SHS test, and during the seismic cycle, are part of the reason that frictional healing depends on loading velocity. This can also be seen by noting that for both laws, the change in the frictional state is larger for the 10 mm s1 case than for the 1 mm s1 case (Figure 15(b) and 15(d)). The simulation results in Figure 15 show that friction increases approximately along a line of constant state upon reloading in an SHS test. Comparison of the friction curves with the fiducial lines of slope a ¼ 0.01 shows that friction begins to depart from a line of constant state at about twothirds of the peak friction value for this range of stiffness and friction parameters. For an SHS test, the RSF laws predict that peak friction occurs at the point at which slip velocity reaches Vs/r, as discussed earlier. This is clear in Figure 15(b) and 15(d). We note also that frictional overshoot, to velocities higher than the initial velocity, occurs for this range of parameters at both Vs/r ¼ 1 and 10 mm s1. This is seen by noting that V exceeds Vo after the peak in friction; then, V reaches a maximum value, followed by a minimum in friction and a local maximum at V ¼ Vs/r before friction and velocity drop back to their initial values (Figure 15). Note that in the recovery following overshoot, the friction curve increases at a higher rate than predicted for constant state. This is consistent with the combined effects of the direct effect and increased frictional state; friction increases faster than predicted by just the direct effect because of appreciable changes in state. In addition to its role in defining the upper stability transition along tectonic faults (e.g., Figure 3), elastic stiffness also has an important effect on frictional healing and creep frictional relaxation (Figure 16). We show simulation results using values of the RSF parameters determined from velocity step tests, with some rounding to produce average values. Healing data are plotted versus a nondimensional stiffness given by k/Kc. Each data point in Figure 16 represents a full numerical simulation of an SHS test computed with a given hold time and elastic stiffness. Note that for k < Kc, frictional instability results, with V tending to large values (e.g., Ranjith and Rice, 1999). However, instability occurs after peak (static) friction is reached; thus, we obtain values of the healing parameters prior to numerical instability. The simulation results show that frictional healing is larger for the Dieterich law (Figure 16), consistent with expectations from Figures 13 and 14. For values of k  Kc, healing increases with increasing stiffness for both laws (Figure 16(a)). For the Dieterich law, the stiffness effect on healing begins to saturate above values of about 50–100 Kc, whereas the Ruina law shows an optimum stiffness for healing with lesser healing for lower and higher stiffnesses (Figure 16(a)). For the Ruina law, maximum healing occurs at a stiffness of 5 Kc, and healing tends toward zero for larger stiffnesses (Figure 16(a)). These results are consistent with the fact that the rate of state evolution goes to zero as V tends to zero for the Ruina law. For very large

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Lines of constant state

Slide-hold-slide simulations 100 s holds

4

129

Dieterich law

(m − mo )/b

3

2 Increasing state

1

0

−1 0

10

30

20

(a)

−8

−6

−4

−2

0

2

4

6

8

−6

−4

−2

0

2

4

6

8

(b)

3 Ruina law

Vs/r

2

1 μm s−1 10 μm s−1

(m − mo )/b

1

0

−1 k = 1 ´ 10−3 μm−1 b = 0.009, b − a = 0.001 Dc = 5 μm, k¢ = 5

−2

−3 (c)

0

10

20

30

Slip/Dc

−8 (d)

ln(V/Vo)

Figure 15 Numerical simulations of 100 s holds at two loading rates. Left-hand panels, (a) and (c), show friction versus displacement. Holds begin at slip ¼ 0. Right-hand panels, (b) and (d), are phase plane plots showing corresponding data of friction versus slip velocity for the same holds shown in (a) and (c). In the phase plane plots, holds begin at V ¼ Vo. After the hold, when friction has regained steady state, V returns to Vo. Note that the y-scale is the same in each row of plots. Simulations with the Dieterich law (a) and (b) show that loading rate has little affect on creep relaxation but that it has a pronounced affect on peak friction, whereas the Ruina law shows appreciably more slip and creep compaction at the higher initial loading rate. Dashed lines in the phase plane plots show lines of constant frictional state.

values of elastic stiffness, V drops to near zero very quickly for even a small frictional relaxation. This may be seen in Figure 16 (b), which shows frictional relaxation △mc during the hold period. The values of △mc are largest for the Ruina law and increase rapidly as stiffness increases (Figure 16(b)). This has important implications for natural faults, in the sense that the extent of healing predicted by the two friction laws is fundamentally different – and therefore would predict distinct interseismic healing behaviors as functions of loading system stiffness and therefore depth (e.g., Figure 3). The influence of loading velocity on frictional healing is clear (e.g., Figure 9), but the origin of this effect, and the roles of the friction direct effect and state evolution, is complex (Figure 17). Each data point in Figure 17 represents a full

numerical simulation of a 100 s SHS test computed with a given set of parameters. Six sets of parameters were used to explore the role of the friction parameters a, b, and a  b. We varied a at constant b, and b at constant a, with corresponding changes in a  b, and we also varied a and b so as to keep a  b constant (Figure 17). The numerical results show that the origin of the velocity dependence of frictional healing differs for the two state evolution laws. For the Ruina law, the velocity dependence of healing is due primarily to the direct effect (Figure 17(a)). For low values of a (a ¼ 8  105), healing is essentially independent of slip velocity, and neither b nor a  b has an appreciable effect on healing or creep relaxation (Figure 17). However, for a larger value of the direct effect (a ¼ 8  103), both healing and creep

130

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

10 0.08

8 0.06

Δm

Δm/b

6 0.04

4

0.02

0

0

0.10

Dieterich law

Ruina law

10

102 s 104 s

0.08

Δmc = mo-mmin

th

8

b = 0.009, b - a = 0.001 Dc = 5 μm, Vs/r = 1 μm s-1

0.06

6 0.04

4

0.02

0 (b)

Δmc /b

(a)

2

2

10-1

100

101 k¢ = (k Dc)/(b - a)

102

103

0

Figure 16 Numerical simulations showing the affect of stiffness on (a) healing and (b) creep relaxation for two hold times and each friction state evolution law. Each data point represents a full numerical simulation of a SHS test. All simulations were done using the RSF parameters given in panel (b). Stiffness is normalized by the critical stiffness kc. For a given hold time and stiffness, frictional healing is larger for the Dieterich law and creep relaxation is larger for the Ruina law. For the Ruina law, healing reaches a maximum and decreases for stiffness greater than about 10kc.

relaxation increase significantly. For the range of velocities studied, frictional healing and the magnitude of creep relaxation vary directly with a. These results are consistent with the fact that the rate of state evolution tends to zero as V goes to zero for the Ruina law. Comparison of the friction phase plane plot for two cases with different values of the direct effect and constant b illuminates the role of the direct effect (Figure 18 (a)). The simulations in Figure 18 show that frictional creep is negligible for the lower value of the direct effect (a ¼ 8  104); hence, slip velocity becomes very low and state does not change appreciably. In contrast, for a larger value of the direct effect (a ¼ 8  103), creep relaxation is greater and thus slip velocity takes longer to drop, which means that state evolves faster and reaches higher values. For these simulations, with Dc ¼ 5 mm and initial slip velocity of 10 mm s1, the state term is initially 0.5 (Figure 18). For the case of a ¼ 8  104, state is 1.04 at the end of the hold and 1.11 when peak friction is reached upon the reload. In contrast,

for a ¼ 8  103, state is 24.5 at the end of the hold and 6.1 when peak friction is reached upon the reload. Recall that state increases toward the upper left corner in a phase plane plot, but the rate of state change is scaled by the direct effect a (Figure 18). For the Ruina law, the direct effect influences frictional healing in an SHS test because creep relaxation scales directly with a and the rate of state evolution during the hold scales inversely with a. The velocity dependence of healing is produced largely by the direct effect a because larger values of a result in greater state evolution during the initial phase of deceleration, before velocity becomes negligible (Figure 18). For the Dieterich law, both a and b influence the velocity dependence of healing (Figure 17). However, the effect of b is much greater than that of a and the signs of the effects differ. For the parameters in Figure 17, an increase of b of a factor of 9 produces an increase in healing of nearly a factor of 10 at 1 mm s1. On the other hand, an increase of a by a factor of 100 results in a decrease in healing of only roughly 10% for the

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Dieterich law

Ruina law

a 8 ´ 10-3 8 ´ 10-5

131

b-a

b

14

9 ´ 10-3 1 ´ 10-3 9 ´ 10-3 8.92 ´ 10-3

8 ´ 10-5 1.08 ´ 10-3 1 ´ 10-3

12

100 s hold 10

Healing, Dm

k = 1 ´ 10-3 μm-1 Dc = 5 μm

8

6

0.05

Δm /(b = 0.009)

0.10

4

2

(a)

0 3

0

0.02

Δm /(b = 0.009)

Relaxation, Dm c

2

0.01

0 (b)

1

100

101

102

103

0 104

Loading velocity, Vs/r (mm s-1)

Figure 17 Numerical simulations showing the affect of RSF parameters and loading velocity on (a) healing and (b) creep relaxation for 100 s holds. Each data point represents a full numerical simulation of an SHS test. All simulations were done using the RSF parameters given in panel (a). Note that the three cases allow assessment of each RSF parameter: the friction direct effect a, the evolution effect, b, and the friction rate parameter a  b. For the Dieterich law, healing is dominated by the evolution parameter b and affected little by the direct effect a. For the Ruina law, healing is dominated by the direct effect a and affected little by the evolution effect b. For both laws, creep relaxation is affected primarily by a.

same velocity. These results are consistent with the fact that the rate of state evolution tends to 1.0 as V goes to zero for the Dieterich law. This means that the state term in eqn [7] continues to increase with hold time during an SHS test (Figure 17). As for the Ruina law, frictional creep scales directly with the friction direct effect for the Dieterich law (Figure 17(b)). However, because state continues to evolve as velocity goes to zero for the Dieterich law, state becomes large

during the hold period and thus lower values of the direct effect, which cause a more rapid drop in slip velocity (Figure 18), produce larger frictional healing. For the simulations in Figure 18, state begins at a value of 0.5, and for the a ¼ 8  104 case, state is 100.4 at the end of the hold and 93.5 when peak friction is reached upon the reload. In contrast, for a ¼ 8  103, state is 85.5 at the end of the hold and 28.0 when peak friction is reached upon the reload (Figure 18).

132

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Coefficient of friction

0.62

Runia law Vs/r = 10 mm s-1 100 s k = 1 ´ 10-3 mm-1 Dc= 5 mm b = 0.009

a = 0.008

a = 0.0008 0.60

0.58 -12

-8

-4

4

8

ln(V/Vo)

(a)

k = 1 ´ 10-3 mm-1 Dc = 5 mm b = 0.009

Dieterich law Vs/r = 10 mm s-1 100 s 0.64

Coefficient of friction

0

0.62 a = 0.0008 a = 0.008

0.60

0.58 -60

-50

-40

(b)

-30 -20 ln(V/Vo)

-10

0

10

Figure 18 Numerical simulations using the (a) Ruina law and (b) Dieterich law to evaluate the affect of the friction direct effect on frictional healing and creep relaxation. See also Figures 15 and 16. Plots show the complete friction–velocity history for a 100 s SHS test. Comparison of the phase plane trajectories shows that frictional creep is negligible for the lower value of the direct effect (a ¼ 8  104). Slip velocity during the creep relaxation stage of the SHS drops significantly in both cases, but the drop is particularly large for the Dieterich law where the velocity is 1033 m s1 at the end of the hold. The friction direct effect has a much larger influence on healing for the Ruina law than for the Dieterich law.

For the Dieterich law, healing varies inversely with the friction direct effect because larger values of a result in larger creep relaxation, which lowers the rate of deceleration and thus slows the rate of friction state evolution.

4.05.4.3 Recovery of Rate State Friction Parameters from SHS Tests The RSF laws were originally developed from velocity step tests, and many previous works have documented the approach to recovering RSF parameters in these cases. However, RSF parameters can also be recovered from SHS tests (Figure 19). In this

approach, the initial conditions for friction and state are taken from steady-state sliding prior to the beginning of the hold period. Then, frictional state and slip velocity are calculated during the hold and reload periods until sliding is once again at steady state. The numerical approach is essentially that of modeling two coupled velocity step tests: from Vs/r to zero and back to Vs/r. The key difference is that the second ‘velocity step,’ which corresponds to the reload after the hold period, does not begin at steady state, but rather with values of state and velocity given by the hold. Figure 19 shows detailed fits of the RSF laws to SHS time series. For clarity, we show model fits to data for a 10 s hold

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

133

Data Simulation

0.02

R

m−mo

0.01

a = 0.0107 b = 0.0086 Dc = 3.5 μm

a = 0088 b = 0.0094 Dc = 9.0 μm

a = 0.0091 b = 0.0094 Dc = 8.3 μm

0.005 D

a = 0.0108 b = 0.0081 Dc = 3.1 μm

0 R 10 s hold Dieterich law simulation

−0.01

−0.02

30 s hold Ruina law simulation

D

m 051 10 s

(a)

Time

30 s

R a = 0.0092 b = 0.0094 Dc = 6.8 μm

0.005

Friction residual (data - model)

D

D

10 s

a = 0.0078 b = 0.0100 Dc=14.4 μm

D

a = 0.0107 b = 0.0085 Dc = 3.1 μm

Time

a = 0.0094 b = 0.0067 Dc = 4.0 μm

R

a = 0.0092 b = 0.0090 Dc = 5.9 μm

D

a = 0.0076 b = 0.0108 Dc = 17.9 μm

0.005

a = 0.0107 b = 0.0086 Dc = 3.5 μm

R

a = 0.0105 b = 0.0080 Dc = 3.2 μm

(c)

R

(b)

a = 0.0088 b = 0.0094 Dc = 9.0 μm

100 s

a = 0.0096 b = 0.0069 Dc = 3.4 μm

(d)

Figure 19 Friction data for SHS tests and best-fit numerical simulations using the RSF laws. (a) The change in friction from the steady state is shown versus time for two SHS tests. The 10 s hold is fit with the Dieterich law and the 30 s hold is fit with the Ruina law. RSF parameters are given for each case. Note that the RSF simulations fit the data quite well throughout nearly the entire SHS. Panels (b)–(d) show residuals computed by finding a least-squares best-fit RSF simulation to an SHS test, as in (a), and then subtracting the fit from the friction data. Each panel shows data for two SHS tests and the fits for the Ruina law and the Dieterich law. Arrows denote the beginning and end of the hold and the peak in friction – see panel (a) for corresponding arrows in the time series for the 10 s SHS. Note that the best-fit RSF parameters differ somewhat as a function of hold time (compare values in panels (b)–(d)) but that they are relatively consistent. In each case, the best-fit RSF parameters a and Dc are larger for the Dieterich law compared to the Ruina law, and the evolution parameter b is larger for the Ruina law.

and a 30 s hold together and offset the data so that the hold begins at the same time in each case (Figure 19(a)). The arrows denote the beginning and end of the hold period for the 10 s hold. Note that the friction creep curves, from the initiation of the SHS test, are identical (Figure 19(a)) and that each data set is reasonably well fit by the RSF simulation. We also show fits to six additional SHS tests, two for 10, 30, and 100 s holds, respectively (Figure 19). In each case, the same SHS data were fit with each state evolution law and we show the residual (data – model) as a function of time. Figure 19(b) shows data for two separate 10 s SHS tests, and the best-fit RSF parameters are quite similar between the data, for a given evolution law. Note that a ¼ 0.0092 for both of the Ruina law fits (Figure 19(b)). The best-fit RSF parameters vary somewhat between the data for 10, 30, and 100 s, but they are reasonably close.

We may use the best-fit RSF parameters recovered from individual SHS tests to assess the extent to which they predict the effect of loading rate on frictional healing and creep relaxation. Figure 20 shows the laboratory data from Figure 9 plotted versus a normalized hold time formed by the product of the slip velocity Vs/r and hold time th. The x-axis in Figure 20 has units of displacement, which we could normalize by Dc, but we chose not to because we do not know Dc a priori. The loading rate effect on healing and creep relaxation is quite well described by the normalized hold time. We also show two curves in Figure 20. These are predictions of the healing behavior computed from the RSF parameters obtained from fits to individual SHS tests (as in Figure 19). The predicted curves are computed from averaged values (Figure 19), and thus, they are expected to fit the healing (Figure 20(a)) and creep relaxation

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Data Vs/r (μm s-1)

0.04

1 3 10 30 100

Dm

0.03

0.02

Data Vs/r (μm s-1)

0.04

1 3 10 30 100

0.03 Dm

134

0.02

0.01

0.01 m 074+

(a)

0

(a)

Simulations

Simulations

0.04

Dieterich

0.04

Ruina

a = 0.0081, b = 0.0089 Dc = 6.1 μm

0.03

0.03 Dmc

Dmc

m 074+ 0

0.02

Dieterich

Ruina

a = 0.0087 b = 0.0062 Dc = 4.6 μm

a = 0.0075 b = 0.0101 Dc = 20.3 μm

101

102

0.02

0.01 0.01 0 100

101

102

(b)

103 (th Vs/r)

104

105

0 0 10 (b)

103 (th Vs/r)

104

105

0.04 0.04 0.03

Dmc

Dmc

0.03 0.02

0.02

0.01 0.01

Dm = Dmc 0 (c)

0

0.01

0.02 Dm

0.03

Dm = Dmc

0.04

Figure 20 Friction data for SHS tests are plotted versus (a and b) the product of hold time and the slide-reload velocity Vs/r. Panel (c) shows healing data plotted versus creep relaxation. For each panel, lines show RSF predictions based on the parameters given in (b) obtained by fits to the individual SHS test (see Figure 19). Note that the effect of loading rate Vs/r on healing is accounted for by the effective healing displacement (th Vs/r). The Ruina law prediction is closer to the trend of the laboratory data for each plot.

(Figure 20(b)) data, at least approximately. While neither curve matches the data perfectly, the Ruina law prediction is much closer to the trend given by the laboratory data for both healing (Figure 20(a)) and creep relaxation (Figure 20(b)). Figure 20(c) shows data and model fits for creep relaxation versus healing. This representation shows that creep relaxation is

0 (c)

0

0.01

0.02 Dm

0.03

0.04

Figure 21 Same data as Figure 20. Curves show best-fit RSF predictions for each law. Note that the RSF parameters are similar to those in Figure 20, but these curves do a much better job of fitting the overall trend of the data. The Ruina law prediction is closer to the trend of the laboratory data for each plot.

larger than healing for small hold times but that creep tends to saturate for longer times, while healing continues to increase. This trend is captured by simulations with both state evolution laws, but the Ruina law is closer to the data trend. One can also find best-fit RSF parameters for the complete SHS data set, at all loading rates, and we show such fits in Figure 21. The constitutive parameter values are not too

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

different from those in Figure 20, which is consistent with the trends shown in Figure 20.

4.05.4.4 Microphysical Explanation of Frictional Healing and Rate Dependence If we assume that shearing in granular quartz occurs primarily by granular flow mechanisms, then it is reasonable that steadystate porosity is higher at higher sliding velocities, which would result in more compaction and relaxation during holds. The observations of greater healing, and, to a lesser extent, increased healing rates, could then be attributed to the larger magnitude of dilation upon reshear. Moreover, the observed changes in healing, compaction, and relaxation with increasing shear displacement (Figures 7 and 8; Richardson and Marone, 1999) are consistent with shear-driven comminution, grain size reduction, and shear localization.

4.05.5

Conclusions

Elastic stiffness has an important effect on frictional healing and creep frictional relaxation (Figure 16). We show simulation results using values of the RSF parameters determined from velocity step tests, with some rounding to produce average values. Healing data are plotted versus a nondimensional stiffness given by k/Kc. Each data point in Figure 16 represents a full numerical simulation of an SHS test computed with a given hold time and elastic stiffness. Note that for k < Kc, frictional instability results with velocity tending toward large values. Fundamentally, the processes within tectonic faults that drive changes in friction constitutive properties (e.g., rate dependence or Dc) with depth, such as lithification, cementation, mineral transformation, consolidation, or shear localization, are also likely to affect the loading system stiffness and fault healing behavior (cf., Figure 3) (e.g., Ikari et al., 2007; Moore and Saffer, 2001). Thus, three interlinked factors are necessary for effecting the transition from stable aseismic deformation, at shallow depths, to the seismic cycle of repeated frictional failure: (1) changes in frictional properties such that the critical stiffness Kc increases with depth in such a way that the criterion for instability is met (eqn [3]; Figure 3), (2) consolidation of fault gouge and surrounding material so that elastic stresses can be supported, and (3) the rate of frictional healing that must be sufficient to allow fault strength to increase at a rate that is higher than the rate of stress relaxation by creep and plastic deformation.

Acknowledgments We thank N. Beeler, B. Carpenter, M. Ikari, A. Niemeijer, and M. Scuderi for discussions and A. Niemeijer and N. Beeler for thorough, thoughtful reviews that improved the manuscript. This work was supported by NSF grants OCE-0648331, EAR0746192, and EAR-0950517 and IGPP/LANL grant 162505-1.

135

References Angevine CL, Turcotte DL, and Furnish MD (1982) Pressure solution lithification as a mechanism for the stick-slip behavior of faults. Tectonics 1: 151–160. Baumberger T and Gauthier L (1996) Creep like relaxation at the interface between rough solids under shear. Journal de Physique I France 6: 1021–1030. Baumberger T, Heslot F, and Perrin B (1994) Crossover from creep to inertial motion in friction dynamics. Nature 367: 544–546. Baumberger T, Berthoud P, and Caroli C (1999) Physical analysis of the state- and ratedependent friction laws: II. Dynamic friction. Physical Review B 60: 3928–3939. Baumberger T and Caroli C (2006) Solid friction from stick slip down to pinning and aging. Advances in Physics 55(3): 279–348. http://dx.doi.org/ 10.1080/00018730600732186. Bayart E, Rubin A, and Marone C (2006) Evolution of fault friction following large velocity jumps. EOS, Transactions, American Geophysical Union, F 2006. Beeler NM (2007) Laboratory-observed faulting in intrinsically and apparently weak materials: Strength, seismic coupling, dilatancy, and pore fluid pressure. In: Dixon TH and Moore JC (eds.) The Seismogenic Zone of Subduction Thrust Faults, p. 692. New York: Columbia University Press, ISBN: 978-0-231-13866-6. Beeler NM and Tullis TE (1997) The roles of time and displacement in velocity dependent volumetric strain of fault zones. Journal of Geophysical Research 102(22): 595–22609. Beeler NM, Tullis TE, and Weeks JD (1996) Frictional behavior of large displacement experimental faults. Journal of Geophysical Research 101: 8697–8715. Beeler NM, Tullis TE, and Weeks JD (1994) The roles of time and displacement in the evolution effect in rock friction. Geophysical Research Letters 21: 1987–1990. Beeler NM, Tullis TE, and Goldsby DL (2008) Constitutive relationships and physical basis of fault strength due to flash heating. Journal of Geophysical Research 113. http://dx.doi.org/10.1029/2007JB004988. Behnsen J and Faulkner DR (2012) The effect of mineralogy and effective normal stress on frictional strength of sheet silicates. Journal of Structural Geology 42: 49–61. http://dx.doi.org/10.1016/j.jsg.2012.06.015. Belardinelli ME, Cocco M, Coutant O, and Cotton R (1999) Redistribution of dynamic stress during coseismic ruptures; evidence for fault interaction and earthquake triggering. Journal of Geophysical Research 104: 14925–14945. Ben-David O, Cohen G, and Fineberg J (2010a) The dynamics of the onset of frictional slip. Science 330: 211–214. Ben-David O, Rubinstein SM, and Fineberg J (2010b) Slip-stick and the evolution of frictional strength. Nature 463: 76–79. Ben-David O and Fineberg J (2011) Static friction coefficient is not a material constant. Physical Review Letters 106: 254301. Beroza G and Ide S (2011) Slow earthquakes and non-volcanic tremor. Annual Review of Earth and Planetary Sciences 39: 271–296. Berthoud P and Baumberger T (1998) Role of asperity creep in time- and velocitydependent friction of a polymer glass. Europhysics Letters 41: 617–622. Berthoud P, Baumberger T, G’Sell C, and Hiver J-M (1999) State- and rate-dependent friction at a multicontact interface between polymer glasses: I. Static friction. Physical Review B 59: 14313–14327. Bishop AW (1954) Discussion. Geotechnique 4: 43–45. Blanpied ML and Tullis TE (1986) The stability and behavior of a frictional system with a two state variable constitutive law. Pure and Applied Geophysics 124: 415–430. Boettcher MS and Marone C (2004) The effect of normal force vibrations on the strength and stability of steadily creeping faults. Journal of Geophysical Research 109: B03406. http://dx.doi.org/10.1029/2003JB002824. Bocquet L, Charlaix E, Ciliberto S, and Crassous J (1998) Moisture-induced ageing in granular media and the kinetics of capillary condensation. Nature 396: 735–737. Bos B and Spiers CJ (2000) Effect of phyllosilicates on fluid-assisted healing of gougebearing faults. Earth and Planetary Science Letters 184: 199–210. http://dx.doi.org/ 10.1016/S0012-821X(00)00304-6. Bos B and Spiers CJ (2002) Fluid-assisted processes in gouge-bearing faults: Insights from experiments on a rock analogue system. Pure and Applied Geophysics 159: 2537–2566. http://dx.doi.org/10.1007/s00024-002-8747-2. Boulton C, Carpenter BM, Toy V, and Marone C (2012) Physical properties of surface outcrop cataclastic fault rocks, Alpine Fault, New Zealand. Geochemistry, Geophysics, Geosystems 13: Q01018. http://dx.doi.org/10.1029/2011GC003872. Brockley CA and Davis HR (1968) The time-dependence of static friction. Journal of Tribology 90: 35–41. http://dx.doi.org/10.1115/1.3601558. Brown KM, Kopf A, Underwood MB, and Weinberger JL (2003) Compositional and fluid pressure controls on the state of stress on the Nankai subduction thrust. Earth and Planetary Science Letters 21: 589–603. Bhushan B, Israelachvili JN, and Landman U (1995) Nanotribology: Friction, wear and lubrication at the atomic scale. Nature 374: 607–616.

136

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Byrne DE, Davis DM, and Sykes LR (1988) Loci and maximum size of thrust earthquakes and the mechanics of the shallow region of subduction zones. Tectonics 7: 833–857. Carpenter BM, Marone C, and Saffer DM (2011) Weakness of the San Andreas Fault revealed by samples from the active fault zone. Nature Geoscience 4: 251–254. http://dx.doi.org/10.1038/ngeo1089. Carpenter BM, Saffer DM, and Marone C (2012) Frictional properties and sliding stability of the San Andreas Fault from deep drill core. Geology 40. http://dx.doi. org/10.1130/G33007.1. Chester FM (1994) Effects of temperature on friction: Constitutive equations and experiments with quartz gouge. Journal of Geophysical Research 99: 7247–7261. Chester FM and Chester JS (1998) Ultracataclasite structure and friction processes of the Punchbowl fault, San Andreas system, California. Tectonophysics 295: 199–221. Chester FM and Higgs NG (1992) Multimechanism friction constitutive model for ultrafine quartz gouge at hypocentral conditions. Journal of Geophysical Research 97: 1859–1870. Collettini C, Niemeijer A, Viti C, and Marone C (2009) Fault zone fabric and fault weakness. Nature 462: 907–910. http://dx.doi.org/10.1038/nature08585. Collettini C, Niemeijer A, Viti C, Smith SAF, and Marone C (2011) Fault zone fabric and fault weakness. Earth and Planetary Science Letters 311: 316–327. http://dx.doi. org/10.1016/j.epsl.2011.09.020. Coulomb CA (1785) The´orie des machines simples, en ayant e´gard au frotement de leurs parties, et a la roideur des cordages. Me´moires de Mathematique & de Physique, X, Paris, pp. 161–342. Davis D, Suppe J, and Dahlen FA (1983) Mechanics of fold-and-thrust belts and accretionary wedges. Journal of Geophysical Research 88: 1153–1172. den Hartog SAM, Niemeijer AR, and Spiers CJ (2012a) New constraints on megathrust slip stability under subduction zone P–T conditions. Earth and Planetary Science Letters 353: 240–252. http://dx.doi.org/10.1016/j.epsl.2012.08.022. den Hartog SAM, Niemeijer AR, and Spiers CJ (2013) Friction on subduction megathrust faults: beyond the illite-muscovite transition. Earth and Planetary Science Letters 373: 8–19. den Hartog SAM, Peach CJ, De Winter DAM, Spiers CJ, and Shimamoto T (2012b) Frictional properties of megathrust fault gouges at low sliding velocities: New data on effects of normal stress and temperature. Journal of Structural Geology 38: 156–171. den Hartog SAM and Spiers CJ (2013) Influence of subduction zone conditions and gouge composition on frictional slip stability of megathrust faults. Tectonophysics 600: 75–90. http://dx.doi.org/10.1016/j.tecto.2012.11.006. Deng X and Underwood MB (2001) Abundance of smectite and the location of a plateboundary fault, Barbados accretionary prism. Geological Society of America Bulletin 113: 495–507. Di Toro G, Han R, Hirose T, et al. (2011) Fault lubrication during earthquakes. Nature 471: 494–498. http://dx.doi.org/10.1038/nature09. Di Toro G, Hirose T, Nielsen S, Pennacchioni G, and Shimamoto T (2006) Natural and experimental evidence of melt lubrication of faults during earthquakes. Science 311: 647–649. Dieterich JH (1972) Time dependent friction in rocks. Journal of Geophysical Research 77: 3690–3697. Dieterich JH (1978) Time-dependent friction and the mechanics of stick-slip. Pure and Applied Geophysics 116: 790–805. Dieterich JH (1979) Modeling of rock friction: Experimental results and constitutive equations. Journal of Geophysical Research 84: 2161–2168. Dieterich JH (1981) Constitutive properties of faults with simulated gouge. In: Carter NL, Friedman M, Logan JM, and Sterns DW (eds.) Mechanical Behavior of Crustal Rocks, AGU Monograph, vol. 24, pp. 103–120. Dieterich JH (1992) Earthquake nucleation on faults with rate- and state-dependent friction. Tectonophysics 211: 115–134. Dieterich JH (1994) A constitutive law for rate of earthquake production and its application to earthquake clustering. Journal of Geophysical Research 99: 2601–2618. Dieterich JH and Kilgore BD (1994) Direct observation of frictional contacts: New insights for state-dependent properties. Pure and Applied Geophysics 143: 283–302. http://dx.doi.org/10.1007/BF00874332. Dieterich JH and Kilgore B (1996a) Implications of fault constitutive properties for earthquake prediction. Proceedings of the National Academy of Sciences 93: 3787–3794. Dieterich JH and Kilgore B (1996b) Imaging surface contacts; power law contact distributions and contact stresses in quartz, calcite, glass, and acrylic plastic. Tectonophysics 256: 219–239. Dowson D (1979) History of Tribology. New York: Longman.

Edmond JM and Paterson MS (1972) Volume changes during the deformation of rocks at high pressures. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 9: 161–182. Fredrich JT and Evans B (1992) Strength recovery along simulated faults by solution transfer processes. Proceedings of the 33rd U.S. Symposium on Rock Mechanics, pp. 121–130. Frye KM and Marone C (2002) The effect of humidity on granular friction at room temperature. Journal of Geophysical Research 107(11): 2309. http://dx.doi.org/ 10.1029/2001JB000654. Ge´minard J-C, Losert W, and Gollub JP (1999) Frictional mechanics of wet granular material. Physical Review E 59: 5881–5890. Gettemy GL and Tobin HJ (2003) Tectonic signatures in centimeter-scale velocity– porosity relationships of Costa Rica convergent margin sediments. Journal of Geophysical Research 108: 2494. http://dx.doi.org/10.1029/2001JB00073. Goldsby DL and Tullis TE (2011) Flash heating leads to low frictional strength of crustal rocks at earthquake slip rates. Science 334: 216–218. Gu J-C, Rice JR, Ruina AL, and Tse ST (1984) Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction. Journal of the Mechanics and Physics of Solids 32: 167–196. Haines SH, van der Pluijm BA, Ikari MJ, Saffer DM, and Marone C (2009) Clay fabric intensity in natural and artificial fault gouges: Implications for brittle fault zone processes and sedimentary basin clay formation. Journal of Geophysical Research 114: B05406. http://dx.doi.org/10.1029/2008JB005866. He G, Mu¨ser MH, and Robbins MO (1999) Adsorbed layers and the origin of static friction. Science 284: 1650–1652. Heslot F, Baumberger T, Perrin B, Caroli B, and Caroli C (1994) Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Physical Review E 49: 4973–4988. Hong T and Marone C (2005) Effects of normal stress perturbations on the frictional properties of simulated faults. Geochemistry, Geophysics, Geosystems 6: Q03012. http://dx.doi.org/10.1029/2004GC000821. Hoskins ER, Jaeger JE, and Rosengren KJ (1968) A medium scale direct friction experiment. International Journal of Rock Mechanics and Mining Sciences 5: 143–154. Hyndman RD, Yamano M, and Oleskevich DA (1997) The seismogenic zone of subduction thrust faults. The Island Arc 6: 244–260. Ida Y (1972) Cohesive force across the tip of a longitudinalshear crack and Griffith’s specific surface energy. Journal of Geophysical Research 77: 3796–3805. Ide S, Beroza GC, Shelly DR, and Uchide T (2007) A scaling law for slow earthquakes. Nature 447: 76–79. http://dx.doi.org/10.1038/nature05780. Ikari MJ, Saffer DM, and Marone C (2007) Effect of hydration state on the frictional properties of montmorillonite-based fault gouge. Journal of Geophysical Research 112: B06423. http://dx.doi.org/10.1029/2006JB004748. Ikari MJ, Saffer DM, and Marone C (2009) Frictional and hydrologic properties of clayrich fault gouge. Journal of Geophysical Research 114: B05409. http://dx.doi.org/ 10.1029/2008JB006089. Ikari M, Marone C, and Saffer DM (2011) On the relation between fault strength and frictional stability. Geology 39: 83–86. http://dx.doi.org/10.1130/G31416. Johnson KL (1996) Continuum mechanics modelling of adhesion and friction. Langmuir 12: 4510–4513. Johnson KL (1997) Adhesion and friction between a smooth elastic spherical asperity and a plane surface. Proceedings of the Royal Society of London A 453: 163–179. Johnson T (1981) Time dependent friction of granite: implications for precursory slip on faults. Journal of Geophysical Research 86: 6017–6028. Karner SL, Marone C, and Evans B (1997) Laboratory study of fault healing and lithification in simulated fault gouge under hydrothermal conditions. Tectonophysics 277: 41–55. Karner SL and Marone C (1998) The effect of shear load on frictional healing in simulated fault gouge. Geophysical Research Letters 25: 4561–4564. Karner SL and Marone C (2001) Frictional restrengthening in simulated fault gouge: effect of shear load perturbations. Journal of Geophysical Research 106: 19319–19337. Kanagawa K, Cox SF, and Zhang S (2000) Effects of dissolution-precipitation processes on the strength and mechanical behavior of quartz gouge at high-temperature hydrothermal conditions. Journal of Geophysical Research 105(B5): 11115–11126. Kato N, Yamamoto K, Yamamoto H, and Hirasawa T (1992) Strain-rate effects on frictional strength and the slip nucleation process. Tectonophysics 211: 269–282. Kilgore B, Lozos J, Beeler N, and Oglesby D (2012) Laboratory observations of fault strength in response to changes in normal stress. Journal of Applied Mechanics 79: 031007. King DSH and Marone C (2012) Frictional properties of olivine at high temperature with applications to the strength and dynamics of the oceanic lithosphere. Journal of Geophysical Research 117: B12203. http://dx.doi.org/10.1029/2012JB009511.

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Knuth MW, Tobin HJ, and Marone C (2013) Ultrasonic velocity of sheared granular material: Implications for the evolution of dynamic elastic moduli with shear strain. Granular Matter 15: 499–515. http://dx.doi.org/10.1007/s10035-013-0420-1. Li YG, Vidale JE, Aki K, Xu F, and Burdette T (1998) Evidence of shallow fault zone strengthening after the 1992 M7.5 Landers, California, earthquake. Science 279: 217–219. Li Y-G and Vidale JE (2001) Healing of the shallow fault zone from 1994–1998 after the 1992 M7.5 Landers, California, earthquake. Geophysical Research Letters 28. http://dx.doi.org/10.1029/2001GL012922. Li Q, Tullis TE, Goldsby D, and Carpick RW (2011) Frictional ageing from interfacial bonding and the origins of rate and state friction. Nature 480: 233–236. Linker MF and Dieterich JH (1992) Effects of variable normal stress on rock friction: Observations and constitutive equations. Journal of Geophysical Research 97: 4923–4940. Liu Y (2013) Numerical simulations on megathrust rupture stabilized under strong dilatancy strengthening in slow slip region. Geophysical Research Letters 40: 1311–1316. http://dx.doi.org/10.1002/grl.50298. Losert W, Ge´minard J-C, Nansuno S, and Gollub J-P (2000) Mechanisms for slow strengthening in granular materials. Physical Review E 61: 4060. Mair K and Marone C (1999) Friction of simulated fault gouge for a wide range of velocities and normal stresses. Journal of Geophysical Research 104: 28899–28914. Marone C (1991) A note on the stress–dilatancy relation for simulated fault gouge. Pure and Applied Geophysics 137: 409–419. Marone C (1998a) Laboratory-derived friction constitutive laws and their application to seismic faulting. Annual Review of Earth and Planetary Sciences 26: 643–696. Marone C (1998b) The effect of loading rate on static friction and the rate of fault healing during the earthquake cycle. Nature 391: 69–72. Marone C and Scholz C (1988) The depth of seismic faulting and the upper transition from stable to unstable slip regimes. Geophysical Research Letters 15: 621–624. Marone C and Scholz CH (1989) Particle-size distribution and microstructures within simulated fault gouge. Journal of Structural Geology 11: 799–814. Marone C and Kilgore B (1993) Scaling of the critical slip distance for seismic faulting with shear strain in fault zones. Nature 362: 618–621. Marone C and Cox SJD (1994) Scaling of rock friction constitutive parameters: the effects of surface roughness and cumulative offset on friction of gabbro. Pure and Applied Geophysics 143: 359–386. Marone C and Saffer DM (2007) Fault friction and the upper transition from seismic to aseismic faulting. In: Dixon TH and Moore JC (eds.) The Seismogenic Zone of Subduction Thrust Faults, p. 692. New York: Columbia University Press, ISBN: 9780-231-13866-6. Marone C, Raleigh CB, and Scholz CH (1990) Frictional behavior and constitutive modeling of simulated fault gouge. Journal of Geophysical Research 95: 7007–7025. Marone C, Hobbs BE, and Ord A (1992) Coulomb constitutive laws for friction: Contrasts in frictional behavior for distributed and localized shear. Pure and Applied Geophysics 139: 195–214. Marone C, Vidale JE, and Ellsworth W (1995) Fault healing inferred from time dependent variations in source properties of repeating earthquakes. Geophysical Research Letters 22: 3095–3098. Marone C, Cocco M, Richardson E, and Tinti E (2009) The critical slip distance for seismic and aseismic fault zones of finite width. In: Fukuyama E (ed.) Fault-zone Properties and Earthquake Rupture Dynamics. International Geophysics Series, vol. 94, pp. 135–162. Burlington, MA: Elsevier. McLaskey GC, Thomas AM, Glaser SD, and Nadeau RM (2012) Fault healing promotes high-frequency earthquakes in laboratory experiments and on natural faults. Nature 491: 101–104. http://dx.doi.org/10.1038/nature11512. Montgomery DR and Jones DL (1992) How wide is the Calaveras fault zone? Evidence for distributed shear along a major fault in central California. Geology 20: 55–58. http://dx.doi.org/10.1130/0091-7613. Moore JC and Saffer DM (2001) Updip limit of the seismogenic zone beneath the accretionary prism of Southwest Japan: An effect of diagenetic to low-grade metamorphic processes and increasing effective stress. Geology 29: 183–186. Muhuri SK, Dewers TA, Scott TE Jr., and Reches Z (2003) Interseismic fault strengthening and earthquake-slip instability: Friction or cohesion? Geology 31: 881–884. http://dx.doi.org/10.1130/G19601.1. Nakatani M and Mochizuki H (1996) Effects of shear stress applied to surfaces in stationary contact on rock friction. Geophysical Research Letters 23: 869–872. Newman AV, Schwartz SY, Gonzalez V, DeShon HR, Protti JM, and Dorman LM (2002) Along-strike variability in the seismogenic zone below Nicoya Peninsula, Costa Rica. Geophysical Research Letters 29(20): 1977.

137

Niemeijer AR and Spiers CJ (2005) Influence of phyllosilicates on fault strength in the brittle-ductile transition: Insights from rock analogue experiments. Geological Society of London, Special Publication 245: 303–327. Niemeijer AR and Spiers CJ (2006) Velocity dependence of strength and healing behavior in simulated phyllosilicate-bearing fault gouge. Tectonophysics 427: 231–253. http://dx.doi.org/10.1016/j.tecto.2006.03.048. Niemeijer AR and Spiers CJ (2007) A microphysical model for strong velocity weakening in phyllosilicate-bearing fault gouges. Journal of Geophysical Research 112: B10405. http://dx.doi.org/10.1029/2007JB005008. Niemeijer AR, Marone C, and Elsworth D (2008) Healing of simulated fault gouges aided by pressure solution: Results from rock analogue experiments. Journal of Geophysical Research 113: B04204. http://dx.doi.org/ 10.1029/2007JB005376. Niemeijer A, Marone C, and Elsworth D (2010) Frictional strength and strain weakening in simulated fault gouge: Competition between geometrical weakening and chemical strengthening. Journal of Geophysical Research 115: B10207. http:// dx.doi.org/10.1029/2009JB000838. Niemeijer A, Di Toro G, Nielsen S, and Di Felice F (2011) Frictional melting of gabbro under extreme experimental conditions of normal stress, acceleration, and sliding velocity. Journal of Geophysical Research 116: B07404. http://dx.doi.org/ 10.1029/2010JB008181. Noda H and Shimamoto T (2009) Constitutive properties of clayey fault gouge from the Hanaore fault zone, southwest Japan. Journal of Geophysical Research 114: B04409. http://dx.doi.org/10.1029/2008JB005683. Obara K (2002) Nonvolcanic deep tremor associated with subduction in southwest Japan. Science 296: 1679–1681. Ohnaka M, Akatsu M, Mochizuki H, Odedra A, Tagashira F, and Yamamoto Y (1997) A constitutive law for the shear failure of rock under lithospheric conditions. Tectonophysics 277: 1–27. Olsen MP, Scholz CH, and Le´ger A (1998) Healing and sealing of a simulated fault gouge under hydrothermal conditions: Implications for fault healing. Journal of Geophysical Research 103(B4): 7421–7430. Palmer AC and Rice JR (1973) The growth of slip surfaces in the progressive failure of over-consolidated clay. Proceedings of the Royal Society of London, Series A 332: 527–548. Paterson MS (1995) A theory for granular flow accommodated by material transfer via an intergranular fluid. Tectonophysics 245(3–4): 135–151. Peng Z and Gomberg J (2010) An integrated perspective of the continuum between earthquakes 187 and slow-slip phenomena. Nature Geoscience 3: 599–607. Peng Z, Vidale JE, Marone C, and Rubin A (2005) Systematic variations in recurrence interval and moment of repeating aftershocks. Geophysical Research Letters 32: L15301. http://dx.doi.org/10.1029/2005GL022626. Perrin G, Rice JR, and Zheng G (1995) Self-healing slip pulse on a frictional surface. Journal of the Mechanics and Physics of Solids 43: 1461–1495. Persson BNJ (1998) Sliding Friction, Physical Principles and Applications. Berlin/ Heidelberg/NewYork: Springer-Verlag, ISBN: 3-540-63296-4. Rabinowicz E (1951) The nature of static and kinetic coefficients of friction. Journal of Applied Physics 22: 1373–1379. Rabinowicz E (1956) Stick and slip. Scientific American 194: 109–118. Rabinowicz E (1958) The intrinsic variables affecting the stick-slip process. Proceedings of the Physical Society of London 71: 668–675. Ranjith K and Rice JR (1999) Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction. Journal of the Mechanics and Physics of Solids 47: 1207–1218. Rathbun AP and Marone C (2010) Effect of strain localization on frictional behavior of sheared granular materials. Journal of Geophysical Research 115: B01204. http:// dx.doi.org/10.1029/2009JB006466. Rathbun AP and Marone C (2013) Symmetry and the critical slip distance in rate and state friction laws. Journal of Geophysical Research 118. http://dx.doi.org/10.1002/ jgrb.50224. Reinen LA, Weeks JD, and Tullis TE (1994) The frictional behavior of lizardite and antigorite serpentinites: Experiments, constitutive models, and implications for natural faults. Pure and Applied Geophysics 143: 317–358. Renard F, Beaupretre S, Voisin C, et al. (2012) Strength evolution of a reactive frictional interface is controlled by the dynamics of contacts and chemical effects. Earth and Planetary Science Letters 341: 20–34. http://dx.doi.org/10.1016/ j.epsl.2012.04.048. Rice JR (1983) Constitutive relations for fault slip and earthquake instabilities. Pure and Applied Geophysics 121: 443–475. Rice JR (1993) Spatio-temporal complexity of slip on a fault. Journal of Geophysical Research 98: 9885–9907. Rice JR and Ruina AL (1983) Stability of steady frictional slipping. Journal of Applied Mechanics 105: 343–349.

138

The Mechanics of Frictional Healing and Slip Instability During the Seismic Cycle

Rice JR, Lapusta N, and Ranjith K (2001) Rate and state dependent friction and the stability of sliding between elastically deformable solids. Journal of the Mechanics and Physics of Solids 49(9): 1865–1898. Richardson E and Marone C (1999) Effects of normal stress vibrations on frictional healing. Journal of Geophysical Research 104: 28859–28878. Roy M and Marone C (1996) Earthquake nucleation on model faults with rate and state dependent friction: the effects of inertia. Journal of Geophysical Research 101: 13919–13932. Rubin AM (2008) Episodic slow slip events and rate-and-state friction. Journal of Geophysical Research 113: B11414. http://dx.doi.org/10.1029/2008JB005642. Rubin AM and Ampuero JP (2005) Earthquake nucleation on (aging) rate and state faults. Journal of Geophysical Research 110(B11312). http://dx.doi.org/ 10.1029/2005jb003686. Rubinstein SM, Cohen G, and Fineberg J (2004) Detachment fronts and the onset of dynamic friction. Nature 430: 1005–1009. Ruina A (1983) Slip instability and state variable friction laws. Journal of Geophysical Research 88: 10359–10370. Ruina AL (1980) Friction laws and instabilities: A quasistatic analysis of some dry frictional behavior. PhD Thesis, Brown University. Saffer DM and Marone C (2003) Comparison of smectite and illite frictional properties: Application to the updip limit of the seismogenic zone along subduction megathrusts. Earth and Planetary Science Letters 215: 219–235. Saffer DM, Frye KM, Marone C, and Mair K (2001) Laboratory results indicating complex and potentially unstable frictional behavior of smectite clay. Geophysical Research Letters 28: 2297–2300. http://dx.doi.org/ 10.1029/2001GL012869. Schleicher A, van der Pluijm B, Solum J, and Warr L (2006) Origin and significance of clay-coated fractures in mudrock fragments of the SAFOD borehole (Parkfield California). Geophysical Research Letters 33. http://dx.doi.org/ 10.1029/2006GL026505. Scholz CH (1988a) The critical slip distance for seismic faulting. Nature 336: 761–763. Scholz CH (1988b) The brittle-plastic transition and the depth of seismic faulting. In: Zankl H, Delliere J, and Prashnowsky A (eds.) Detachment and Shear. Geologische Rundschau 77: 319–328 Scholz CH (1998) Earthquakes and friction laws. Nature 391: 37–42. Scholz CH (2002) The Mechanics of Earthquakes and Faulting, p. 439. New York: Cambridge. Scholz CH, Molnar P, and Johnson T (1972) Detailed studies of frictional sliding of granite and implications for the earthquake mechanism. Journal of Geophysical Research 77: 6392–6406. Scott D, Marone C, and Sammis C (1994) The apparent friction of granular fault gouge in sheared layers. Journal of Geophysical Research 99: 7231–7247. Segall P and Rice JR (1995) Dilatancy, compaction, and slip instability of a fluid infiltrated fault. Journal of Geophysical Research 100: 22155–22173. Shelly DR, Beroza GC, and Ide S (2007) Non-volcanic tremor and low frequency earthquake swarms. Nature 446: 305–307. Shimamoto T and Logan JM (1981a) Effects of simulated clay gouges on the sliding behavior of Tennessee sandstone. Tectonophysics 75: 243–255. http://dx.doi.org/ 10.1016/0040-1951(81)90276-6.

Shimamoto T and Logan JM (1981b) Effects of simulated fault gouge on the sliding behavior of Tennessee Sandstone: Nonclay gouges. Journal of Geophysical Research 86: 2902–2914. http://dx.doi.org/10.1029/JB086iB04p02902. Sleep NH, Richardson E, and Marone C (2000) Physics of friction and strain rate localization in simulated fault gouge. Journal of Geophysical Research 105: 25875–25890. Tadokoro K and Ando M (2002) Evidence for rapid fault healing derived from temporal changes in S wave splitting. Geophysical Research Letters 29. http://dx.doi.org/ 10.1029/2001GL013644. Tenthorey E, Cox SF, and Todd HF (2003) Evolution of strength recovery and permeability during fluid-rock reaction in experimental fault zones. Earth and Planetary Science Letters 206: 161–172. Tenthorey E and Cox SF (2006) Cohesive strengthening of fault zones during the interseismic period: An experimental study. Journal of Geophysical Research 111: B09202. http://dx.doi.org/10.1029/2005JB004122. Tesei T, Collettini C, Carpenter BM, Viti C, and Marone C (2012) Frictional strength and healing behavior of phyllosilicate-rich faults. Journal of Geophysical Research 117: B09402. http://dx.doi.org/10.1029/2012JB009204. Tse ST and Rice JR (1986) Crustal earthquake instability in relation to the depth variation of frictional slip properties. Journal of Geophysical Research 91: 9452–9472. Tullis TE (1988) Rock friction constitutive behavior from laboratory experiments and its implications for an earthquake prediction field monitoring program. Pure and Applied Geophysics 126: 555–588. Tullis TE (1996) Rock friction and its implications for earthquake prediction examined via models of Parkfield earthquakes. Proceedings of the National Academy of Sciences of the United States of America 93: 3803–3810. Unsworth M, Malin P, Egbert G, and Booker J (1997) Internal structure of the San Andreas fault at Parkfield, CA. Geology 25: 359–362. Vidale JE, Ellsworth WL, Cole A, and Marone C (1994) Variations in rupture process with recurrence interval in a repeated small earthquake. Nature 368: 624–626. http://dx.doi.org/10.1038/368624a0. Vrolijk P (1990) On the mechanical role of smectite in subduction zones. Geology 18: 703–707. Vrolijk P and van der Pluijm B (1999) Clay gouge. Journal of Structural Geology 21: 1039–1048. Yasuhara H, Elsworth D, and Polak A (2003) A mechanistic model for compaction of granular aggregates moderated by pressure solution. Journal of Geophysical Research 108(B11): 2530. http://dx.doi.org/10.1029/2003JB002536. Yasuhara H, Marone C, and Elsworth D (2005) Fault zone restrengthening and frictional healing: The role of pressure solution. Journal of Geophysical Research: Solid Earth 110: B06310. Zheng B and Elsworth D (2012) Evolution of permeability in heterogeneous granular aggregates during chemical compaction: Granular mechanics models. Journal of Geophysical Research: Solid Earth 117: B03206. Zheng B and Elsworth D (2013) Strength evolution in heterogeneous granular aggregates during chemo-mechanical compaction. International Journal of Rock Mechanics and Mining Sciences 60. Zheng G and Rice JR (1998) Conditions under which velocity-weakening friction allows a self-healing versus a cracklike mode of rupture. Bulletin of the Seismological Society of America 88: 1466–1483.