G 3. AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical Society

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1 Geosystems G 3 AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical Society Article Volume 7, Number October 2006 Q10008, doi: /2006gc ISSN: Frictional dilatancy Norman H. Sleep Department of, Stanford University, 397 Panama Mall, Stanford, California 94305, USA (norm@geo.stanford.edu) [1] Frictional sliding dilates gouge or increases the separation of sliding surfaces. A common hypothesis is that the rate of dilatational strain scales linearly with the rate of shear strain. The proportionality constant is called the dilatancy coefficient. Real contact theory of friction may explain this feature. Moving contact asperities produce damaged regions with elastic strains scaling to the ratio of the real strength of asperities to the elastic modulus. Balance between this local strain-energy production rate and macroscopic work against normal traction indicates that the dilatancy coefficient scales with this ratio. So do the average slopes on a mature rough sliding surface if opening-mode cracks are unimportant. The result is compatible with the observed dilatancy coefficient of quartz gouge, 4%. Components: 4613 words, 4 figures. Keywords: tribology; asperity; exponential creep; fault gouge. Index Terms: 5104 Physical Properties of Rocks: Fracture and flow; 5120 Physical Properties of Rocks: Plasticity, diffusion, and creep; 7209 Seismology: Earthquake dynamics (1242). Received 29 May 2006; Revised 21 August 2006; Accepted 30 August 2006; Published 11 October Sleep, N. H. (2006), Frictional dilatancy, Geochem. Geophys. Geosyst., 7, Q10008, doi: /2006gc Introduction [2] The gouge on major faults accommodates numerous earthquakes. Over time the gouge approaches a quasi steady state where production in porosity during earthquakes balances the compaction of gouge between earthquakes. Rock physicists have documented analogous effects on laboratory samples. Shear strain increases the porosity of laboratory fault gouge, reducing its coefficient of friction. The sample heals, that is it becomes stronger by compaction during very slow sliding. In general, the frictional strength of a sample depends on both its history and the current sliding velocity. [3] Rate and state friction mathematically represents these effects with the state variable y representing the history. Segall and Rice [1995] included dilatancy in this formalism by expressing the state variable of a fault in terms of porosity. Sleep [1997] and Sleep et al. [2000] showed that their relationship has simple kinematic interpretation that dilatancy produces porosity at a rate proportional to the shear strain rate. The treatment of rate and state friction by Beeler and Tullis [1997] also has this feature. While this hypothesis seems reasonable, no theory is available that expresses the ratio of shear strain rate to dilatant strain rate in terms of measurable asperity-scale physical parameters. The purpose of the paper is to examine this issue. [4] I begin with a short review of rate and state friction to put the relevant frictional parameters in context. I then discuss kinematic dilatancy at very low normal traction. I then apply the physics of Copyright 2006 by the American Geophysical Union 1 of 9

2 creep at contact asperities to obtain dimensional scaling relationships for dilatancy. 2. Rate and State Friction [5] As noted in the introduction, the friction on a sliding surface depends on its past history. It also depends on the instantaneous sliding velocity. Ruina [1980] showed that using strain rate rather than macroscopic sliding velocity facilitates discussion of strain rate localization and aids comparison with other flow laws. I thus use a convenient form for instantaneous friction in terms of the instantaneous strain rate t ¼ P m 0 þ a ln e 0 =e 0 0 þ b ln ð y=ynorm Þ ; ð1þ where m 0 is the coefficient of friction at reference conditions, a and b are small dimensionless constants, the strain rate is e 0 V/W where V is sliding velocity and W is the thickness of the shear zone the reference strain rate is e 0 0 V 0 /W, where V 0 is a reference velocity, and y norm is normalizing value for the state variable, which I discuss below. [6] It is also necessary to have an evolution law that describes the combined effects of damage from sliding and healing. Dieterich s [1979] evolution law represents these effects explicitly with ¼ e0 0 Pa=b ye0 ; e int P a=b e int 0 where the first term represents healing and the second damage. The variable t is time, the intrinsic strain is e int D c /W, where D c is the critical displacement to significantly change the properties of the sliding surface, a is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from Linker and Dieterich [1992], and P 0 is a reference normal traction. It arises formally from the steady state value of the state variable ð2þ y ss ¼ Pa=b e 0 0 P a=b 0 e 0 : ð3þ If the steady state coefficient of friction is independent of normal traction y norm ¼ Pa=b : ð4þ P a=b 0 Segall and Rice [1995] represent the state variable in terms of the porosity f as y ¼ exp f f ; ð5þ C e where f is a reference porosity and C e is a dimensionless material property. [7] The Dieterich [1979] evolution law then has simple kinematic implications f 0 ¼ C ee 0 e int C ee 0 0 Pa=b ; ð6þ ye int P a=b 0 where the first term represents that the increase of porosity from dilatancy is proportional to the shear strain rate. This defines a dilatancy coefficient b C e /e int. Beeler and Tullis [1997] obtain an equivalent result. The second term represents power law compaction creep during holds where the exponent is n = a/b. [8] The Ruina [1983] porosity evolution law does not have a simple implication " f 0 ¼ e 0 C!# e ln þ e0 e int e 0 ln ypa=b 0 : ð7þ 0 P a=b Still, it can be derived with the assumption that the dilatancy rate scales linearly with the shear strain rate [Sleep, 2005, 2006]. [9] To reiterate in terms of macroscopic processes, seismic dilatancy has an important direct effect on the physics of earthquakes. From (5), porous gouge is weak. This weakness is essential to unstable sliding. The steady state coefficient of friction in (1) and (3) decreases with strain rate if a > b. Dilatancy may also affect earthquake nucleation; increased porosity decreases the fluid pressure in a sealed fault zone. This effect increases the effective stress and increases the friction on the fault, tending to quench earthquakes. One would thus like to know how dilatancy arises before exporting laboratory friction to crustal faults. [10] I begin by reviewing familiar processes that are probably not relevant to mature fault surfaces in section 3. I then relate dilatancy to elasticity and asperity strength in section Kinematic and Crack Dilatancy [11] Pore space is needed of a rigid material to deform at low normal traction. Conversely, broken surfaces of brittle materials, like china, do not fit 2of9

3 Geosystems G 3 sleep: frictional dilatancy /2006GC Figure 1. A square block pivoting between two sheets illustrates features of kinematic dilatancy. The opening displacement Dz equals the shear displacement Dx. back together. It is tempting but probably incorrect to attribute frictional dilatancy to this geometrical effect. There are two problems with this approach. [12] First, the dilatancy coefficient is too high, that is, of the order of one. To see this, consider a rigid square between two rigid sheets. The sheets need to come apart for the block to roll. When the block pivots about a corner, the instantaneous shear velocity of the plates and separation velocity are the same (Figure 1). [13] Second, there is no rate dependence of the steady state velocity as implied by (5), (6), and (7). The room needed for rigid blocks to roll and move around each other depends only on their shape. [14] Neither is frictional dilatancy likely to be related to the behavior of cracks in an intact elastic material. This effect arises, for example, in pure shear when the mean stress stays constant. All the cracks oriented perpendicular the most compressive the principal stress close when it reaches a critical value and further increase in this stress causes no more crack compaction. Increasing tensional stress in the other direction continues to open cracks, leading to a net dilatancy at constant mean stress. This is a quasistatic process with no simple rate dependence. The strains are in principle recoverable when the stresses are removed, but some kinematic dilatancy occurs because crack surfaces may not fit back together. 4. Real Contact Theory and Dilatancy [15] Modern theories of friction postulate that thermally active creep occurs at asperity contacts where the real stresses are several GPa [e.g., Berthoud et al., 1999; Baumberger et al., 1999; Rice et al., 2001; Nakatani, 2001; Nakatani and Scholz, 2004]. Beeler [2004] develops an analogous derivation term terms of high stresses at crack tips. Asperities are more convenient mathematically for present purposes. [16] With forethought, I begin with contact theory that yields the first-order expression t = m 0 P. This result suffices for dimensionally obtaining the dilatancy coefficient. The real strain rate at an asperity depends exponentially on the real stress. The scalar equation is h e 0 ¼ e 0 0 exp t i ; ð8þ s where s is a stress scale of the order of 0.1 GPa and e 0 0 is a constant with dimensions of strain rate. The real stress t is proportional to the second invariant pffiffiffiffiffiffiffiffiffi of the deviatoric stress t ij t ij. I normalize it so that it is the resolved shear stress in twodimensional plane strain. The commonly p ffiffiffi used von Mises stress is this stress times 3. [17] I stay dimensional (until section 4.3) so that I do not obscure simple relationships with complicated geometrical factors. I start with the wellknown derivation for the constant coefficient of friction m 0, the leading term in (1). Failure (for example in quartz) occurs with the real shear traction of the order of 10 GPa from indenter data by Goldsby et al. [2004]. The quantity is large enough compared to s to warrant approximating the flow law with failure at a shear yield stress t y. A yield stress for normal traction P y arises similarly. 3of9

4 Figure 2. (a) A cylindrical asperity of radius A has a circular footprint on the substrate. It damages an area scaling to 2VAt in time t. (b) A cross section perpendicular to the track. The damaged area in cross section scales with A 2. The yield stress P y acts on the asperity and the underlying substrate. (The effects of macroscopic shear and normal traction both act on an asperity and effect the stress invariant t within the asperity so there is an implicit geometrical assumption about the contact, which needs to be considered to obtain the evolution laws (6) and (7) [Sleep, 2005, 2006].) In the traditional derivation, microscopic properties scale to the macroscopic ones, the coefficient of friction is m 0 t P ¼ t y P y : The derivation to this point is well known. I extend it to include anelastic strains within regions overrun by contact asperities. I apply the dimensional result that the real shear and normal tractions are comparable and that they scale with macroscopic properties Elastic Strain Energy [18] I include elasticity with some forethought and with the justification that an infinitely rigid material stores no elastic strain energy. I justify ignoring inertia at the end of this section. I follow the effects of the normal traction P as I am after a dimensional ð9þ result. That is, I do not distinguish at the stage between t and P, whose ratio is the order of 1. I initially let the shear modulus G dimensionally represent the elastic modulus for the actual geometry. I implicitly apply Saint-Venant s principle [Sokolnikoff, 1956, pp ] throughout my discussion, the difference between loading by asperities and uniform loading by the macroscopic stresses away from the asperity being examined does not matter. [19] I consider friction between two surfaces to avoid the geometrical complexity of gouge. That is, I associate the separation rate of the plates with dilatancy in gouge. I follow the path of a representative asperity of radius A (Figure 2). For bookkeeping purposes, my asperity stays attached to the upper sheet in the figure. The lower sheet is initially planar and parallel to the upper sheet. The contact bears the macroscopic stresses over a region of radius R. That is, pa 2 P y ¼ pr 2 P: ð10þ The asperity crosses a surface area of the lower substrate sheet of 2AVt over time t. The yield stresses act on the base of the asperity, causing the immediate substrate to yield downward to a depth scaling with the radius A. The volume of damaged substrate produced over the time is dimensionally VA 2 t. [20] Stresses in the substrate change as the asperity moves. The aftermath of its passage on the lower sheet is relevant. For simplicity, I consider that the asperity slides on a planar surface parallel to the sheets. The flat position of the deformed material exposed after the passage of the asperity is that with the asperity on it with normal traction P y. With the load of the asperity removed, the exposed substrate material springs back, but stresses remain. The deformation is comparable to that from a negative normal traction scaling to P y over the exposed area. I use this feature as a statistical average to obtain macroscopic scaling relationships for dilatancy. [21] I begin with microscopic energies. The elastic strain energy per volume in the region damaged by passage of the asperity scales as N ¼ P2 y G : ð11þ This assumes that the stresses within the damaged region are comparable to the yield stress. See 4of9

5 Appendix A for a simpler example. The rate that the moving asperity creates elastic strain energy is dimensionally Q ¼ VA 2 N ¼ VA2 P 2 y G : ð12þ More mathematically, the equilibrium elastic strain energy of an unloaded region is a minimum compatible with the internal anelastic strains within the material and the macroscopic boundary forces (here the normal traction) [e.g., Sokolnikoff, 1956, p. 383]. I assume on the basis of the example in Appendix A that comparable fractions of the available energy go into local elastic strains and work against normal traction (dilatancy). I use the total energy Q to dimensionally represent both the local strain energy and the energy associated with dilatancy. [22] I now equate microscopic and macroscopic effects. From (10), the macroscopic production rate of elastic energy is Q ¼ VR2 P y P G ; ð13þ which is independent of the asperity radius. I compare this rate with the rate of dissipation of energy over this area from macroscopic friction, which is Q f ¼ VR 2 mp: ð14þ This yields that the ratio of elastic strain energy production to frictional energy dissipation scales as where the coefficient of friction m is set to 1 because I have already dropped other factors of this order in my dimensional presentation. [23] As already noted, the elastic strain energy deforms the sliding surface. It will also deform a gouge. Part of the elastic strain energy is available to push surfaces and grains apart. It can also cause spalling/cracking. Some of the elastic strain thus acts macroscopically to dilate the gap between the surfaces with the macroscopic velocity V z. [24] Dimensionally applying the variation principle of minimum elastic energy [e.g., Sokolnikoff, 1956, p. 383], the energy available for work on macroscopic boundaries scales with the elastic strain energy. See Appendix A. Kinematically, the macroscopic work per time from dilatancy over the area is Q d ¼ V z R 2 P: ð16þ As Q d scales with the available energy Q, comparing (14) and (16), yields the ratio of work done to dilate and work done by friction is Q d /Q f Q/Q f P y /G. As I did not retain factors of the order of unity and as the coefficient of friction is near its reference value m 0, I do not distinguish this energy result m 0 PV z = btv x from the kinematic statement in (6)V z = bv x that the dilatational velocity (or strain rate) is proportional to the shear velocity. I return to the value of the dilatancy coefficient in the section 4.3. [25] Note that inertial effects vanish in the limit of slow sliding. The forces from the tractions P y and t y are essentially independent of the sliding velocity. The vertical velocity of the surface scales with the surface slope times the horizontal velocity; its sign changes on the timescale A/V for an asperity to traverse a point. The acceleration thus scales as V 2 /A Surface Slopes and Roughness [26] The concept that the stresses within unloaded anelastically damaged regions scale with the yield stresses t y and P y associated with loading by asperities leads to further geometrical results. Consider, first a substrate region just vacated by an asperity of radius A. The effective negative load then scales as pa 2 P y. Treating this as a point source, the vertical displacement is Q Q f P y G ; ð15þ U ¼ P ya 2 ðl þ 2GÞ 4Gðl þ GÞr ; ð17þ where l is the second Lamé constant and r is the radial distance along the substrate surface from the center of the point load [Sokolnikoff, 1956, p. 340]. Timoshenko and Goodier [1970, p. 404] give the full expressions for a distributed circular load in terms of elliptic integrals. The radial slope at the edge of the disk r = A in ¼ P yðl þ 2GÞ 4Gðl þ GÞ ¼ 3P y 8G ; ð18þ where G = l gives the second equality. This equation provides a scaling for strains in the damaged surface. It also provides a scaling for the slopes over which asperities pass (Figure 3). 5of9

6 Figure 3. A pointed asperity moves up a slope, pushing the surfaces apart. Typical slopes scale with t y /G. [27] The derivation has implications to the roughness of a surface. In terms of equivalent stresses, a Poisson distribution in 2 dimensions might represent sparse damaged regions on a nearly fresh surface. Gaussian white noise at wavelengths longer than the asperity dimension, A, might represent the equivalent stresses acting on a mature surface. In both cases, the spectrum of stress is flat [e.g., Turcotte, 1992, pp ] in the wavelength range between contact dimension and macroscopic scale (say of a laboratory sample). For shorter wavelengths, one needs to consider the distribution of stresses beneath the asperities and the tendency for asperities to smear along the sliding direction. [28] Distributed white-noise stress has a simple geometrical consequence. As elastic displacement at constant stress scales inversely to wavelength, the surface height to wavelength is self-similar [e.g., Turcotte, 1992, p. 83; Batrouni et al., 2002]. This is equivalent to the elastic strains and slopes being independent of wavelength. This feature applies to most of the surface. Random roughness keeps surfaces apart and produces stress concentrations at protrusions [Batrouni et al., 2002]. A process (here anelastic strains from damage beneath asperities) creates surface roughness thus also produces dilatancy. [29] Note that pointy asperity contacts are likely to exist. Fracture (including spalling) produces sharp protruding corners. At low normal tractions, such protrusions will form one side of many contact asperities. That is, the contacts will locally be indenter against flat surface as assumed in the models Appraisal and Application to Quartz [30] The dimensional equation (15) and roughness equation (18) yield estimates of the dilatancy coefficient in terms of parameters that can be measured independently of friction. I appraise the derivation with quartz. It is a well-characterized laboratory material that occurs within natural faults. The shear modulus for quartz is 44 GPa. [31] I use invariants, as the elastic strain energy in (11) is proportional to e ij t ij which is proportional to t ij t ij. As friction involves shear, I use deviatoric stress normalized to give resolved shear stress in two dimensions t and the shear modulus G. The strain energy per volume is then t 2 /2G and the dilatancy coefficient is approximately t y /2G. The actual value should be somewhat less that this as only part of the elastic strain energy does work against macroscopic normal traction. See Appendix A. One would need numerical calculations to obtain a more precise estimate. [32] I need to convert with the real strengths in the literature to strengths in shear. Tribologists determine real strength in different ways and express it with various normalizations. Poirier [1990, p. 38] gives a rule of thumb theoretical strength limit t/g of 1/10, which give a real strength of 4.4 GPa. Dieterich and Kilgore [1996] measured real stresses on quartz surfaces in contact and obtained a range of 7 to 14 GPa. Intuitively upper limit represents the maximum strength. There are two ways that might be converted to shear strength. It could represent the real normal traction in shear where the real shear traction is m 0 times it or 10 GPa. Alternatively, failure occurs in uniaxial compres- 6of9

7 sion of the contacts. The equivalent shear stress is the scales as G/E in an incompressible substance or by a factor of 1/3. This implies that a real strength of 4.7 GPa. Goldsby et al. [2004] obtain Vickers hardnesses of 12 to 14 GPa from nano-indentation tests, which they consider compatible with Dieterich and Kilgore s [1996] data. Finite element codes exist for obtaining the stress invariant at indentation failure [e.g., Bouzakis and Michailidis, 2006], but I am aware of no work on quartz or a similar silicate. [33] Given this uncertainty, I let the real shear strength of quartz range from 4 to 10 GPa. The predicted dilatancy coefficient t y /2G is then %. The lower end of this range is reasonable, 5% is traditional rounded estimate for the dilatancy coefficient of gouge. Experiments where the normal traction is increased are effective for determining the dilatancy coefficient as compaction occurs during holds as implied by the Dieterich [1979] evolution law (6). Laboratory results are not strongly affected by strain rate localization. Sleep et al. [2000] obtained %. The work of Segall and Rice [1995] yields 3.7%. 5. Conclusions [34] Simple dimensional arguments suggest that moving contact asperities leave anelastic strains scaling with the ratio of real strength to elastic modulus t y /G in their wake. Energy balance indicates that the dilatancy coefficient scales with this ratio. Poirier [1990, p. 38] notes that this ratio does not vary a lot in macroscopically brittle substances. Physically the ratio represents a strain that significantly deforms a crystal lattice. My hypothesis has the attractive attribute that sliding continually produces surface roughness as well as porosity. The sliding surface thus does not get worn smooth with all places in contact. [35] The hypothesis is in principle testable in the laboratory. One could use hard metal, mica, or clay where the ratio t y /G is lower than in strong silicates. The hypothesis can be modeled numerically with viscoelastic code that can handle exponential creep over small regions and the free surfaces associated with porosity. I am not aware of a code that suffices. An elastic-plastic code would provide an approximate model. [36] I conclude with some caveats that lead to possible physical experiments. First, I presumed that the derivation extends to gouge. That is, I equate gouge to a complicated set of sliding contacts where the physics of a sliding surface locally holds. It is conceivable that there is also reversible dilatancy in gouge (N. Beeler, personal communication, 2006). The elastic normal strain across the gouge may depend on the contact geometry and hence the shear traction and sliding velocity. I also obtained the fractal roughness of a sliding surface. I did not consider fracture explicitly. N. Beeler (personal communication, 2006) suggests that fracture (that is, opening-mode cracks) may dominate over asperity contacts in producing roughness. Appendix A: Bent Beam Analogy [37] The deformation in the wake of moving asperities is too complicated to expect an analytical solution. I use a bent beam (Figure A1) to illustrate the concepts of the paper. [38] The beam is bent far beyond its elastic limit. The fiber stress within the beam is the yield stress in both the upper and lower halves. The bending moment (per length in cross section) is M ¼ Z Z0 Z 0 P y jjdz z ¼ P y Z0 2 ; ða1þ where Z 0 is the half thickness of the beam, z is the dummy variable for distance from the centerline, and the fiber stress (tension positive by convention) is the yield stress P y. The elastic strain energy (per length) is W ¼ 1 B Z Z0 Z 0 Py 2 dz ¼ 2Py 2Z 0 B ; ða2þ where B is an elastic modulus. The beam springs back some when the load is removed. The new bending moment is Z Z0 M ¼ P y jj z c z2 Z 0 Z 0 dz ¼ P y Z c ; ða3þ where the dimensionless coefficient c multiplies a fiber stress from bending that is proportional to the distance from the center line. If unloaded, the beam flexes so the c = 3/2. The elastic strain energy is W ¼ 1 Z " Z0 Py 2 signðþ c z z # 2 dz: B Z 0 Z 0 ða4þ 7of9

8 work scales to the total elastic strain energy at the elastic limit as assumed in (13). Here 3 = 4 of the original elastic strain energy is available for external work. This justifies my assumption that the available fraction on frictional surfaces and gouge is of the order of 1. Acknowledgments [39] This research was in part supported by NSF grant EAR This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR and USGS Cooperative Agreement 02HQAG0008. The SCEC contribution number for this paper is The 2004 and 2005 SCEC meetings showed the need to understand the physical basis of friction. David Goldsby and Jim Rice promptly answered my questions. Nick Beeler and Masao Nakatani provided helpful reviews. References Figure A1. A beam bent far beyond its elastic limit illustrates the rebound of a highly stressed region. (a) The fiber stress in the loaded beam is tensional above and compressional below. (b) The fiber stress in the loaded beam is everywhere at the yield stress except at the centerline. (c) The fiber stress in the relaxed beam has minimum elastic strain energy and no net bending moment. The energy going from Figure A1b to A1c is available to do external work. The minimum as expected from the variation principle [e.g., Sokolnikoff, 1956, p. 383] occurs at c = 3/2 where there is no net bending moment. It is W min ¼ P2 y Z 0 2B ; ða5þ This expression illustrates the inference that the amount of energy in the beam available for external Batrouni, G. G., A. Hansen, and J. Schmittbuhl (2002), Elastic response of rough surfaces in partial contact, Europhys. Lett., 60(5), Baumberger, T., P. Berthoud, and C. Caroli (1999), Physical analysis of state- and rate-dependent friction law, II. Dynamic friction, Phys. Rev. B, 60(6), Beeler, N. M. (2004), Review of the physical basis of laboratory-derived relations for brittle failure and their implications for earthquake occurrence and earthquake nucleation, Pure Appl. Geophys., 161(9 10), Beeler, N. M., and T. E. Tullis (1997), The roles of time and displacement in velocity-dependent volumetric strain of fault zones, J. Geophys. Res., 102, 22,595 22,609. Berthoud,P.,T.Baumberger,C.G Sell,andJ.-M.Hiver (1999), Physical analysis of the state- and rate-dependent friction law: Static friction, Phys. Rev. B., 59(22), 14,313 14,327. Bouzakis, K. D., and N. Michailidis (2006), Indenter surface area and hardness determination by means of a FEMsupported simulation of nanoindentation, Thin Solid Films, 494, Dieterich, J. H. (1979), Modeling of rock friction: 1. Experimental results and constitutive equations, J. Geophys. Res., 84(B5), Dieterich, J. H., and B. D. Kilgore (1996), Imaging surface contacts: Power law contact distributions and contact stresses in quartz, calcite, glass and acrylic plastic, Tectonophysics, 256, Goldsby, D. L., A. Rar, G. M. Pharr, and T. E. Tullis (2004), Nanoindentation creep of quartz, with implications for rateand state-variable friction laws relevant to earthquakes mechanics, J. Mater. Res., 19(1), Linker, M. F., and J. H. Dieterich (1992), Effects of variable normal traction on rock friction: Observations and constitutive equations, J. Geophys. Res., 97(B4), Nakatani, M. (2001), Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology, J. Geophys. Res., 106(B7), 13,347 13,380. Nakatani, M., and C. H. Scholz (2004), Frictional healing of quartz gouge under hydrothermal conditions: 2. Quantitative 8of9

9 interpretation with a physical model, J. Geophys. Res., 109, B07202, doi: /2003jb Poirier, J.-P. (1990), Creep of Crystals, High-Temperature Deformation Processes in Metals, Seamics and Minerals, 260 pp., Cambridge Univ. Press, New York. Rice, J. R., N. Lapusta, and K. Ranjith (2001), Rate and state dependent friction and the stability of sliding between deformable solids, J. Mech. Phys. Solids, 49(9), Ruina, A. (1980), Friction laws and instabilities: A quasistatic analysis of some dry frictional material, Ph.D. thesis, 99 pp., Brown Univ., Providence. Ruina, A. (1983), Slip instability and state variable laws, J. Geophys. Res, 88(B12), 10,359 10,370. Segall, P., and J. R. Rice (1995), Dilatancy, compaction, and slip-instability of a fluid-penetrated fault, J. Geophys. Res., 100(B11), 22,155 22,171. Sleep, N. H. (1997), Application of a unified rate and state friction theory to the mechanics of fault zones with strain localization, J. Geophys. Res., 102(B2), Sleep, N. H. (2005), Physical basis of evolution laws for rate and state friction, Geochem. Geophys. Geosyst., 6, Q11008, doi: /2005gc Sleep, N. H. (2006), Real contacts and evolution laws for rate and state friction, Geochem. Geophys. Geosyst., 7, Q08012, doi: /2005gc Sleep, N. H., E. Richardson, and C. Marone (2000), Physics of strain localization in synthetic fault gouge, J. Geophys. Res., 105(B11), 25,875 25,890. Sokolnikoff, I. S. (1956), Mathematical Theory of Elasticity, 476 pp., McGraw-Hill, New York. Timoshenko, S. P., and J. N. Goodier (1970), Theory of Elasticity, 567 pp., McGraw-Hill, New York. Turcotte, D. L. (1992), Fractals and Chaos in Geology and, 221 pp., Cambridge Univ. Press, New York. 9of9