Microscopic elasticity and rate and state friction evolution laws



[1] Rate and state friction formalism represents the dependence of macroscopic shear traction τM on sliding velocity V and the history of the sliding surface. In macroscopic terms, τM = PN[μ0 + a ln(V/Vref) + b ln(ψ/ψnorm)], where PN is normal traction, μ0 is the coefficient of friction, a ≈ b ≈ 0.01 are small dimensionless parameters, Vref is a reference velocity, ψ is the state variable that depends on history, and ψnorm represents the effect of changes in normal traction. This representation does not consider microscopic elasticity and is inadequate over very small times. The apparent value of a just after a small decrease in shear traction is a factor of a few larger than its traditional value of ∼0.01. Changes of microscopic elastic strain energy may cause this effect. Microscopic elasticity affects friction after changes in normal traction. Shear traction does not change instantly after a sudden change in normal traction because time is required for real contact area to change. A hybrid of the aging law (where ψincreases linearly with time during holds) and the slip law behavior (where the state variable does not change in the limit of zero sliding velocity) is necessary. Slip-law behavior dominates near steady state and also applies to sudden initial sliding where the state variable and porosity are far from steady state. Porosity increases from its initial value toward the steady state value over slip scaling with the critical displacementDc. The ratio of dilatant to shear strain in low porosity material is a modest fraction of 1 and related to the construct of dilatancy angle in engineering.

1. Introduction

[2] Geophysicists apply the construct of rate and state friction to modeling laboratory experiments and crustal faults. They would like to consistently represent steady state fault creep, earthquake nucleation, rupture tips including those with rapid variations in normal traction, dilatancy, healing of faults between earthquakes, and the initial failure of intact rock. To do this, tribologists use a state variable ψ to represent the past history of the frictional material. Recent experiments by Nagata et al. [2012] bear on the appropriate form for an evolution law for the state variable and the micromechanics of friction. In particular, the traditional formalism is deficient at very short times after a sudden change in shear traction and sudden changes in normal traction. The purposes of this paper are to reinterpret the short time results of Nagata et al. [2012] in terms of microscopic elasticity following Sleep [2010] and to discuss their work on evolution laws expanding on the works of Kato and Tullis [2001] and Sleep [2005, 2006a, 2006b]. In section 4, I discuss construct of dilatancy angle of intact rocks [e.g., Alejano and Alonso, 2005] with the intent of constraining the evolution law for initial sliding of low-porosity gouge and intact rocks. Overall objectives are to suggest feasible new experiments are well as to interpret existing ones.

2. Rate and State Formalism

[3] The well-known formalism of rate and state friction represents changes between static and sliding friction with a state variableψ that depends on the past history of sliding. I use the strain rate form [Ruina, 1980] of a unified theory compiled by Sleep [1997, 2006a] and Sleep et al. [2000] to facilitate comparison with microscopic affects. The instantaneous shear traction τM as a function of normal traction PN is then

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where the subscript M indicates macroscopic and mesoscopic (scale of a few grains) properties with forethought to distinguish them from microscopic quantities. The dominant term μ0 ≈ τM/PN represents the approximation that the coefficient of friction has a constant value for a given surface (Amontons' law), a and b are small ∼0.01 dimensionless constants, math formula is the shear strain rate, V is sliding velocity, W is the width of the sliding zone, math formula is a reference strain rate that may set to a convenient value thereby setting μ0, and the ψnorm is the normalizing value of the state variable which I constrain in the next paragraph; the reference velocity is then math formula.

[4] As discussed by Ruina [1980], alternative formulations of the state variable and equation (1) yield mathematically equivalent physical predictions. Some tribologists including Nagata et al. [2012] use state variables that depend on ln(ψ), in their case: Φ ≡ τM − aPNln(V/Vref) = PNμ0 + PNbln(ψ). Using Ψ ≡ Φ/PN removes the explicit dependence of this state variable on normal traction, compacting notation when normal traction changes with time.

2.1. Traditional Evolution Laws

[5] Tribologists have suggested various evolution laws to represent changes in the state variable over time. The aging law has kinematically explicit terms [Dieterich, 1979]

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where the first term represents healing and the second damage that is proportional to the strain rate. The variable t is time, εint is the strain to significantly change the properties of the sliding surface (it is the critical displacement Dc divided by the width W of the slipping zone), α is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from the work of Linker and Dieterich [1992], and Pref is a reference normal traction. The state variable increases linearly with time during a hold math formula. An alternative slip evolution law by Ruina [1983] has the property that no healing occurs when sliding is stopped. In the notation of this paper,

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The steady state value of the state variable by intent is the same in (2) and (3)

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Moreover, the change in the state variable after a small step-like change in velocity very similar in(2) and (3) (Figure 1). With regard to changes in normal traction, assuming that the coefficient of friction at a given strain rate is independent of normal traction yields that the normalized value of the state variable in (1) is

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I explicitly consider sudden changes in normal traction in section 4.4.

Figure 1.

Change in the state variable over time since a change of slip velocity from 1 to 1.4 (normalized to Dc per unit time) versus state variable (normalized so that it is 1 at steady state velocity 1). The aging law (dotted) differs slightly from the slip law (solid). Hybrid law (8) is identical to the slip law if V ≫ Vmin. Hybrid law (9a) (dashed) is essentially the same. Thus, experiments where velocity slightly changes should yield the same value of Dc for all these evolution laws and hence the same value of εint.

[6] It is well known that low-porosity gouge dilates during frictional sliding and that laboratory gouge compacts during holds.Segall and Rice [1995] obtained a simple relationship between porosity f and state

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where ϕ is a reference porosity and Cε is a dimensionless material property. Note that shear traction in (1) and porosity in (6) both depend on the logarithm of the state variable. The aging law in terms of porosity is

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When ψ is much larger than its steady state value, the ratio of dilatational to shear strain is Cε/εint. This quantity is insensitive to strain localization [Sleep et al., 2000]. I discuss this convenient relationship in terms of real contact theory in section 3 and its inapplicability for starting friction within intact rock in section 4.

2.2. Hybrid Evolution Laws

[7] Nagata et al. [2012] reviewed literature on evolution laws. They pointed out that the slip law appears to better represent friction after moderate changes in sliding velocity and that the aging law better represents the effect of long holds. Kato and Tullis [2001] obtained a hybrid law that addresses this issue. In macroscopic notation at given normal constant traction, it is

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where Vmin is an empirical parameter with the dimensions of velocity and θ distinguishes this state variable from ψ. At high sliding velocities, the first term is zero and the second term is the slip law (3). During holds V = 0, the equation reduces to the aging law (2).

[8] Sleep [2005] suggested an alternative hybrid law,

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The parameter cage is the relative fraction of evolution governed by the aging law. The strain εage should be close to the strain εslipso that the observed slip-law behavior for moderate changes in strain rate is retained. Otherwisecage ≪ 1, so that aging has little effect on expected behavior during holds, negating the motivation of the relationship. In terms of porosity to facilitate comparison with microscopic processes and to examine the behavior of intact rock, (9a) is

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This equation implies that changes of porosity from aging law and slip law occur in parallel at different places within the gouge. Appendix A shows that (8) gives a better representation of behavior at the start of holds than does (9a).

[9] Evolution laws (2), (3), (8) and (9a) may be written in a more general form

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where the coefficients cage, cA, and cslip may be functions of normal traction and strain rate or sliding velocity. Equation (8) is part of a general class where cage is a function with a rapid transition from 1 at high velocity to 0 at low velocity, cA = 0, and cslip = 1. This class can be modified so that the steady state value of the state variable is independent of velocity by setting cA = 1. Equation (9a) differs from that class in that cage is constant and less than 1.

[10] Nagata et al. [2012] used P wave transmissivity to infer the variation in ln(ψ). They proposed another evolution law by modifying the aging law (2). In macroscopic notation,

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where c ≈ 2 is an empirical dimensionless constant. This expression should not be called a hybrid law; I call it the stress-rate law after its final term. It has the feature of the slip law that the change of shear stress with displacement is nearly symmetric for step-like velocity increases and decreases. It requires thata ≈ b ≈ 0.05 (at a minimum 0.035) rather than the traditional value of ∼0.01. It is not obvious how to modify the final term of the equation so that one can represent sudden changes in normal traction. In particular, the final term does not affect the steady state value of the state variable so the simple derivation of (5) does not carry through.

[11] Hybrid laws (8) and (9a)and stress-rate law(10) imply very similar behavior when the sliding velocity is ∼0. Laboratory holds represent fault behavior between earthquakes where healing occurs in the absence of slip. The state variable increases linearly with time, which becomes evident once the state variable is significantly greater than its initial value as observed (Figure 2). Evolution during resumed sliding after a long hold represents the commencement of slip in an earthquake. State evolves much more rapidly with the slip law (3) and hybrid laws (8) and (9a) than with the aging law (Figure 3). Equation (8) gives identical results to the slip law, when renewed sliding is fast. Equation (9a) gives results close to the slip law. I discuss (8), (9b) and (10) in terms of micromechanics in the section 3.

Figure 2.

The computed change of the state variable over time during a hold of a sliding surface originally in steady state for hybrid law (8). The state variable is normalized so its predicted value is 1 at a sliding velocity of 1 μm s−1. The value of Dc is 1 μm from experiments by Nagata et al. [2012] assuming a slip law for velocity steps. Nagata et al. [2012] measured P wave transmissivity curves during holds (rough curves). I converted their coefficient with log10(state) = transmissivity * 65. − 48.85. The dots mark the place where the logarithmic derivative is 1/2. These points are predicted by (8) but not (9a). See Appendix A.

Figure 3.

The change in the state variable from its initial values of 103 and 1012 following an increase of strain rate to 1 (εint per unit time) where the equilibrium state variable is 1. The slip law and hybrid law (8) give identical results. Hybrid law (9a) differs slightly from the slip law. The aging law curves are parallel lines and the 103slip-law curve is 1/4 of the 1012 curve.

3. Fractal Elastic Strain Energy and the Rate Parameter a

[12] Nagata et al. [2012] experimentally showed that the apparent value of rate parameter a in (1) shortly after an imposed decrease of shear traction is greater than its traditional value of ∼0.01. They included elastic deformation within the sliding surface in their possible causes. I discuss this topic in terms of rate and state friction and include the micromechanics of hybrid evolution laws.

3.1. Macroscopic Stress Change From Velocity Increase

[13] Following Nagata et al. [2012], I consider a small sudden change of sliding velocity on a macroscopic surface that imposes a mesoscopic change of strain rate within the deforming material. The material evolves toward a new steady state. To obtain predictions, I presume that the traditional slip evolution law (2) is correct for small changes in strain rate and that the traditional value a ≈ 0.01 is also correct. For brevity, I let a = bso that only short-term stress changes in shear traction occur.

[14] I begin by obtaining the additional work per volume in (1) following a sudden increase in strain rate. With (2), the state evolves to its new steady state value and the shear traction returns to its previous value over a strain scaling with εint. The additional frictional work per volume is

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where the strain rate changes from math formula to math formula. For a small velocity change, this result applies to slip, aging, and hybrid laws. Measuring ΔτM just after a velocity change gives an estimate of a, as does measuring the transient energy budget. The additional shear traction as expected is ΔτM ≈ ΔQ/εint. I estimate the change in microscopic elastic strain energy ΔQE in sections 3.2 and 3.3. This strain energy accumulates over some strain εE. The additional shear traction needed to store this energy is ΔτE = ΔQE/εE. The initial apparent value of a depends on the sum ΔτM + ΔτE. I discuss the necessary properties of εE for elastic strain energy to have a significant effect in section 3.4.

3.2. Micromechanical Strain Rates

[15] I apply real contact theory to obtain the change in microscopic elastic strain energy per volume for comparison with (11). Microscopic strain occurs as thermally activated creep on real contracts where the stress is a few GPa [e.g., Berthoud et al., 1999; Baumberger et al., 1999; Rice et al., 2001; Nakatani, 2001; Nakatani and Scholz, 2004; Beeler, 2004; Sleep, 2005, 2006a; Nagata et al., 2012]. That is, the micromechanics of rate and state friction involve thermodynamic averages while the mesoscopic and macroscopic processes involve averages (or expectation functions) over numerous grains and finite time.

[16] Exponential creep occurs within highly stressed contact regions. Following Sleep [2010], the microscopic deviatoric strain rate tensor in isotropic materials is formally

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where math formula is a material property with dimensions of strain rate, τij is the deviatoric stress tensor, math formula is the second invariant of the deviatoric stress tensor normalized to give shear stress in simple shear, and s is the change in stress that changes creep rate by factor of e.

[17] Following Sleep [2005, 2006a, 2006b], I consider deformation at a tabular contact on a sliding surface to show how the slip and hybrid evolution laws arise and to introduce the distribution of microscopic stresses. I show only key steps and condense the full derivation. The contact thickness is 2Z in the z direction perpendicular to the sliding surface, 2X in the x-direction of sliding, and 2Y in the y-direction perpendicular to sliding in the plane of the surface. The contact deforms under microscopic normal tractionP and shear traction τ. The normal traction extrudes material perpendicular to the contact causing macroscopic compaction of pore space. The deviatoric stress invariant is dimensionally

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for contacts that are very elongated in the x-direction of slidingX ≫ Y and

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for contacts where extrusion occurs in the x-direction and is aided by slip in favored quadrants of the contact. In both cases, the slip law arises when the microscopic contact stresses scale with macroscopic onesτ/P ≈ τM/PN and deformation velocities across the contact scale with macroscopic velocities and mesoscopic strain rates. The total velocity perpendicular to the contact is the sum of the effects of shear dilatancy and compaction

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β is the dilatancy coefficient. The velocity of shear deformation is

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[18] At steady state, compaction from normal traction balances dilatancy from slip,

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where the third expression presumes that microscopic stresses scale to macroscopic ones. In terms of macroscopic parameters, the aspect ratio of the contact is

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Sleep et al. [2000] used μ0 = 0.7 and dilatancy parameter β = 0.05 for gouge to obtain a contact aspect ratio Z/Y is 1/5.34, confirming the assumption of tabular geometry.

[19] I relate microscopic strain rate in (12) and (15) to the mesoscopic strain rate in (1) following Nakatani [2001]. Microscopic shear traction τ is the dominant term in the invariant r in (12) when sliding is actually occurring, so the argument of the exponential is r/s ≈ τ/s. Solving (1) for math formula yields that the argument of the exponential is τM/PNa. Microscopic strain and stresses rates scale with mesoscopic ones. Hence the proportional increase in both strain rates is the same and the arguments of the exponentials are equal. Solving yields that 1/a = (r/s)/(τM/PN). If sliding is actually occurring r ≈ rY, the yield value of the invariant where deformation occurs at a significant rate and τM/PN ≈ μ0. This gives the useful estimate that rY/μ0s ≈ 1/a ≈ 100.

[20] Sleep [2006a] obtained the slip law from (14) by assuming that microscopic stresses scale to microscopic ones. Qualitatively, the exponential term dominates in dilatancy term (14) and the shear strain rate term (15). A strong drop in sliding velocity from steady state to a hold decreases the dilatancy term in (14) to ∼0. In addition, drop in τ associated with a strong decrease in sliding velocity in (15) thus decreases r in (14) so that predicted compaction velocity on the contact and hence the macroscopic compaction rate approach zero as macroscopic sliding velocity approaches zero.

3.3. Hybrid Evolution Laws

[21] I qualitatively illustrate the plausibility for hybrid evolution laws by somewhat relaxing the assumption that microscopic stresses everywhere scale faithfully with macroscopic ones. I consider the regions of high stress in deforming region that creep at a finite rate. With forethought, the dimensional results suffice to introduce the concept of a microscopic yield stress rY, where deformation occurs at a reasonable strain rate. The yield stress in silicates ∼4 GPa is typically ∼0.1 of the stress modulus of ∼40 GPa of quartz [Poirier, 1990, p. 38]. Stresses with r ≫ rY quickly relax at extreme strain rates and thus rarely occur. Negligible microscopic strain rate occurs in domains with much lower stresses than rY and one may ignore this deformation in the macroscopic average, but not in computing microscopic strain energy, as shown in section 3.4.

[22] The decrease in macroscopic shear traction τM during a hold will in general decrease the microscopic stresses in r. However, certain domains will be shaped and oriented so that macroscopic normal traction has a larger effect on the invariant r than macroscopic shear traction. During sliding, such domains are not evident as they have slow strain rates where r is several s less than rY. (Otherwise they would have already compacted quickly and not be present.) Once sliding ceases, these slow strain rates are significant and the material behaves as if it is compacting under ambient normal traction. Equation (8) presumes that such contacts exist only at very low sliding velocities and holds [Kato and Tullis, 2001]. Equation (9a) presumes that these contacts exist at all sliding velocities but become evident only during low sliding velocities and holds.

[23] With forethought to discussing the aspect ratios of contacts, aging laws in porosity (7) and (9b) reduce to power law creep during a hold, where the compaction rate depends on the normal traction to the α/b power. I apply the general relationship between exponentials and power laws expanded about x = xref where xref/s ≫ 1,

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to compare macroscopic and macroscopic expressions for this property. For the traditional value of a ≈ 0.01, the sliding rate in (15) implies a power of ∼100. The exponential is the dominant term in (14) for compaction rate and extrusion in the direction of shear traction in (13b) dominates during holds. Some shear traction is typically present on contacts even during holds. Nagata et al. [2012] maintained finite macroscopic shear traction just below that for observable creep during their holds, so their microscopic ratio τ/P was modestly less than μ0.

[24] The power ∼α/b ≈ Z/aX from differentiating r using (13b) is then much less than the traditional value ∼100 of 1/a for tabular contacts. The compaction power is measurable in the laboratory by varying normal traction during compaction experiments. Laboratory values of α/b cover a range. Real contacts are expected to evolve to very tabular aspect ratios for isotropic compaction. The laboratory value of α/b for sandstone is ∼10 implying Z/X is ∼10 for the traditional value of a [Hagin et al., 2007; Sleep, 2010]. The value of α/b for mature gouge during holds is 16–32 [Sleep et al., 2000], implying an aspect ratio of 6–3. Linker and Dieterich [1992] obtained the parameter α ≈ 0.3 from experiments where normal traction suddenly changed while sliding velocity remained constant. Sleep [1999] showed that this parameter also represents starting friction in intact rock. Sleep [2011] obtained 0.15 for sandstone and 0.3 for granite. These values of α are consistent with the above range of α/b provided that b is near its traditional value of ∼0.01. I return to this issue in section 4.1.

[25] Nagata et al. [2012] associated their value of a ≈ b ≈ 0.05 with the intrinsic value rY/μ0s ≈ 1/a ≈ 20 that would be obtained from thermodynamics. It is not clear how to modify (10) so that it represents behavior during holds. I present the simplest approach to illustrate the implications of (10). I ignore the ∂τM/∂t term in (10), as it is 0 during holds. The evolution law (7) then applies for porosity during holds. Then, 1/a ≈ α/b. This situation does arise using the invariant in (13b) when contacts are equidimensional Z ≈ X. However, Nagata et al. [2012] require that a ≈ b so that the steady state value of shear traction in (1) does not very greatly with sliding velocity. Mathematically, these modifications require that b/a ≈ α ≈ 1 contrary to observations, where normal traction changes suddenly at constant sliding velocity [Linker and Dieterich, 1992].

3.4. Microscopic Strain Energy

[26] The stress-rate law(10)is hence problematic if the short-term value ofa is associated with its thermodynamic value. I thus present an alternative explanation to the large apparent values of a observed after small decreases in shear traction [Nagata et al., 2012] invoking microscopic elasticity. I discuss a traditional experiment where sliding velocity suddenly increases for convenience, rather than the actual experiment by Nagata et al. [2012] where shear traction decreased.

[27] A sudden (but small) increase in strain rate requires an increase in microscopic stresses in (11) as well as macroscopic shear traction. I present a derivation of microscopic stresses based on this concept. The geometry of the grain lattice does not instantly evolve so the microscopic stresses where sliding occurs increase proportionally to the macroscopic ones

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where rY is the ∼4 GPa approximate value of the invariant for significant creep and rY/μ0s ≈ 1/a, with μ0 ≈ 1 to avoid spurious precision. We do not have to precisely know the before and after invariant stresses rold and rnew to apply (19).

[28] Qualitatively, one expects significant microscopic stress within an actively deforming region. Inelastic deformation within a geometrically complex medium leaves residual stresses that scale with the yield stress in its wake. The elastic strain energy from these residual stresses in part drives dilatancy, so an attractive value of the dilatancy coefficient β is a fraction of the yield strain rY/G [Sleep, 2006b]. Furthermore, the orientation of the microscopic stress tensor is somewhat random, for example, so grains can slide past each other. The macroscopic shear traction on an imaginary surface through gouge is the sum of numerous large-amplitude traction vectors that nearly cancel out. Antithetic stresses and strains occur locally as grains jostle. Local GPa tensile stresses open crack-tips and break grains, thereby producing rate and state behavior [Beeler, 2004]. The large normal traction vectors from compaction and opening also nearly cancel out.

[29] Returning to mathematics, I obtain the elastic strain energy by following Marsan [2005] and Sleep [2010] from a fractal distribution of microscopic stresses. I ignore factors of 1 − finvolving porosity and compressional elastic strain energy for compactness and to avoid implying excessive precision in a dimensional treatment. Considerable notational simplification accrues if the fractal exponent is exactly −2, the near-field value appropriate for deforming gouge. The number of domains or the fraction of the volume occupied by domains where the stress is less thanr is

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where rnorm is a normalizing constant with the dimensions of stress and σ is a dummy variable for stress. The lower limit of the integral rmin is mathematically necessary for the integral to be bounded; it can be set so that the integral over the full range of stress is 1, yielding volume fractions. Mathematically,

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where rmax is the maximum stress present. Physically, the lower limit represents that the entire volume is near real contacts and subjected to macroscopic stresses, so places where the stress is essentially zero are quite rare and the lower truncation of the distribution is an acceptable approximation. Additionally, rmax ≫ rmin when a truncated fractal distribution is a relevant approximation. Then rmin ≈ rnorm.

[30] The total elastic strain energy per volume is

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where the shear modulus G is the intrinsic value ∼40 GPa. There is equal elastic strain energy in each stress interval so the sharp truncation of the distribution at the limits of the integral has only a minor effect.

[31] The change in both the maximum stress and the minimum stress are relevant to a sudden increase in sliding rate. Equation (19) gives the change in maximum stress Δr. The minimum stress scales with the macroscopic stress rmin ≈ τM and its change is from (1) math formula, where μ ≈ 1. The total change in strain energy is

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This result follows with the value of ΔQE reduced by a factor of 2 if only the upper limit is considered. This energy per volume in (23) is significant compared to the frictional energy per volume in (11). Their ratio is

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where the yield strain is εY ≡ rY/G = ∼0.1. The brackets facilitate showing that this ratio is near unity. The first bracket is the coefficient of friction, which has already been sent to 1 in the dimensional derivation. The intrinsic strain is also near 0.1 so the second bracket is also ∼1, which I use in the remainder of this section; Sleep et al. [2000] (correcting for strain localization in gouge) obtained a range of 0.06 to 0.12 for εint. Nagata et al. [2012] studied initially polished sliding surfaces so an appropriate value of εint is not obvious in their experiments as the effective thickness of gouge is unknown.

[32] The elastic strain energy changes over some strain εE causing a shear traction change of ΔQE/εE that adds to the apparent value of a. From (24), the effect of this change depends on the ratio εint/εE ≈ εY/εE. The effect is strong if the ratio is large, modest if it is ∼1, and small if it is small. The yield strain and the intrinsic strain are attractive scales for εE, so it is plausible that the ratio is greater than or comparable to 1 and microscopic elastic strain does affect the apparent value of a.

[33] I do not have a good way to determine εY independent of the data of Nagata et al. [2012]. I instead infer the properties of εY from their experiments and give qualitative arguments. Nagata et al. [2012] obtained a larger apparent value of a ∼ 0.05 from their experiment where shear traction (and hence velocity) decreased slightly, than they did it the experiment with a larger stress and velocity change. Intuitively, a very small velocity change does not reorganize the slip geometry in the lattice. Stresses in (22) change proportionally to the macroscopic change over the full distribution over a small strain εE; the apparent value of a is large while this strain is occurring. A large change in sliding velocity reorganizes the geometry of slip. The macroscopic shear traction increases and the microscopic shear traction increases along a throughgoing network that accommodates the sliding. Stresses intermediate between these limits do not significant aid macroscopic or microscopic slip. Rather, they are residual stresses associated with past deformation; it takes significant strain in the gouge for these stresses to adjust to the new situation. The apparent value of a thus changes modestly from the frictional value for a large velocity step. The accumulation of microscopic strain energy in terms of math formula is similar both large and small increases of sliding velocity; the accumulation for the large increase is spread out over a large slip distance and hence more difficult to observe. The state variable evolves and macroscopic shear traction decreases over this slip distance, probably decreasing the additional elastic strain energy. I give plausibly to this argument in section 4 by showing that other microscopic elastic effects arise in rate and state friction.

4. Dilatancy of Sliding Gouge and Intact Rock

[34] It is desirable that rate and state relationships apply over the range of sliding velocities that are relevant to the Earth and for both intact rock and gouge. I discuss the relationship of the slip law to the construct of dilatancy angle [e.g., Alejano and Alonso, 2005] and the Linker and Dieterich [1992] relation for intact rock. One objective is to show that microscopic elasticity likely affects the latter empirical relationship to provide analogy with the discussion of sudden small changes in sliding velocity in section 3.4.

4.1. Predicted Porosity Variation

[35] Both porosity in (6) and P wave transmissivity [Nagata et al., 2012] scale with ln(ψ). It is not a priori evident, which measurable parameter is the better proxy for state in (1). I discuss porosity over the relevant ranges of sliding velocities and normal tractions to show some limitations of (6).

[36] Beginning with sliding velocity, rate and state friction does not apply in its basic form at high sliding velocity where real contacts become hot enough to affect their rheology, above ∼10−1 m s−1 [Beeler et al., 2008]. At the other end, geologists may study a creeping crustal fault with a velocity of 0.3 mm a−1 (10−11 m s−1). The relevant steady state value of ψ thus varies by ∼10 orders of magnitude at a given normal traction. I use 0.1 as a typical porosity ϕ of laboratory gouge in examples. Nagata et al. [2012] reported laboratory experiments with a sliding velocity of ∼10−6 m s−1 and normal traction of 10 MPa, which I use as reference values. Sleep et al. [2000] obtained Cε from 2.0 to 3.4 × 10−3. For the upper value, porosity is 0.1391 at 10−1 m s−1 and 0.0609 at 10−11 m s−1, which are not unrealistic.

[37] Continuing with normal traction, the steady state porosity from (4) and (6) at the reference velocity does imply an unrealistic range of porosity at steady state at the reference velocity

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Measured values of α in sliding experiments where normal traction is suddenly changed are typically 0.2–0.30 [Linker and Dieterich, 1992]. The coefficient of the logarithm ranges thus from 0.040 to 0.102 for the traditional value ∼0.01 of b. The applicable range of pressure is from ∼1500 MPa where the coefficient of friction of intact granite approaches that of steady state gouge (Figure 4) and ∼0.1 MPa for very shallow rock. The predicted porosity range (with Pref = 10 MPa for the lower estimate 0.04 of the coefficient) is −0.10 to 0.23, which is not possible. That is, the inferred coefficient αCε/bis too large. Reducing it by a factor of two would make the high-pressure estimate of gouge porosity positive.

Figure 4.

The starting internal friction of intact Westerly granite and the steady state sliding friction of granite gouge converge at a normal traction of ∼1500 MPa. The Linker and Dieterich [1992] relationship curve (26) assumes α = 0.30, Pref = 1500 MPa, and μref = 0.7. Friction coefficients are from the work of Lockner [1995] based on experiments by Byerlee [1967]. The smooth long-dashed curve is discussed inAppendix B. Modified from the work of Sleep [1999, 2011].

[38] It is conceivable that mildly inaccurate estimates of the parameters in αCε/b caused my estimate of its lower limit to be too high. The coefficient α/b arises from studying during compaction during holds. With regard to α, the predicted coefficient of friction does provide a good representation of the starting friction of intact granite (Figure 4) and intact sandstone [Sleep, 1997, 2011],

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where the coefficient is μref at normal traction Pref. Experiments appraising this relationship yield estimates of α 0.15–0.30 [Sleep, 2011]. My assumed value coefficient of the term Cε is somewhat suspect over the full pressure and porosity ranges; I return to this issue in section 4.2.

[39] Note the coefficient of friction of gouge varies somewhat with normal traction in Figure 4. This micromechanical behavior of gouge may be related to the fact that real contact area increases at high normal tractions [Byerlee, 1967]. See Appendix B.

4.2. Micromechanics of Real Contact Area

[40] I relate (6) to micromechanics of real contacts with the intent of showing its limitations. With forethought, the differential relationship between state and porosity in (6) is

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where in general Cεneed not be constant. I continue the well-known behavior of a sliding surface with areas of real contact. The microscopic shear traction is inversely related to the areaA of real contact

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where AM is the macroscopic contact area. Nagata et al. [2012] inferred relative variations in A/AM with P wave transitivity on a sliding surface (Figure 2). The logarithm of the microscopic strain rate with τ ≈ r in (15) becomes

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where the second term dominates. At constant sliding microscopic sliding velocity, the second term stays approximately constant C

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I compare (30) with (1) evaluated at constant velocity and normal traction. The change in macroscopic stress for a change in porosity in (6) is

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That is, (6) is an attractive linearization were the fraction of the sliding surface covered by real contacts scales negatively with porosity.

[41] The coefficient Cε need not be constant over the physical range of porosities away from those in gouge where it was measured. Sleep [2011] suggested a percolation theory form of (6) that allows porosity to reach an upper limit fmax in rapid sliding at very low normal traction,

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where ψP → 0 at the limiting porosity fmax and ψP ≈ ψ in (6) for f ≈ ϕ. Qualitatively, a gouge or a sliding surface loses all strength when contact is lost.

[42] Note that the argument of the exponential relationship between porosity and state in (27) using (32) could be modified so that the state variable becomes very large in the limit of zero porosity, say by multiplying (32) by ϕ/f raised to some positive power of that is a modest fraction 1, so that it has little effect otherwise. This modification allows keeping track of dilatancy during the initial stage of fault nucleation in very low porosity rock. Section 4.3 presents the alternative that the inferred value of Cε of 2.0 × 10−3 is too high so that (32) does not need to be modified to low porosities.

4.3. Initial Dilatancy of Intact Rock

[43] I continue with the initial behavior of rock with the intent of considering microscopic elasticity in section 4.4. Hybrid evolution law (9b) (for simplicity at PN = Pref and math formula where the steady state porosity is fss) becomes

display math

where finit is the initial porosity. I assume that εint ≈ εage ≈ εslip so that (9b) represents velocity stepping experiments in gouge.

[44] The slip term dominates when sliding suddenly starts, as εint is ∼0.1, and fss ≈ ϕ is ∼0.1 in typical gouge experiments and does not explicitly involve the coefficient Cε. Qualitatively, strain of εint restores porosity finit to its steady state fss. In addition, the coefficient fss/εint for low porosity rock is ∼1 at low normal traction. So the ratio of dilation strain rate ∂f/∂t to shear strain math formula is also ∼1 not β ≈ 0.05 for intact rocks. The expansion of pore space against the normal traction and inelastic deformation of strong real contacts do comparable amounts of work that becomes macroscopic friction, so the result has the appearance of equal partition of energy. The macroscopic coefficient of friction increases by (fss − fint)/εint ≈ 1. The starting coefficient of granite [Lockner, 1995] at low normal tractions extrapolated using (26) approaches 2 (Figure 4) so there is no blatant contradiction [Sleep, 1999, 2010].

[45] Rock damage in the shallow subsurface by strong seismic waves bears on this issue. Sleep and Hagin [2008]examined the energy budget of this process within sandstone near Parkfield California. They concluded that damage that transiently reduced the low-amplitude S-wave velocity occurred on crack surfaces. Shear sliding and crack dilatation against lithostatic pressure dissipated comparable amounts of seismic energy. As the coefficient of frictions is ∼1, shear strain and dilatational strain were comparable as suggested in the previous paragraph. This study did not resolve the depth distributions of damage and nonlinear attenuation of the incident strong wave.

[46] In geological engineering, the ratio R of dilatant to shear strain on initial sliding is the dilatancy angle for intact rocks [e.g., Alejano and Alonso, 2005]. For rate and state friction, it is expected to vary with normal traction from (25) with the finite steady state porosity fss,

display math

where the coefficients are those for steady state gouge and cage = 0 for brevity. The coefficient of the logarithm αCε/int is 0.4 to 1.02 for my assumed parameter values, including b = 0.01. Ribacchi [2000] measured dilatancy angles on jointed detrital limestone (Figure 5). The observed relationship is in fact logarithmic [Alejano and Alonso, 2005, equation 27] and the observed slope related to the coefficient αCε/int is 0.11. In general, dilatancy angle decreases with confining pressure for a wide variety of geological materials [Zhao and Cai, 2010].

Figure 5.

The dilatancy angle of limestone measured by Ribacchi [2000] from the work of Alejano and Alonso [2005, p. 487]. As noted by Alejano and Alonso [2005], the relationship is approximately linear, shown by eyeball line.

[47] The reason for the difference between the observed and predicted slope in Figure 5 is not obvious. As noted in section 4.2, my estimated value of αCε/b is too high in that it implies an unrealistic range of porosity for steady state sliding at applicable normal tractions. The parameter set α = 0.15, b = 0.01, Cε = 0.73 × 10−3 yields the observed value of 0.11. The predicted steady state porosity at 1500 MPa for ϕ = 0.1 and Pref = 10 MPa is 0.0063. This value is attractive as steady state porosity becomes small in the normal traction range were gouge and intact rock behave similarly. In addition, my rate and state parameters come from experiments on quartz gouge, sandstone, and granite while Ribacchi [2000] studied detrital limestones. The appropriate values of α, b, and Cε were not measured and they need not be the same as those for gouge and sandstone. Measurement of these parameters and dilatancy angle on a common set of samples is warranted.

[48] Physically, shear dilatancy of an initially granular substance like sandstone and detrital limestone differs from that of an almost nonporous material like intact granite. Shear deformation in sandstone transforms equidimensional porosity into crack-like porosity, thereby weakening the materials without net dilatation. Initial (square wheel) rotation of grains dilates material at very low normal tractions and is not explicitly included in rate and state formalism.

[49] Two other obvious effects qualitatively reduce the nominal dependence of dilatancy angle on normal traction. First, the initial porosity finit is likely to decrease with the ambient normal traction in real experiments where it is not measured and where some unknown aging has occurred. Thus, fss − finit decreases more slowly with the logarithm of normal traction than the predicted value of αCε/int for just fss. Second, joints accommodated the porosity within the rocks studied by Ribacchi [2000]. At very low normal traction, much of the surface of a joint is not in mesoscopic contact; the normal traction of domains of the joint with macroscopic contact is well above its nominal macroscopic value. As normal traction increases, much more of the surface of the joint comes into mesoscopic contact; mesoscopic and macroscopic normal traction converge. Thus the actual change in mesoscopic normal traction in Figure 5 is less than the nominal change. The real slope of the line may be somewhat greater than its measured value of 0.11.

[50] These results make unification of rate and state friction with dilatancy angle formalism attractive. Experiments with gouge at low normal tractions that has been aged to high normal tractions involve a high initial value of the state variable and low porosity. The predicted difference between the aging and slip laws is large for known critical displacement (Figure 3).

4.4. State Variable of Intact Rock

[51] The Linker and Dieterich [1992] relationship (26) predicts the starting coefficient of friction of intact rock if the state variable remains constant (Figure 4). This observation bears on the effects of microscopic elasticity on friction and the relationship of the logarithm of the state variable to real contact area.

[52] First, constant state variable as defined in this paper over a range normal traction does not imply a constant fraction of the sliding surface is covered with real contacts with the yield stress rY. Constant area of real contact implies nearly constant shear traction. Rather, the observed shear traction varies from ∼260 to 900 MPa at normal tractions from 200 to 1500 MPa (Figure 4). The initial shear traction within a flawless material would be the full yield stress of ∼4 GPa.

[53] A likely qualitative explanation is that (26) includes elastic effects of normal traction. A distribution of stresses exists within real material. At ∼1500 MPa normal traction, there is little macroscopic porosity but flaws with weak microscopic contact and normal traction transiently accommodate opening and shear. When normal traction is lowered, these flaws dilate into cracks and the fraction of the surface covered by real contacts decreases. The normal traction of the experiments in Figure 4 varied slowly enough that the material lattice had time to respond elastically.

[54] Second, this formulation in (5)implies that shear traction changes instantaneously with normal traction, which leads to an ill-posed physical problem [Perfettini et al., 2001]. Equation (5) thus cannot be correct at very short times after normal traction changes. Rather real contact area and shear traction do not change instantaneously; the instantaneous value of α is 1 at this limit. Elastic deformation that takes finite time is required for α to have its laboratory value of 0.1–0.3. Perfettini et al. [2001] and Sleep [2005] suggested modified Linker and Dieterich [1992] relationships where shear traction does not change instantaneously and where (5) is approximately correct at moderate times. Experiments confirm that shear traction does not change instantly with normal traction [Kilgore et al., 2010].

5. Practical Applications and Conclusions

[55] Earth and materials scientists would like to represent frictional failure of intact material and gouge with self-consistent phenomenological laws. Ideally, these laws should be based on thermodynamics and the collective behavior of the material lattice. The widely applied construct of rate and state friction strives to meet these criteria. I consider the implications of this paper to a variety of geological and tribological situations.

[56] The experiments of Nagata et al. [2012] show that the decrease in sliding velocity in (1) immediately after a sudden decrease in shear traction is significantly higher than that expected from the traditional value a ≈ 0.01. I attribute this effect to elastic strain energy on the microscopic scale of real contacts in (24). In general, traditional rate and state friction does not explicitly include elasticity, which is permissible in the limit of significant anelastic displacements and strains. Elasticity likely affects the behavior of laboratory gouge just after changes in sliding rate and normal traction and faults in the Earth during high frequency seismic shaking. The Linker and Dieterich [1992] relationship (5) also implicitly includes the effects of elasticity and incorrectly predicts that shear traction varies instantaneously with normal traction.

[57] Hybrid evolution laws (8) and (9a) (as well as the generalized law (9c)) represent both the near equilibrium behavior of gouge, the aging of faults during the interseismic interval, and subsequent earthquake nucleation. The details of friction do matter even at constant normal traction for nucleation and initial rupture [e.g., Rubin, 2011]. The derivation of the slip law by Sleep [2006a] assumed that microscopic stresses on deforming real contacts scale with macroscopic stresses and that perturbations were not far from steady state. As expected, the slip law is good approximation near steady state. It is not evident from the derivation whether slip or aging laws provide a better approximation far from steady state. Hybrid slip law (8) which reduces to an aging law during holds better represents such behavior than (9a) (Appendix A).

[58] The initial ratio of dilatant strain rate to shear strain rate is a moderate fraction of 1 and equivalent to the dilatancy angle in geological engineering [e.g., Alejano and Alonso, 2005]. Geological engineering experiments grossly calibrate this parameter as a function of normal traction. The slip law (3) and hybrid laws (8) and (9b) imply this behavior when sliding suddenly starts within intact rock, but the aging law (2)does not. One would count the dilatancy-angle effect twice by including it with slip-law or hybrid rate and state dilatancy. Engineering studies may well provide better predictions at very low normal tractions than extrapolations of rate and state experiments from somewhat higher normal tractions. Measurement on the initial friction and dilatancy of strongly compacted gouge are warranted. The pore pressure thus decreases upon initial slip within undrained material, tending to quench earthquake instabilities.

[59] With regard to rupture tips, the energy per surface area dissipated depends on the area under the curves in Figure 3. The curves do not differ strongly from linearity so slip weakening is a reasonable approximation in a lengthy calculation. The duration of the peak (in strain, macroscopic slip in terms of Dc, and time) for given strain εint is much less the slip law than the aging law at high initial values of the state variable. The peak strength depends on the initial value of the state variable and hence requires a hybrid evolution law to model the interseismic period or intact rock.

[60] Very short-term variations of shear traction associated with normal traction changes need to be correctly included in earthquake rupture simulations [Perfettini et al., 2001]. Stress-rate evolution law(10)similarly represents very short-term changes in sliding velocity associated with changes in shear traction, but does not attempt to represent changing normal traction. It has not yet been used in dynamic models so its effect on rupture is not known. In general, there is a trade-off between representing physics and tractability where the modeler evaluates equations for normal traction, shear traction, and velocity at a vast number of time steps and spatial nodes.Equation (10) implies terms with τ then on both sides of the stress equation (1)that might increase computation time. I do not have useful computer-friendly alternative.

[61] Finally and qualitatively with regard to hybrid evolution laws, it is helpful to consider macroscopic work done on the gouge lattice. Shear traction does most of the work near steady state and with the sudden initiation of sliding. The slip law applies in these cases. Normal traction does most of the work for compaction during holds. Hybrid law (8) conveniently demarcates and represents these domains, but its parameter Vmin is not readily obtained by experiment or theory and the normal traction dependence of the parameter Vmin is not known. A formulation involving work is attractive, but I have not found a compact way of doing this that would be friendly to numerical modeling.

Appendix A:: Appraisal of Hybrid Evolution Laws

[62] I briefly discuss microscopic mechanisms that might cause frictional deformation at low velocities to differ from thus at high velocities to put hybrid laws in context. I exclude very high sliding velocities >0.1 m s−1 where contacts become hot [e.g., Beeler et al., 2008]. Kato and Tullis [2001] based their evolution law (8) on the concept that chemical bonds between sliding surfaces take time to form. Nakatani and Scholz [2006] proposed an intrinsic cutoff time related to reaction rates on real contacts, which resulted in minimum values of state and frictional strength at high sliding velocity and given normal traction. Their analysis involved experiments on gouge. Sleep et al. [2000] attributed these observed effects to strain localization and delocalization. The sliding surfaces studied by Nagata et al. [2012] should be reasonably free of these gouge effects. Li et al. [2011] considered reaction rates at indentor contacts. I continue in a phenomenological manner.

[63] The theoretical curves in Figure 2 allow appraisal evolution laws (8) and (9a). I refer to the generalized law (9c) to put these examples in context. My procedure follows the cutoff time analysis of Nakatani and Scholz [2006]. I assume that the previous sliding velocity in (8) is much greater than Vmin so its term is 0 during sliding and 1 during the hold. In macroscopic terms, the previous steady state value of the state variable is then θss = Dc/V for (8) and ψss = Vref/V for (9a) at the reference normal traction. During a hold, the state variable in (8) is

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and the state variable for (9a) is

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[64] A stable method of interpreting Figure 2 involves the break in slope of the curves. At long times ∂ln(ψ)/∂ln(t) = ∂ln(θ)/∂ln(t) = 1. This logarithmic derivative is 1/2 when t = Dc/V in (A1) and t = Dc/(cageV) in (A2). The critical displacement is ∼1 μm in velocity stepping experiments interpreted with standard slip law (3) [Nagata et al., 2012]. The predicted times for (A1) and (8) for velocities of 0.01 and 0.1 μm s−1 is 100 and 10 s, as observed. The predicted time for 1 μm s−1 is 1 s and the observed time is somewhat shorter. I do not attempt to measure the break in slope for 10 μm s−1. The predicted times are longer by 1/cage for (A2) and (9a). Thus during holds, hybrid law (8) and (9c) with cage = 1 gives the better representation that (9a) and (9c) with cage ≪ 1.

[65] Experiments are warranted to obtain the parameter Vmin and to constrain behavior when normal traction changes and in general the function math formula in (9c). The parameter Vmin in (8) may not be easy measure. It has the effect of changing its term from ∼1 to ∼0 over an order of magnitude of sliding velocity. This parameter may be important on creeping crustal faults, but still may be too low to tractably measure in the laboratory.

[66] The Vmin term ((9c) with cA = 0) only slightly increases the steady state value of the steady state state variable at low sliding velocities from its high velocity limit. The argument of the logarithm is 1/Ω = 1.76 …, when the high velocity argument 1, so that the coefficient of friction in (1) increases by − b ln(Ω), which would be difficult to observe. The state variable relative to its steady state value at long times is a factor of cage lower for (9a) than (8) so the coefficient of friction differs by only b ln(cage).

Appendix B:: Sliding Friction in Gouge at High Normal Tractions

[67] It is not possible to obtain a clear function for the steady state coefficient of friction of gouge from the data in Figure 4. The smooth long dashed curve and even a straight line that quickly extrapolates to 0 are possible representations. It is necessary to consider additional data collected at high normal tractions. Byerlee [1978] used data for normal tractions up to 1800 MPa to obtain the continuous curve,

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That is, the coefficient of friction should extrapolate to ∼0.6 at high normal traction and not exactly merge with the intact rock curve at 1500 MPa.

[68] I provide the smooth (long-dashed) curve inFigure 4 to facilitate deducing the functional form of an applicable expression:

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which is continuous. In contrast, replacing μ0 in (1) with a term of the obvious form μ0 − ζ ln(PN/Pref) (where ζ is another small dimensionless parameter) in analogy to (26) gives a very poor fit to the gouge data. The change in the logarithm and hence the predicted change in the coefficient of friction from 200 to 1500 MPa normal traction is the same as the change from 27 to 200 MPa. The observed coefficient of friction decreases negligibly in the higher interval but significantly in the lower interval.

[69] Byerlee [1967] suggested that the surface saturated with real contacts around 200 MPa normal traction. The real shear strength of contacts is likely around 4000 MPa in shear using that shear strength is ∼0.1 shear modulus [Poirier, 1990], so that hypothesis is inapplicable. Rather macroscopic porosity becomes small at ∼1500 MPa normal traction, but the real stresses on the contact surface are spatially variable. For a real contact shear traction of 4000 MPa and a coefficient of friction of 0.6, 22.5% of the surface is covered by real contacts at 1500 MPa normal traction. The percentage at the transition in (B1) 200 MPa normal traction with μ0 = 0.85, is 4.25%. At 15 MPa normal traction, contacts cover only 0.32%. A random distribution of contact points does not give any obvious significance to the transition normal traction of ∼200 MPa; only 0.0425*0.0425 = 0.0018 of the contacts overlap.

[70] I present a qualitatively explanation for (B1) and (B2). Areas of contact episodically detach from the sliding surface by moving over the roughness of the surface or by being propped apart by adjacent contact points that move over roughness. At steady state, new contacts form as others detach. Residual stresses scaling with the yield stress maintain roughness and drive dilatancy [Sleep, 2006b].

[71] The microscopic normal traction and the microscopic shear traction are ∼0 on a contact that has just touched. Elastic strain accumulates on the contact increasing these stresses until its becomes anelastic. The rate of buildup of microscopic shear traction scales with the sliding rate. The rate of buildup of normal traction scales in absolute value with the sliding rate times the dilatancy coefficient β or equivalently the sliding rate times surface roughness. There is a period of sliding where the shear traction has reached the yield stress τY but the normal traction is well below its microscopic average P. The anelastic behavior of microscopic shear traction also differs from that of microscopic normal traction. The former dominates the invariant in (13a) and (13b), so the shear traction on real contacts is ∼ τY once sliding commences. The microscopic normal traction on sliding contacts may vary somewhat from its mean P on the sliding surface, as it contributes weakly to the invariant r. This process is asymmetric about its mean, again causing a bias to low local values of microscopic normal traction. Large positive values of normal traction are precluded as they do increase the invariant causing rapid compaction. Small values of normal traction including negative ones just before detachment are permitted.

[72] There is thus an excess class of contact area that supports shear traction but not normal traction. Intuitively, this effect is greater at lower macroscopic normal tractions, where contacts on average are likely to be narrow, thin, and persist briefly. The rate of extrusion and elasticity tend to buffer the microscopic normal traction on long-lived wide contacts. The fraction of the time of its existence that a contact slides in shear traction, but has low normal traction thus decreases with macroscopic normal traction. I have not found a way to derive (B2) from these inferences.


[73] Madhur Johri pointed out the dilatancy angle construct. Kohei Nagata patiently answered my questions and provided his data in digital form. Nick Beeler gave helpful comments. An anonymous review and a review by Allan Rubin were thorough and helpful. This research was in part supported by NSF grant EAR-0909319, which grant was funded under the American Recovery and Reinvestment Act of 2009 (ARRA) (Public Law 111-5). This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. The SCEC contribution number for this paper is 1680. The 2011 SCEC meeting showed the need to reconsider rate and state formalism.