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Keywords:

  • asperity;
  • evolution laws;
  • friction;
  • rate and state

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

The well-known formalism of rate and state dependent friction represents the transition between starting friction and sliding friction. It expresses the instantaneous coefficient of friction in macroscopic quantities as μ = μ0 + a ln (V/V0) + b ln (ψ/ψnorm), where μ0 is the first-order coefficient of friction, a and b are small (∼0.01) dimensionless constants, V is sliding velocity, V0 is a reference sliding velocity, and the inverse of the state variable 1/ψ represents damage. The normalizing state variable ψnorm ≡ (P/P0)α/b represents the effect of the normal traction P on stress concentrations, where P0 is a constant with dimensions of pressure and α ≫ b is another dimensionless constant. Evolution equations represent the combined effect of damage from sliding and healing on the state variable. The Ruina (1983) evolution equation implies that the state variable does not change (no healing) during holds when sliding is stopped. It arises from exponential creep within gouge when the concentrated stress at asperities scales with the macroscopic quantities. The parameter ba being positive is a necessary condition for a spring-slider system becoming unstable. Microscopically, this parameter represents the tendency of asperities accommodating shear creep to persist longer than asperities of compaction creep at high sliding velocities. In the Dieterich (1979) evolution law, healing occurs when the sample is at rest. It is a special case where creep that produces shear and creep that produces compaction occur at different microscopic locations at a subgrain scale. It also applies qualitatively for compaction at a shear traction well below that needed for frictional sliding. Chemically, it may apply when shear sliding occurs within weak microscopic regions of hydrated silica while compaction creep occurs within comparatively anhydrous grains.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

The rate and state friction formalism describes the transition between static friction and sliding friction. It provides a good representation of laboratory data by including a term for damage associated with sliding and a term for healing when the slider is slows down or is at rest. These terms result in the attractive property that the formalism can represent repeated earthquake cycles where the fault strengthens during the interseismic period. It is well known that the second-order terms representing damage and healing determine mathematically whether a fault creeps peaceably or fails catastrophically in an earthquake.

I concentrate on healing in this paper. One would like to know the physical basis of the laws that purport to represent it before exporting laboratory results to the earthquake cycle at depth. Troublingly, there are two different evolution laws for keeping track of damage and healing on the fault surface. The Dieterich [1979] evolution law predicts that a fault heals when stopped so that its friction when slipping is resumed is higher than that before the hold. The Ruina [1983] evolution law predicts no healing during a hold. An intriguing result is that the Ruina [1983] behavior occurs at low relative humidity and the Dieterich [1979] behavior at high relative humidity in simulated gouge [Frye and Marone, 2002]. I address the physical basis of rate and state friction with these issues in mind.

2. Rate and State Equations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

The formalism for rate and state friction evolved from laboratory experiments where one measures (or controls) the macroscopic quantities of normal traction P (in general effective pressure, but I stick to drained conditions for simplicity), shear traction τ, and sliding velocity V (using the notation of Sleep et al. [2000]). Scientists have developed several simple semiempirical relationships between these measurable quantities and porosity. They are approximations, to be sure, but the simple forms allow revealing mathematical manipulation. I begin with macroscopic properties as would be measured over a few centimeter-squared contact on a laboratory apparatus (Figure 1). I introduce mesoscopic properties averaged over enough grains that a continuum is meaningful in section 2.1. I introduce microscopic quantities at real contacts in section 2.2. They apply at scales significantly greater than atomic dimensions where a continuum is meaningful, that is, a few nanometers. The stress on a few millimeter wide patch of the macroscopic contact arises in section 3.4.

image

Figure 1. The paper considers stress involved in friction on several scales. Most laboratory experiments record the sliding velocity and the stresses averaged over the contact surface. The stresses on a patch of the surface may differ from these averages. The mesoscopic stress and strain rate within the gouge are useful for representing rate and state equations as flow laws. The microscopic stress and strain rate within the small part of a grain, like an asperity of real contact, are useful for representing deformation as thermally activated creep.

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I begin with a unified theory of rate and state friction compiled by Sleep [1997] and Sleep et al. [2000] based on the Dieterich [1979] evolution law. A state variable ψ represents the condition of the surface or gouge; 1/ψ is a measure of damage. The instantaneous shear traction from friction is in terms of macroscopic variables

  • equation image

where μ0 is the coefficient of friction at reference conditions, a and b are small dimensionless constants, V0 is a reference velocity, and ψnorm is the normalizing value for the state variable, which I discuss below. The state variable evolves with time from Dieterich's [1979] law

  • equation image

where t is time, Dc is the critical displacement to significantly change the state properties of the sliding surface, α is a dimensionless parameter than represents the behavior of the surface after a change in normal traction from Linker and Dieterich [1992], and P0 is a reference normal traction. The first right-hand term represents healing and the second term damage from sliding. It is well know that b > a is a necessary condition for a spring-slider system (including a fault in the elastic Earth) to become unstable.

2.1. Strain Rate Form of Equations

The relationship of frictional sliding to other flow laws becomes more apparent if one uses strain rates rather than macroscopic quantities. One obtains this form of the equations by dividing V, V0, and Dc by the thickness of the sliding zone W. This is a logical testable extension of the macroscopic equations allows modeling strain rate localization [Sleep et al., 2000]. The computed strain rates are then “mesoscale” expectation functions over enough grains for the gouge and enough time to act as a continuum. The mesoscopic or grain scale quantities then are shear strain rate ɛ′ ≡ V/W, reference strain rate ɛ′0V0/W, and the intrinsic strain ɛintDc/W. The friction equation (1) then becomes

  • equation image

and the evolution equation (2) becomes

  • equation image

The steady state value of the state variable in (4) is

  • equation image

If the steady state sliding friction is independent of the normal traction (3) and (5) imply that

  • equation image

(It has not escaped me that a more complex expression could represent the subtle variation of friction with normal traction.) To this point, the equations are mathematically equivalent to rate and state friction with the Dieterich [1979] evolution law and the Linker and Dieterich [1992] formulation for changes in normal traction.

One would like to relate the state variable to a physically measurable parameter. The porosity f is a natural choice. I follow the formulation of Segall and Rice [1995] as it leads to compact expressions, especially when I later introduce the Ruina [1983] evolution law. The state variable is then

  • equation image

where ϕ is a reference porosity and Cɛ is a dimensionless material property. This equation applies to gouges made mainly of hard grains like quartz where the porosity is low enough (<∼10%) that shear strain dilates rather than compacts. I implicitly restrict discussion to processes occurring at constant temperatures and effective normal tractions appropriate for the seismogenic zone, ∼3–15 km depth.

The evolution equation for porosity then becomes

  • equation image

where the first term implies that the dilational strain rate is proportional to the shear strain rate and the second term resembles power law compaction from the normal traction P with the state variable ψ representing the nonlinear effect of porosity on compaction. In a simpler form the equation becomes

  • equation image

The first term indicates that the dilational rate is linearly proportional to the shear strain rate with the constant β. The second term represents compaction with the constants collected into C1. It has the reasonable properties that the compaction rate increases as a power n of normal traction and exponentially with porosity. Note that my use of ψnorm in (4) implies that the porosity in (8) and the state variable in (4) cannot change suddenly when normal traction changes. This is a reasonable assumption or approximation.

The steady state porosity in (8a) is

  • equation image

It is also illustrative to express the friction equation (3), as a flow law for the shear strain rate. Combining (3) and (6) yields

  • equation image

The denominator in the exponential term of (10) is small, as a is of the order of 0.01 and τ/P is of the order of 1. If one defines the constants μ0, ɛ′0, and P0 so they refer to achievable values within a simulated gouge or a slipping crustal fault zone, τ/P is approximately μ0, a constant coefficient of friction (Figure 2). The predicted sliding is undetectably small for significantly lower ratios and unreachably large for larger ratios. As a practical matter, (10) and (3) need to give good predictions only over this narrow range of ratios.

image

Figure 2. Stresses on a plane in shear-traction, normal-traction space. Frictional sliding occurs within the shaded region where the ratio τ/P is approximately μ0. Higher ratios cannot be reached at realizable sliding velocities. The shear strain rate is immeasurably slow at lower ratios, where gouge compacts.

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2.2. Friction as Creep Process

To see how the behavior in (10) arises, I consider the micromechanics of friction. It is generally agreed that rate and state friction is a thermally activated process [e.g., Berthoud et al., 1999; Baumberger et al., 1999; Rice et al., 2001; Nakatani, 2001; Nakatani and Scholz, 2004; Beeler, 2004]. A simple form of the derivation proceeds from real contact theory. Microscopic real contacts with normal traction Preal exist along the mesoscopic material contact. The rest of the mesoscopic surface is voids. The shear traction on the contacts scales to the macroscopic values

  • equation image

The simple interpretation of a coefficient of friction comes from treating τreal as a yield stress, which gives μ0 = τyield/Preal.

A general expression for the thermally activated creep rate of a substance allows relating terms [Sleep, 1997]

  • equation image

where ɛ′base is a material constant, M is the molecular volume of the rate limiting step, R is the gas constant, and T is absolute temperature. In the low-stress limit, this equation reduces to power law creep with the exponent q, which can be seen by taking the Taylor series of the exponential. Indentation tests indicate that an amorphous or highly disordered material forms at real contacts [Goldsby et al., 2004]. This may indicate that a linear material with q ≈ 1 is a better value than the exponent for dislocation creep of 3–5.

Here I am interested in the creep of hard mineral grains at room to seismogenic temperatures. Shear creep can occur at a measurable rate only at high stresses where the exponential term is much greater than 1. Ignoring the 1 yields a simpler expression:

  • equation image

The microscopic strain rate in (12b) should be proportional to the mesoscopic strain rate in (10). I compare the term within the exponential in (12b) with the term within the exponential in (10). (Note the μ0 term in (10) can be expressed as a multiplicative constant outside the exponential.) This yields using (11)

  • equation image

The derivation is adequate to this point and an analogous derivation follows for b [Nakatani and Scholz, 2004]. One needs to explicitly consider the micromechanical interaction of shear-driven creep and compaction creep to understand evolution laws and to get at the stability parameter ba.

3. Ruina Evolution Law

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

An alternative evolution law by Ruina [1983] has the property that no healing occurs when sliding is stopped. It has this form in my notation:

  • equation image

where the steady state is given by (5) and the state variable is dimensionless. My form (14) is mathematically equivalent to the Linker and Dieterich [1992] relationship combined with the Ruina [1983] law. With the shear-traction equation (3) it predicts the same changes of shear traction for changes in normal traction.

Simulated gouge under low relative humidity shows this behavior in the sense that friction does not increase from its previous value when sliding restarts after a hold [Frye and Marone, 2002]. However, the prediction of the porosity-state equation (7) and (14) that porosity does not change during holds is not observed. Frye and Marone's [2002] gouge compacted during low-humidity holds. I consider the physics of a nonlinear granular material to show how the Ruina [1983] law arises. I discuss the observed compaction in section 4.

The different forms of the evolution law in (14) and (4) matter greatly in situations relevant to the earthquake cycle. First, sliding may slow down, allowing healing to occur. I show this effect without loss of generality in normalized form where both the steady state-strain rate before sliding slows and the reference strain rate have values of 1. (We can set the reference strain rate and reference velocity for convenience with the restriction they cannot be 0. The previous sliding velocity is convenient in a laboratory experiment. The long-term geological sliding rate might be convenient for an actual fault.) I let the reference normal traction and the normal traction be equal. The steady state value of the state variable ψ is then 1. Figure 3 shows the time derivative of the state variable immediately after a sudden velocity change. The Ruina [1983] curve is similar to the Dieterich [1979] curve between its maximum at normalized ɛ′ = 1/e and its initial value of 1. That is, the choice of evolution law does not matter much for subtle velocity changes. At lower velocities, the laws are quite different. The Ruina [1983] curve tends to 0, while the Dieterich [1979] curve approaches 1.

image

Figure 3. The normalized time derivative of the state variable after a sudden change in strain rate. The new velocity is normalized to the previous strain rate, that is, ɛ′/ɛ′0 in terms of mesoscopic variables. The Ruina [1983] law tends to zero for small velocities, while the Dieterich [1979] curve tends to 1. The hybrid curve is 1/10 Dieterich [1979] curve plus 9/10 Ruina [1983] curve. It is intended to represent laboratory situations that approximately follow the Ruina [1983] law with modest changes in strain rate and that heal somewhat over long times during holds.

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A second simple case involves the state variable when preseismic creep resumes following a significant length of time after an earthquake. Sliding during the earthquake leaves the state variable at a small and unknown value. With the Dieterich [1979] evolution law, the state variable increases proportionally to Pα/b in (4) with time to a large value where the initial small value is irrelevant. Porosity decreases with the logarithm of time in (8b) at constant P. When the Ruina [1983] law is taken literally, neither the state variable nor the porosity changes from their values immediately after the earthquake until preseismic creep restarts years later.

3.1. Compaction as Nonlinear Creep

Equation (8) models compaction of the fault zone as power law creep. The shear traction does not enter into the final term representing compaction. However, the macroscopic shear traction and the macroscopic normal traction are of the same order as the coefficient of friction is of order 1. One would expect that these tractions could interact within a nonlinear substance. A molecule within a grain cannot tell the macroscopic origin of the microscopic stress field.

To start generally, I represent the microscopic strain rates as tensors. The deviatoric strain rate in an isotropic viscous nonlinear material is [e.g., Schubert et al., 2001, pp. 248–249]

  • equation image

where e′ is the strain rate tensor, ij are tensor indices, η is a material property with dimensions of viscosity, τref is a reference stress, n is the exponent of the power law rheology, σij is the deviatoric stress tensor, and σ ≡ equation image is its second invariant. Equation (15) applies within the mineral grains that are incompressible in irreversible flow. I am interested in the deformation of porous gouge. Both the normal traction P that drives compaction and the shear traction τ lead to deviatoric strain within the mineral grains. Both do macroscopic work. The other components of the stress tensor within the gouge depend on them. The invariant governing creep thus involves both these stresses [e.g., Wong et al., 1997]. The simplest form for the macroscopic shear strain is then

  • equation image

where c is a dimensionless constant of order 1 that takes into account that the gouge is a porous substance and the dimensionless constant from the different definition of the invariant in (15) and (16) can be included in η. The compaction rate of the gouge is similarly

  • equation image

where E is a dimensionless constant that represents the different ductile compliance of the material in compaction than in shear. The final term linearly represents dilatancy as in (8b). It is an anisotropic term arising from the grain geometry of the gouge. Note that I have made the assumption that the gouge is a thin tabular layer so that the rate of approach of the walls of the gouge zone is the negative rate of change of porosity times the gouge zone thickness, −fW.

Strictly, an analogous term to the second right-hand-side term of (17) should be included in (16). It represents compaction creep driving shear creep. That is,

  • equation image

where Ω is a constant with unknown sign of the order of β. If I restrict the analysis to situations in which frictional sliding is actually occurring somewhere near steady state, then both terms in (17) and f′ are of the order of βɛ′, where β is on the order of a few percent. The cross term in (18) is thus of the order β2ɛ′ and can be ignored.

Equations (16) and (17) represent the macroscopic effects of exponential creep at asperities (over a limited range of real stresses) with a power law rheology, before presenting an exponential rheology in section 3.3. The exponent n is large and the invariant dominates in (16) and (17). The creep rate goes from very small to very large over a limited range of the invariant. If one defines failure as the occurrence of interestingly fast irreversible creep, this feature is a simple form of the very useful approximation of an elliptic failure envelope in viscous porous materials [e.g., Wong et al., 1997; Rudnicki, 2004].

With forethought, I represent the compaction rate as a function of the shear strain rate

  • equation image

I wish to associate terms for the case where frictional sliding is occurring and the friction equation (3) is applicable. Using (3) for P/τ yields

  • equation image

where I have used the Taylor series expression 1/(1 + x) ≈ 1 − x to bring the second order terms into the numerator. By assumption, β does not depend on the velocity or the state variable. A steady state then can be reached only if it cancels the constant term in the bracket. That is, I associate β = E/μ0.

3.2. Relationship of Nonlinear Creep to Ruina Law

Making this association, equation (20) becomes

  • equation image

Continuing I express the left hand side of the Ruina [1983] evolution law (14) in terms of the Segall and Rice [1995] porosity relationship (7). Expanding the logarithm yields,

  • equation image

where I use (6) to compact notation. It is intriguing that (21) and (22) have similar forms. They become homologous when a = b. Otherwise the two steady states are not consistent. The implications of (21) and (22) are in any case similar, reducing shear traction not only decreases the shear strain rate as expected, but also the compaction rate.

Note that (22) cannot apply in the limit of intact rock f = 0 . Then using (6) and (7) and letting ϕ be the steady state porosity at feasible ɛ′ = ɛ′0 and P = P0, one gets f′/ɛ′ = ϕ/ɛint, which is of the order of 1 [Sleep et al., 2000]. This unlikely limit does not occur in the Dieterich [1979] law (8), which gives an acceptable description of the initial friction in intact rock as a function of normal traction [Sleep, 1999b].

3.3. Micromechanics of Exponential Creep

I continue the approach of the previous section by explicitly considering the microscopic variation of stress within the gouge and including exponential creep as in (12). This leads to compact expressions, and it is better to use the actual exponential rheology rather than the approximate power law.

It is convenient to express microscopic stresses with parametric equations in θ with analogy to the invariant in (16) (Figure 4),

  • equation image

and

  • equation image

where the invariant is [Berthoud et al., 1999, equation (20)]

  • equation image

(Note that a cross term τmPm is possible if the stresses locally drive flow in the same deviatoric sense; then careful attention needs to be given to signs.) The microscopic shear strain rate is [Berthoud et al., 1999, equation (22)]

  • equation image

where γ′ is a material constant with dimensions of strain rate and s is a material property with dimensions of stress. It is related to RT/M in (12) and is on the order of 100 MPa [Nakatani and Scholz, 2004] as shown at the end of this section. The corresponding expression for compaction is

  • equation image

where a dilatancy term is included as in (17). The strain rates go from extremely small to extremely large over a small range in the invariant r, which is (when considered less precisely) the yield stress or real strength of the material on the order of a few GPa for rocks. That is, the material is at a critical state where many asperities are close to failure. As already noted, the solid grains are incompressible so that only microscopic deviatoric stresses within the grains produce deviatoric creep. I have made the simplifying assumption that this deviatoric creep can be divided into creep that produces mesoscopic shear strain and creep that produces compaction.

image

Figure 4. Microscopic stresses in parametric r, θ space. The red box indicates stresses that drive creep. Its angular width is δ. Lower values of the invariant r do not produce significant creep; such stresses occur, for example, within the interior of grains. Higher values relax and are not present. The parameter λ represents the tendency for shear stress concentrations to persist longer than normal stress concentrations.

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The instantaneous grain scale strain rate is the integral of the actual instantaneous microscopic strain rates over a region in the gouge. I apply this concept to special cases in sections 3.4 and 4. For now, I consider mesoscopic strain rates that are spatial and brief time integrals of the microscopic strain rates over the gouge. I assume that the gouge is chaotic so that one can only predict its statistical properties. This is reasonable for simulated or real gouge with irregular fragments of various sizes. I also assume that the grain and asperity sizes are small enough that the finite thickness of the gouge zone does not create order. My formulation is thus not applicable to systems with high order like aligned rods and sorted spheres capable of rolling studied by Anthony and Marone [2005].

I pose the problem statistically with integrals over a weighting or probability density function g in analogy with Figure 8 of Aharonov and Sparks [2004]. Here I integrate formally over values of r and θ to get the expected values of mesoscopic properties (Figure 4). The mesoscopic shear strain rate over a finite time is formally,

  • equation image

where the dimensions of g are inverse stress squared. The weighting function need only be bounded for small values of r where the strain rate is extremely small (Figure 4). It must go to zero with r faster than the strain rate in (25) so that the integral stays bounded above the yield stress. Physically these conditions represent that one expects to find few extreme stress asperities that would relax quickly and that most of the deformation occurs near the yield stress. Equivalently, the weighting function need only represent the distribution in the domain of stress components where creep actually occurs. The θ dependence needs to represent that the microscopic stresses scale to their mesoscopic and macroscopic quantities. A convenient form of (27) satisfies these inferences

  • equation image

where h is the r -dependent product of the weighting function and the invariant (dimensions of inverse stress squared). Reiterating, the weighting function h is significant only for values of the invariant stress r near the yield stress. Extremely sluggish creep occurs at low values and rapid creep and stress relaxation precludes high values. Otherwise we need not know the rheology in detail. The dimensionless constant δ is proportional to the standard deviation of the Gaussian distribution (Figure 4). I presume it is small so that microscopic parameters scale to macroscopic ones. The other parameters within the bracket are

  • equation image

which represents the tendency of the ratio of the microscopic stresses to scale with the mesoscopic stresses. The parameter λ represents the intuitive tendency of asperities with high normal traction to behave differently than those with high shear traction (Figure 4). Well-aligned shear zones allow an indefinite amount of slip, while compaction closes pore space. In more detail, creep driven by normal traction brings touching grains closer together, increasing the microscopic contact area and decreasing the microscopic stress. Creep driven by shear traction parallel to a contact may even decrease contact area. Thus a high shear traction contact may accommodate more creep than a high normal traction contact during its lifetime. I define λ so that is positive when high shear traction contacts tend to persist longer than high normal traction ones.

As intended, I obtain simplification as r and θ parts of the integral separate. Taking the Taylor series for sin(θ) about θ0 and the limits of the rapidly convergent integral at ±∞ yields simple expressions

  • equation image

The expression for compaction rate is

  • equation image

I assume that we are in the range of parameters where frictional sliding occurs and apply the friction law (3) for τ/P terms from tan (θ0) in (29) so that I can associate terms. This yields that the ratio of compaction rate to shear strain rate as in (19) is

  • equation image

where small Taylor series terms are retained to first order. That is, λ2, a2, b2, aλ, bλ, ab and higher terms are ignored. The term in the brackets multiplying λ is positive so that the effect of λ is to retard compaction relative to shear as expected. As with (20), the constant term represents compaction balanced by dilatation β = E/μ0. The porosity change with this association is

  • equation image

Regrouping the terms yields an expression that provides some insight into the mesoscopic and macroscopic implications of the variable λ,

  • equation image

The first two terms in the bracket yield an expression in the form of the Ruina [1983] evolution law in (22) and its steady state if the latter two terms cancel. That is,

  • equation image

Differentiating yields

  • equation image

That is, λ must change with strain rate to change steady state properties. The parameter ba is related to the tendency of shear-traction concentrations to persist longer relative to normal-traction concentrations with increasing slip rate. This and the definition of λ in (28) are simple kinematic interpretations of rate and state friction.

Continuing, the exponential terms in (25) and (26) represent contact theory as does (12). I obtain a from the effect of a sudden strain rate change. The contact asperity stays the same (for a little while) so the effect is to add microscopic shear traction proportional to the mesoscopic change in strain rate. The change in (3) or (10) is

  • equation image

Differentiating (25) and applying (24) yields

  • equation image

Applying the first-order relationship that τm = μ0Pm to terms in the bracket and letting microscopic quantities scale to mesoscopic ones in the bracket, τm = τPm/P, yields

  • equation image

A condition for (39) is that the assumed first-order relationships hold. Comparing (37) and (39) yields

  • equation image

It is useful to compare the form of (40) with that of the thermally activated creep expression in (13). The microscopic pressure Pm corresponds to the real normal traction when frictional sliding actually occurs and

  • equation image

This allows approximate evaluation of s. The term in the bracket is of order 1. For a laboratory experiment on quartz, T ≈ 300 K, q ≈ 1 (for amorphous damaged material), and M ≈ 2 × 10−5 m3 mol−1. This yields 120 MPa.

3.4. Changes in Normal Traction

The friction equation (3) predicts that shear traction changes instantly when normal traction changes. This feature is impossible, as modeling fault rupture between materials with different elastic constants with this property leads to a physically ill-posed problem [Ranjith and Rice, 2001]. That is, the Linker and Dieterich [1992] law included in (3) applies to “sudden” changes where stress concentrations have had time to self organize at the asperity level and evolve the statistical distribution implied by g in (27). Contact theory provides that shear traction does not change when normal traction instantly changes at constant sliding velocity. The microscopic contact area in (12) does not instantly change so the real shear traction does not change either. That is, the geometry of asperities depends on the past history of normal traction not its instantaneous value [Ranjith and Rice, 2001]. In terms of (10), the pressure and state variable terms represent the tendency of asperities to persist in the lattice. They reflect a time-dependent process and hence cannot change instantly.

Perfettini et al. [2001] discuss how to formulate the evolution law so that shear traction does not suddenly change in such a case. In my notation, the state variable does not change instantaneously and I need to modify the expression for ψnorm in (6). A convenient form is

  • equation image

where P(t0) is the instantaneous normal traction, the instantaneous coefficient of friction is μ ≡ τ/P, and ω has units of 1/time. This expression has the property that an instantaneous change in normal traction does not instantaneously change shear traction. If normal traction varies on times longer than 1/ω, it becomes the Linker and Dieterich [1992] relationship in (6). This can be seen by substituting P = P(t0).

A bulky but more physical delay term would involve strain rather than time. The quick response from (42) and the longer-term response from the evolution equation (4) or (14) separate if the quick strain is much less than ɛint, 0.06–0.12 using the values obtained by Sleep et al. [2000], who accounted for strain rate localization. It has not escaped me that this is similar to the porosity in the gouge.

A related effect is that the real area of asperity contact immediately after a hold is higher than that when friction reaches a peak after a small amount of shear strain [Goldsby et al., 2004]. The contact areas increase logarithmically with hold time during the hold as expected from an exponential rheology, but a first increment of shear strain partly disrupts these contacts without changing porosity. The Segall and Rice [1995] relationship with the friction law (3) is an approximation that applies when enough sliding has occurred for an expectation function to apply.

Qualitatively, there are two effects at real contacts. First, creep from normal traction strengthens an individual contact and brings the grains closer together. This convergence compacts the gouge making more contact area elsewhere. Second, shear creep rearranges the contact where it occurs and makes and breaks other contacts elsewhere on a rough surface. This effect occurs at constant porosity and is distinct from dilatancy.

Thus both shear and compaction strain need to be included in a quantitative model of a gouge where normal traction changes. For example, one might expect that the Linker and Dieterich [1992] parameter α to differ between a sudden increase in normal traction where the gouge layer rapidly compacts and a sudden decrease in normal traction where rapid shear strain rearranges real contacts and produces dilatancy.

Measurements to resolve whether the parameter α depends on the sense of the change in normal traction are inadequate to resolve the issue. One complication is the failure of the patch (millimeter scale, Figure 1) normal traction on a laboratory contact may not be the same as the macroscopic nominal value everywhere on the gouge surface [Sleep, 1999a]. A second complication involves strain rate localization and delocalization [Sleep et al., 2000].

4. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

As shown in section 3, the gross behavior of the Ruina [1983] evolution law arises from exponential creep whenever the stress components at asperities scale to the macroscopic components as was assumed in the derivation of (32). In particular, the macroscopic strain rate depends on the integral of microscopic strain rate or more tractably on an integral of the expectation function for microscopic strain (28), (30), and (31). The rate and state stability parameter ba represents the tendency of shear stress asperities to persist lower at high strain rates relative to normal traction concentrations (36).

My derivation involved little detailed physics and lattice dynamics so it should have some generality. In particular, I did not need to use whether the macroscopically limiting region of exponential creep was at the yield stress near asperities between grains or that at crack tips as assumed by Beeler [2004]. One can do much better using the real tensor compliance properties of grains, cracks tips, and gouge lattices. This computation seems possible but protracted. One might obtain the probability density function, the gross coefficient of friction μ0, and the form of the invariants from a feasible number of grains. One would then integrate to get macroscopic properties. For example, Aharonov and Sparks [2004] obtain a probability density function for stress with a 24 × 25 two-dimensional lattice of grains and a rheology with elastic grains and frictional contacts.

I now return to situations where the Ruina [1983] laws fails to represent the gross behavior of frictional sliding. First, the compaction rate in the Dieterich [1979] evolution law is independent of the shear traction so that healing occurs during holds. As such behavior sometimes happens in the laboratory, it is relevant to see how it arises in real samples. Examination of (17) and (26) provides some insight. As implied by the weighting function g in (27), microscopic components of the stress tensor and the strain rate tensor differ from the mesoscopic tractions at asperities. An extreme case is that the normal traction asperities do not have correspondingly high shear traction concentrations and high shear traction asperities do not have high normal traction concentrations. The high normal traction term thus dominates the microscopic invariant in (17) at the asperities that accommodate compaction. Taking the first order term yields

  • equation image

which is independent of the shear traction. The contact theory in (13) or (40) still applies with an effective yield pressure and a real shear traction.

Returning to physics, I qualitatively compare high humidity where the Dieterich [1979] law appears to hold with low humidity where the Ruina [1983] law appears to hold. High humidity results in hydrated silica, a weak material at asperities accommodating shear strain [Frye and Marone, 2002; Di Toro et al., 2004; Anthony and Marone, 2005; Hong and Marone, 2005]. The microstrain for compaction may occur within the stronger anhydrous silica of the grains. This creep maintains real contact areas. The net effect is to separate shear strain from compaction so that the Dieterich [1979] law applies. Conversely, the Ruina [1983] law applies at low humidity because the essentially anhydrous grains are more homogeneous substance.

More subtly, the Ruina [1983] evolution law does not apply when the gouge has been under low shear traction for a long time so that no frictional sliding has occurred. The friction law (3) used in the derivation then need not apply at astronomically small shear strain rates. Normalizing the compaction rate to the ill-determined minute shear strain rate in (22) and (33) is not warranted. The kinematic difference between shear and compaction is the opposite of that during active sliding. Small amounts of shear strain relax stresses that are not renewed by macroscopic sliding. Compaction can continue at a finite rate until pore space vanishes.

In addition, strain rate localization during sliding lets Dieterich [1979] and Ruina [1983] behavior coexist during a hold. The gouge surrounding a high strain zone is far from frictional failure and can continue to compact as in (27), while the gouge within the high strain rate zone is near frictional failure.

Experiments support the inference that gouge can compact far from frictional failure. For example, Hagin et al. [2005] present data showing that the Dieterich [1979] compaction law (8) applies to sand under isotropic compaction ɛ′ = 0, which is as far from shear failure as one can get in an experiment. Power law creep as in (17) and (43) is a reasonable representation of these data. Note that continuing compaction is kinematically less likely than continuing shear to regenerate high-stress asperities.

Situations intermediate between the Dieterich [1979] and Ruina [1983] evolution laws are expected, complicated, and observed in the laboratory [Boettcher and Marone, 2004]. As a practical matter for modeling real materials, one can represent the evolution law as a sum of a small Dieterich [1979] in (4) and a large Ruina [1983] term in (14),

  • equation image

where ɛD and ɛR are the intrinsic strains for the Dieterich [1979] and Ruina [1983] terms, respectively. The time derivative of the state variable in Figure 3 then reaches a small asymptote at small strain rates rather than going all the way to zero. This expression as given retains the steady state in (5). Kato and Tullis [2001] proposed a different form of a hybrid evolution law with grossly similar implications.

Finally, crustal faults at seismogenic depths are obviously wet. I expect that healing and compaction occur over a long interseismic interval. The Dieterich [1979] evolution law or a hybrid law may provide a good approximation. One may have to explicitly consider pressure solution.

Notation
a

rate coefficient, dimensionless.

b

state coefficient, dimensionless.

c

constant in stress invariant, dimensionless.

C1

grouped constant for healing term, stressn.

Cɛ

material property relating porosity and state variable, dimensionless.

Dc

critical displacement, m.

e

strain rate tensor, s−1.

E

constant for ductile compaction, dimensionless.

f

porosity, dimensionless.

fm

microscopic compaction rate, s−1.

fss

steady state porosity, dimensionless.

g

weighting function, stress−2.

h

weighting function depending on stress invariant, stress−2.

ij

tensor indices, dimensionless.

M

molecular volume, m3 mol−1.

n

power law exponent, dimensionless.

P

normal traction, N m−2.

P0

reference normal traction, N m−2.

Pm

microscopic normal traction, N m−2.

Preal

real normal traction on contact, N m−2.

q

power law exponent at low stress, dimensionless.

R

gas constant, J K−1 mol−1.

r

stress invariant, N m−2.

s

stress scale for exponential creep, N m−2.

t

time, s.

T

absolute temperature, K.

V

sliding velocity, m s−1.

V0

reference sliding velocity, m s−1.

W

thickness of the sliding zone, m.

α

coefficient associated with changes in normal traction, dimensionless.

β

dilatancy coefficient, dimensionless.

γ′

material constant for microscopic creep, s−1.

δ

parameter representing variation of ratio of microscopic shear to normal traction, dimensionless.

η

viscosity for nonlinear material, Pa s.

θ

parametric angle for microscopic stress, dimensionless.

θ0

parametric angle for macroscopic stress, dimensionless.

ɛint

intrinsic strain, dimensionless.

ɛD

Dieterich term intrinsic strain, dimensionless.

ɛR

Ruina term intrinsic strain, dimensionless.

ɛ′

strain rate, s−1.

ɛ′0

reference strain rate, s−1.

ɛ′m

microscopic strain rate, s−1.

ɛ′base

material constant for exponential creep, s−1.

ɛ′real

real strain rate at contact, s−1.

ϕ

reference porosity, dimensionless.

ψ

state variable, dimensionless.

ψnorm

normalizing coefficient for state variable, dimensionless.

ψss

steady state value of state variable, dimensionless.

λ

parameter representing persistence of shear stress concentrations, dimensionless.

μ

coefficient of friction, dimensionless.

μ0

first-order coefficient of friction, dimensionless.

σij

deviatoric stress tensor, N m−2.

τ

shear traction, N m−2.

τm

microscopic shear traction, N m−2.

τreal

real shear traction on contact, N m−2.

τref

reference stress, N m−2.

ω

decay factor in exponential, s−1.

Ω

constant for compaction-driven shear, dimensionless.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References

This research was in part supported by NSF grant EAR-0406658. 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 896. I thank Jim Rice, Chris Marone, and Terry Tullis for discussions at the 2004 SCEC meeting and helpful emails. David Sparks and an anonymous reviewer provided numerous suggestions.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Rate and State Equations
  5. 3. Ruina Evolution Law
  6. 4. Discussion and Conclusions
  7. Acknowledgments
  8. References