Frictional sliding dilates gouge or increases the separation of sliding surfaces. A common hypothesis is that the rate of dilatational strain scales linearly with the rate of shear strain. The proportionality constant is called the dilatancy coefficient. Real contact theory of friction may explain this feature. Moving contact asperities produce damaged regions with elastic strains scaling to the ratio of the real strength of asperities to the elastic modulus. Balance between this local strain-energy production rate and macroscopic work against normal traction indicates that the dilatancy coefficient scales with this ratio. So do the average slopes on a mature rough sliding surface if opening-mode cracks are unimportant. The result is compatible with the observed dilatancy coefficient of quartz gouge, ∼4%.
The gouge on major faults accommodates numerous earthquakes. Over time the gouge approaches a quasi steady state where production in porosity during earthquakes balances the compaction of gouge between earthquakes. Rock physicists have documented analogous effects on laboratory samples. Shear strain increases the porosity of laboratory fault gouge, reducing its coefficient of friction. The sample “heals,” that is it becomes stronger by compaction during very slow sliding. In general, the frictional strength of a sample depends on both its history and the current sliding velocity.
Rate and state friction mathematically represents these effects with the state variable ψ representing the history. Segall and Rice  included dilatancy in this formalism by expressing the state variable of a fault in terms of porosity. Sleep  and Sleep et al.  showed that their relationship has simple kinematic interpretation that dilatancy produces porosity at a rate proportional to the shear strain rate. The treatment of rate and state friction by Beeler and Tullis  also has this feature. While this hypothesis seems reasonable, no theory is available that expresses the ratio of shear strain rate to dilatant strain rate in terms of measurable asperity-scale physical parameters. The purpose of the paper is to examine this issue.
I begin with a short review of rate and state friction to put the relevant frictional parameters in context. I then discuss kinematic dilatancy at very low normal traction. I then apply the physics of creep at contact asperities to obtain dimensional scaling relationships for dilatancy.
2. Rate and State Friction
As noted in the introduction, the friction on a sliding surface depends on its past history. It also depends on the instantaneous sliding velocity. Ruina  showed that using strain rate rather than macroscopic sliding velocity facilitates discussion of strain rate localization and aids comparison with other flow laws. I thus use a convenient form for instantaneous friction in terms of the instantaneous strain rate
where μ0 is the coefficient of friction at reference conditions, a and b are small dimensionless constants, the strain rate is ɛ′ ≡ V/W where V is sliding velocity and W is the thickness of the shear zone the reference strain rate is ɛ′0 ≡ V0/W, where V0 is a reference velocity, and ψnorm is normalizing value for the state variable, which I discuss below.
It is also necessary to have an “evolution law” that describes the combined effects of damage from sliding and healing. Dieterich's  evolution law represents these effects explicitly with separate terms
where the first term represents healing and the second damage. The variable t is time, the intrinsic strain is ɛint ≡ Dc/W, where Dc is the critical displacement to significantly change the properties of the sliding surface, α is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from Linker and Dieterich , and P0 is a reference normal traction. It arises formally from the steady state value of the state variable
If the steady state coefficient of friction is independent of normal traction
where the first term represents that the increase of porosity from dilatancy is proportional to the shear strain rate. This defines a dilatancy coefficient β ≡ Cɛ/ɛint. Beeler and Tullis  obtain an equivalent result. The second term represents power law compaction creep during holds where the exponent is n = α/b.
The Ruina  porosity evolution law does not have a simple implication
Still, it can be derived with the assumption that the dilatancy rate scales linearly with the shear strain rate [Sleep, 2005, 2006].
To reiterate in terms of macroscopic processes, seismic dilatancy has an important direct effect on the physics of earthquakes. From (5), porous gouge is weak. This weakness is essential to unstable sliding. The steady state coefficient of friction in (1) and (3) decreases with strain rate if a > b. Dilatancy may also affect earthquake nucleation; increased porosity decreases the fluid pressure in a sealed fault zone. This effect increases the effective stress and increases the friction on the fault, tending to quench earthquakes. One would thus like to know how dilatancy arises before exporting laboratory friction to crustal faults.
I begin by reviewing familiar processes that are probably not relevant to mature fault surfaces in section 3. I then relate dilatancy to elasticity and asperity strength in section 4.
3. Kinematic and Crack Dilatancy
Pore space is needed of a rigid material to deform at low normal traction. Conversely, broken surfaces of brittle materials, like china, do not fit back together. It is tempting but probably incorrect to attribute frictional dilatancy to this geometrical effect. There are two problems with this approach.
First, the dilatancy coefficient is too high, that is, of the order of one. To see this, consider a rigid square between two rigid sheets. The sheets need to come apart for the block to roll. When the block pivots about a corner, the instantaneous shear velocity of the plates and separation velocity are the same (Figure 1).
Second, there is no rate dependence of the steady state velocity as implied by (5), (6), and (7). The room needed for rigid blocks to roll and move around each other depends only on their shape.
Neither is frictional dilatancy likely to be related to the behavior of cracks in an intact elastic material. This effect arises, for example, in pure shear when the mean stress stays constant. All the cracks oriented perpendicular the most compressive the principal stress close when it reaches a critical value and further increase in this stress causes no more crack compaction. Increasing tensional stress in the other direction continues to open cracks, leading to a net dilatancy at constant mean stress. This is a quasistatic process with no simple rate dependence. The strains are in principle recoverable when the stresses are removed, but some kinematic dilatancy occurs because crack surfaces may not fit back together.
With forethought, I begin with contact theory that yields the first-order expression τ = μ0P. This result suffices for dimensionally obtaining the dilatancy coefficient. The real strain rate at an asperity depends exponentially on the real stress. The scalar equation is
where s is a stress scale of the order of 0.1 GPa and e′0 is a constant with dimensions of strain rate. The real stress τ is proportional to the second invariant of the deviatoric stress . I normalize it so that it is the resolved shear stress in two-dimensional plane strain. The commonly used von Mises stress is this stress times .
I stay dimensional (until section 4.3) so that I do not obscure simple relationships with complicated geometrical factors. I start with the well-known derivation for the constant coefficient of friction μ0, the leading term in (1). Failure (for example in quartz) occurs with the real shear traction of the order of 10 GPa from indenter data by Goldsby et al. . The quantity is large enough compared to s to warrant approximating the flow law with failure at a shear yield stress τy. A yield stress for normal traction Py arises similarly. (The effects of macroscopic shear and normal traction both act on an asperity and effect the stress invariant τ within the asperity so there is an implicit geometrical assumption about the contact, which needs to be considered to obtain the evolution laws (6) and (7) [Sleep, 2005, 2006].) In the traditional derivation, microscopic properties scale to the macroscopic ones, the coefficient of friction is
The derivation to this point is well known. I extend it to include anelastic strains within regions overrun by contact asperities. I apply the dimensional result that the real shear and normal tractions are comparable and that they scale with macroscopic properties.
4.1. Elastic Strain Energy
I include elasticity with some forethought and with the justification that an infinitely rigid material stores no elastic strain energy. I justify ignoring inertia at the end of this section. I follow the effects of the normal traction P as I am after a dimensional result. That is, I do not distinguish at the stage between τ and P, whose ratio is the order of 1. I initially let the shear modulus G dimensionally represent the elastic modulus for the actual geometry. I implicitly apply Saint-Venant's principle [Sokolnikoff, 1956, pp. 89–90] throughout my discussion, the difference between loading by asperities and uniform loading by the macroscopic stresses away from the asperity being examined does not matter.
I consider friction between two surfaces to avoid the geometrical complexity of gouge. That is, I associate the separation rate of the plates with dilatancy in gouge. I follow the path of a representative asperity of radius A (Figure 2). For bookkeeping purposes, my asperity stays attached to the “upper” sheet in the figure. The lower sheet is initially planar and parallel to the upper sheet. The contact bears the macroscopic stresses over a region of radius R. That is,
The asperity crosses a surface area of the lower “substrate” sheet of 2AVt over time t. The yield stresses act on the base of the asperity, causing the immediate substrate to yield downward to a depth scaling with the radius A. The volume of damaged substrate produced over the time is dimensionally VA2t.
Stresses in the substrate change as the asperity moves. The aftermath of its passage on the lower sheet is relevant. For simplicity, I consider that the asperity slides on a planar surface parallel to the sheets. The “flat” position of the deformed material exposed after the passage of the asperity is that with the asperity on it with normal traction Py. With the load of the asperity removed, the exposed substrate material springs back, but stresses remain. The deformation is comparable to that from a negative normal traction scaling to Py over the exposed area. I use this feature as a statistical average to obtain macroscopic scaling relationships for dilatancy.
I begin with microscopic energies. The elastic strain energy per volume in the region damaged by passage of the asperity scales as
This assumes that the stresses within the damaged region are comparable to the yield stress. See Appendix A for a simpler example. The rate that the moving asperity creates elastic strain energy is dimensionally
More mathematically, the equilibrium elastic strain energy of an unloaded region is a minimum compatible with the internal anelastic strains within the material and the macroscopic boundary forces (here the normal traction) [e.g., Sokolnikoff, 1956, p. 383]. I assume on the basis of the example in Appendix A that comparable fractions of the available energy go into local elastic strains and work against normal traction (dilatancy). I use the total energy Q to dimensionally represent both the local strain energy and the energy associated with dilatancy.
I now equate microscopic and macroscopic effects. From (10), the macroscopic production rate of elastic energy is
which is independent of the asperity radius. I compare this rate with the rate of dissipation of energy over this area from macroscopic friction, which is
This yields that the ratio of elastic strain energy production to frictional energy dissipation scales as
where the coefficient of friction μ is set to 1 because I have already dropped other factors of this order in my dimensional presentation.
As already noted, the elastic strain energy deforms the sliding surface. It will also deform a gouge. Part of the elastic strain energy is available to push surfaces and grains apart. It can also cause spalling/cracking. Some of the elastic strain thus acts macroscopically to dilate the gap between the surfaces with the macroscopic velocity Vz.
Dimensionally applying the variation principle of minimum elastic energy [e.g., Sokolnikoff, 1956, p. 383], the energy available for work on macroscopic boundaries scales with the elastic strain energy. See Appendix A. Kinematically, the macroscopic work per time from dilatancy over the area is
As Qd scales with the available energy Q, comparing (14) and (16), yields the ratio of work done to dilate and work done by friction is Qd/Qf ≈ Q/Qf ≈ Py/G. As I did not retain factors of the order of unity and as the coefficient of friction is near its reference value μ0, I do not distinguish this energy result μ0PVz = βτVx from the kinematic statement in (6)Vz = βVx that the dilatational velocity (or strain rate) is proportional to the shear velocity. I return to the value of the dilatancy coefficient in the section 4.3.
Note that inertial effects vanish in the limit of slow sliding. The forces from the tractions Py and τy are essentially independent of the sliding velocity. The vertical velocity of the surface scales with the surface slope times the horizontal velocity; its sign changes on the timescale A/V for an asperity to traverse a point. The acceleration thus scales as V2/A.
4.2. Surface Slopes and Roughness
The concept that the stresses within unloaded anelastically damaged regions scale with the yield stresses τy and Py associated with loading by asperities leads to further geometrical results. Consider, first a substrate region just vacated by an asperity of radius A. The effective negative load then scales as πA2Py. Treating this as a point source, the vertical displacement is
where λ is the second Lamé constant and r is the radial distance along the substrate surface from the center of the point load [Sokolnikoff, 1956, p. 340]. Timoshenko and Goodier [1970, p. 404] give the full expressions for a distributed circular load in terms of elliptic integrals. The radial slope at the edge of the disk r = A in (17) is
where G = λ gives the second equality. This equation provides a scaling for strains in the damaged surface. It also provides a scaling for the slopes over which asperities pass (Figure 3).
The derivation has implications to the roughness of a surface. In terms of equivalent stresses, a Poisson distribution in 2 dimensions might represent sparse damaged regions on a nearly fresh surface. Gaussian white noise at wavelengths longer than the asperity dimension, A, might represent the equivalent stresses acting on a mature surface. In both cases, the spectrum of stress is flat [e.g., Turcotte, 1992, pp. 78–79] in the wavelength range between contact dimension and macroscopic scale (say of a laboratory sample). For shorter wavelengths, one needs to consider the distribution of stresses beneath the asperities and the tendency for asperities to smear along the sliding direction.
Distributed white-noise stress has a simple geometrical consequence. As elastic displacement at constant stress scales inversely to wavelength, the surface height to wavelength is self-similar [e.g., Turcotte, 1992, p. 83; Batrouni et al., 2002]. This is equivalent to the elastic strains and slopes being independent of wavelength. This feature applies to most of the surface. Random roughness keeps surfaces apart and produces stress concentrations at protrusions [Batrouni et al., 2002]. A process (here anelastic strains from damage beneath asperities) creates surface roughness thus also produces dilatancy.
Note that pointy asperity contacts are likely to exist. Fracture (including spalling) produces sharp protruding corners. At low normal tractions, such protrusions will form one side of many contact asperities. That is, the contacts will locally be indenter against flat surface as assumed in the models.
4.3. Appraisal and Application to Quartz
The dimensional equation (15) and roughness equation (18) yield estimates of the dilatancy coefficient in terms of parameters that can be measured independently of friction. I appraise the derivation with quartz. It is a well-characterized laboratory material that occurs within natural faults. The shear modulus for quartz is 44 GPa.
I use invariants, as the elastic strain energy in (11) is proportional to ɛijτij which is proportional to τijτij. As friction involves shear, I use deviatoric stress normalized to give resolved shear stress in two dimensions τ and the shear modulus G. The strain energy per volume is then τ2/2G and the dilatancy coefficient is approximately τy/2G. The actual value should be somewhat less that this as only part of the elastic strain energy does work against macroscopic normal traction. See Appendix A. One would need numerical calculations to obtain a more precise estimate.
I need to convert with the real strengths in the literature to strengths in shear. Tribologists determine real strength in different ways and express it with various normalizations. Poirier [1990, p. 38] gives a rule of thumb theoretical strength limit τ/G of 1/10, which give a real strength of 4.4 GPa. Dieterich and Kilgore  measured real stresses on quartz surfaces in contact and obtained a range of 7 to 14 GPa. Intuitively upper limit represents the maximum strength. There are two ways that might be converted to shear strength. It could represent the real normal traction in shear where the real shear traction is μ0 times it or 10 GPa. Alternatively, failure occurs in uniaxial compression of the contacts. The equivalent shear stress is the scales as G/E in an incompressible substance or by a factor of ∼1/3. This implies that a real strength of ∼4.7 GPa. Goldsby et al.  obtain Vickers hardnesses of 12 to 14 GPa from nano-indentation tests, which they consider compatible with Dieterich and Kilgore's  data. Finite element codes exist for obtaining the stress invariant at indentation failure [e.g., Bouzakis and Michailidis, 2006], but I am aware of no work on quartz or a similar silicate.
Given this uncertainty, I let the real shear strength of quartz range from 4 to 10 GPa. The predicted dilatancy coefficient τy/2G is then 4.5–11%. The lower end of this range is reasonable, ∼5% is traditional rounded estimate for the dilatancy coefficient of gouge. Experiments where the normal traction is increased are effective for determining the dilatancy coefficient as compaction occurs during holds as implied by the Dieterich  evolution law (6). Laboratory results are not strongly affected by strain rate localization. Sleep et al.  obtained 2.8–5.6%. The work of Segall and Rice  yields 3.7%.
Simple dimensional arguments suggest that moving contact asperities leave anelastic strains scaling with the ratio of real strength to elastic modulus τy/G in their wake. Energy balance indicates that the dilatancy coefficient scales with this ratio. Poirier [1990, p. 38] notes that this ratio does not vary a lot in macroscopically brittle substances. Physically the ratio represents a strain that significantly deforms a crystal lattice. My hypothesis has the attractive attribute that sliding continually produces surface roughness as well as porosity. The sliding surface thus does not get worn smooth with all places in contact.
The hypothesis is in principle testable in the laboratory. One could use “hard” metal, mica, or clay where the ratio τy/G is lower than in strong silicates. The hypothesis can be modeled numerically with viscoelastic code that can handle exponential creep over small regions and the free surfaces associated with porosity. I am not aware of a code that suffices. An elastic-plastic code would provide an approximate model.
I conclude with some caveats that lead to possible physical experiments. First, I presumed that the derivation extends to gouge. That is, I equate gouge to a complicated set of sliding contacts where the physics of a sliding surface locally holds. It is conceivable that there is also reversible dilatancy in gouge (N. Beeler, personal communication, 2006). The elastic normal strain across the gouge may depend on the contact geometry and hence the shear traction and sliding velocity. I also obtained the fractal roughness of a sliding surface. I did not consider fracture explicitly. N. Beeler (personal communication, 2006) suggests that fracture (that is, opening-mode cracks) may dominate over asperity contacts in producing roughness.
Appendix A:: Bent Beam Analogy
The deformation in the wake of moving asperities is too complicated to expect an analytical solution. I use a bent beam (Figure A1) to illustrate the concepts of the paper.
The beam is bent far beyond its elastic limit. The fiber stress within the beam is the yield stress in both the upper and lower halves. The bending moment (per length in cross section) is
where Z0 is the half thickness of the beam, z is the dummy variable for distance from the centerline, and the fiber stress (tension positive by convention) is the yield stress Py. The elastic strain energy (per length) is
where B is an elastic modulus. The beam springs back some when the load is removed. The new bending moment is
where the dimensionless coefficient χ multiplies a fiber stress from bending that is proportional to the distance from the center line. If unloaded, the beam flexes so the χ = 3/2. The elastic strain energy is
The minimum as expected from the variation principle [e.g., Sokolnikoff, 1956, p. 383] occurs at χ = 3/2 where there is no net bending moment. It is
This expression illustrates the inference that the amount of energy in the beam available for external work scales to the total elastic strain energy at the elastic limit as assumed in (13). Here of the original elastic strain energy is available for external work. This justifies my assumption that the available fraction on frictional surfaces and gouge is of the order of 1.
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 1030. The 2004 and 2005 SCEC meetings showed the need to understand the physical basis of friction. David Goldsby and Jim Rice promptly answered my questions. Nick Beeler and Masao Nakatani provided helpful reviews.