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Proceedings of the National Academy of Sciences of the United States of America
FIG. 1. Effect of velocity steps on coefficient of friction μ in various materials, from Dieterich and Kilgore (7). The top curve gives the response predicted by the rate- and state-dependent constitutive formulation.
acteristic slip, Dc and seeks the steady-state value Under conditions of stationary contact, Eq. 2 has the property that θ increases with the time of stationary contact. This results in strengthening of surfaces by the logarithm of time, which is also observed in experiments (1, 7, 11). At constant normal stress, θ increases whenever , which provides for the recovery of frictional strength after unstable slip events.
It is beyond the scope of this brief review to discuss the physical mechanisms that give rise to the rate- and state-dependent behavior. Observations of micromechanical processes for bare surfaces are described by Dieterich and Kilgore (7). Controlling mechanisms for gouge layers are somewhat more problematic (10, 12).
Relationship to Other Constitutive Characterizations
A variety of other fault constitutive formulations have been employed in analyses of earthquake processes. These include velocity-weakening laws, displacement weakening at the onset of slip, apparent fracture energy at a rupture front, and the rudimental concept of a static and sliding friction. Each of these representations can be described as an approximate limiting-case characterization of the rate- and state-dependent formulation.
The dependence of steady-state fault strength on sliding speed is obtained from Eq. 1 by taking dθ/dt=0 in evolution law (2). Under conditions of constant normal stress, the steady-state condition is , which gives from Eq. 1,
where the definition has been used. If the transient evolution effects observed when sliding speed changes (Fig. 1) are ignored, Eq. 3 represents a constitutive law for decreasing strength with increasing slip speed (velocity weakening) provided B>A (Fig. 2A).
Fig. 2b illustrates the sliding resistance as a function of slip displacement of a fault that was previously stationary and then is constrained to slip at a constant sliding speed. Because slip speed is constant, only the evolution of the state variable governs the displacement-weakening behavior. From Eq. 2, the evolution of state at constant slip speed is
where θ0 is state at , and σ and are held constant. Substitution of Eq. 4 for θ in Eq. 1 yields displacement weakening from the peak strength τp to the steady-state strength τss (Fig. 2b). The change of resistance from τp to τss is governed by the evolution of state from θ0 to the steady-state value ,
which gives from Eq. 3
Slip weakening comparable to that of Fig. 2b has been documented at the front of dynamically propagating slip instabilities where the slip speed is approximately constant (13, 14). The apparent fracture energy at a rupture front is defined