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FIG. 1. Response of friction to an abrupt increase in sliding velocity. The parameters a, b, and Dc are those in Eqs. 1a, 1b, and 1c. The behavior given by the equations for a sudden velocity change is illustrated graphically. The magnitude of the direct effect is measured by a and of the evolution effect is measured by b.

However, a complete understanding of what occurs at the contacting asperities does not yet exist.

The other effect seen in all friction experiments is an initial increase in the resistance to sliding that occurs when the velocity of sliding is abruptly increased (Fig. 1). This is termed the direct effect because the change of resistance occurs instantaneously and in the same sense as the change in velocity.

Stability of sliding. The absolute value of frictional resistance is unimportant for controlling the stability of sliding. However, the time, velocity, and displacement dependence of friction interact with the elastic stiffness of the surroundings to produce either stable or unstable sliding. The stability has been analyzed by using the constitutive laws described in the following section (14, 16, 17, 1924).

The stability of sliding is ultimately controlled by an interaction between the stiffness of the loading system and the dependence of frictional resistance on displacement. In systems for which the frictional behavior is as illustrated in Figs. 1 and 2, the dependence of frictional resistance on displacement is itself a function of how friction depends on velocity (15, pp. 566–569). If the frictional resistance decreases with increasing velocity, a behavior termed velocity weakening, then unstable sliding is possible. This unstable sliding generally is not possible for a velocity strengthening material, one in which the resistance increases with slip speed. This generality must be modified if the resistance does not show a monotonic change after the direct peak (26).

Predictability of unstable sliding. The onset of an unstable sliding event in the laboratory is typically preceded by some nonlinearity in the loading curve and by accelerating slip. Whether this is a universal feature of all laboratory experiments is not known, since detecting this behavior may require high precision measurement of the stress and perhaps measurement of the displacement using a transducer mounted

FIG. 2. Increase of static friction as a function of the time period of holding static.

inside the pressure vessel immediately adjacent to the sample. In Fig. 3 is shown a typical sequence of stick slip events measured with internal stress and displacement transducers in our rotary shear apparatus (26). At first glance (Fig. 3A), the events look as if they occur without warning, but successively closer examinations (Fig. 3 B-D) show that there is accelerating slip and an associated nonlinearity in the stress-time curve that foretells each unstable event. There is no reason to believe that this precursory accelerating slip should not occur in the earth. The real question is whether it will be large enough to be usefully detected and so provide the basis for a short-term earthquake prediction. In order to answer this we need to learn how to extrapolate the laboratory results to the earth. The constitutive laws discussed in a following section form the present basis for doing this.

Constitutive Laws. Empirical constitutive laws have been used by many workers to fit the frictional behavior described (e.g., refs. 13, 17, 2638). Although a variety of functions have been presented, two laws are most commonly used, and even these can be cast in slightly different ways. The form used in ref. 13 is convenient because the state variable has dimensions of time in both laws. Both laws represent friction as a function of velocity and a state variable by the same equation:

μ=μ0+a ln(V/V0)+b ln(θV0/Dc). [1a]

The direct effect is contained in the term a ln(V/V*) and the evolution effect in the term b ln(θDc/V*). The nature of the evolution is what differs in the two laws. In the law termed the slip law, because slip is required for evolution, the state variable evolves according to:

dθ/dt=(θV/Dc) ln(θV/Dc). [1b]

In the law termed the slowness law, because the evolution depends on the slowness (inverse of velocity) or on time, the evolution is given by:

dθ/dt=1−θV/Dc. [1c]

Neither of these two commonly used laws fits all aspects of experiments data (39), and a better law is needed. Both laws do a reasonable job of fitting data until one looks carefully at the details. If the processes that cause the observed behavior can be understood, then the correct form of this law may be found, and we can have more confidence in extrapolating its behavior outside the range of existing laboratory data.

Lack of Data and Constitutive Laws for Dynamic Slip Under Realistic Conditions. No laboratory experiments combine the large displacement, high slip rate, high normal stress, and presence of pressurized pore fluids that characterize dynamic earthquake slip. Most experiments involving high slip velocity (e.g., refs. 4042) have been done at low total displacement and low normal stress, without the presence of pore fluids. These failings mean that processes that may occur during dynamic slip in earthquakes have not been explored experimentally. Chief among these are shear heating and associated possible melting or increase in pore fluid pressures. Some experiments at low normal stress at relatively high velocity and large total displacement have produced shear melting (4345). Experiments on foam rubber suggest the importance of reduction of normal stress during dynamic slip (46).

Thus, although we have considerable data that may be relevant to the accelerating slip that occurs prior to an earthquake, we have no experimental data that is really useful for characterizing the velocity or displacement dependence of frictional resistance that might cause self-healing slip pulses (4749), the lack of a heat flow anomaly in area of faults that typically slip in seismic events (5052), or other evidence for the absolute and relative weakness of major faults (5355).

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