Depinning with dynamic stress overshoots: A hybrid of critical and pseudohysteretic behavior

A model of an elastic manifold driven through a random medium by an applied force F is introduced and studied. The focus is on the effects of inertia and elastic waves, in particular stress overshoots in which motion of one segment of the manifold causes a temporary stress on its neighboring segments in addition to the static stress. Such stress overshoots decrease the critical force for depinning and make the depinning transition hysteretic with static and pinned configurations coexisting with the steadily moving phase for a range of F. We find that the steady-state velocity of the moving phase is, nevertheless, history independent and the critical behavior as the force is decreased is in the same universality class as in the absence of stress overshoots-the dissipative limit in which hysteresis cannot occur and theoretical analysis has been possible. To reach this conclusion, finite-size scaling analyses have been performed and a variety of quantities studied, including velocities, roughnesses, distributions of critical forces, and universal amplitude ratios. If the force is increased slowly from zero, the behavior is complicated with a spectrum of avalanche sizes occurring that seems to be quite different from the dissipative limit. Related behavior is seen as the force is increased back up again to restart the motion of samples that have been stopped from the moving phase. The restarting process itself involves both fractal-like and bubblelike nucleation. Hysteresis loops in small- and intermediate-size samples can be understood in terms of a depletion layer caused by the stress overshoots. Surprisingly, in the limit of very large samples the hysteresis loops vanish. Although complicated crossovers complicate the analysis, we argue that the underlying universality class governing this pseudohysteresis and avalanches is again that of the apparently very different dissipative limit. But there are history dependent amplitudes-associated with the depletion layer-that cause striking differences over wide ranges of length scales. Consequences of this picture for the statistics and dynamics of earthquakes on geological faults are briefly discussed.