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A model of an elastic manifold driven through a random medium by an appliedforce F is studied focussing on the effects of inertia and elastic waves, inparticular {\it stress overshoots} in which motion of one segment of themanifold causes a temporary stress on its neighboring segments in addition tothe static stress. Such stress overshoots decrease the critical force fordepinning and make the depinning transition hysteretic. We find that the steadystate velocity of the moving phase is nevertheless history independent and thecritical behavior as the force is decreased is in the same universality classas in the absence of stress overshoots: the dissipative limit which has beenstudied analytically. To reach this conclusion, finite-size scaling analyses ofa variety of quantities have been supplemented by heuristic arguments.  If the force is increased slowly from zero, the spectrum of avalanche sizesthat occurs appears to be quite different from the dissipative limit. Afterstopping from the moving phase, the restarting involves both fractal andbubble-like nucleation. Hysteresis loops can be understood in terms of adepletion layer caused by the stress overshoots, but surprisingly, in the limitof very large samples the hysteresis loops vanish. We argue that, althoughthere can be striking differences over a wide range of length scales, theuniversality class governing this pseudohysteresis is again that of thedissipative limit. Consequences of this picture for the statistics and dynamicsof earthquakes on geological faults are briefly discussed.Comment: 43 pages, 57 figures (yes, that's a five followed by a seven), revte

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