Wave-modulated orbits in rate-and-state friction

A frictional spring-block system has been widely used historically as a model to display some of the features of two slabs in sliding frictional contact. Putelat et al. (2008) demonstrated that equations governing the sliding of two slabs could be approximated by spring-block equations, and studied relaxation oscillations for two slabs driven by uniform relative motion at their outer surfaces, employing this approximation. The present work revisits this problem. The equations of motion are first formulated exactly, with full allowance for wave reflections. Since the sliding is restricted to be independent of position on the interface, this leads to a set of differential-difference equations in the time domain. Formal but systematic asymptotic expansions reduce the equations to differential equations. Truncation of the differential system at the lowest non-trivial order reproduces a classical spring-block system, but with a slightly different "equivalent mass" than was obtained in the earlier work. Retention of the next term gives a new system, of higher order, that contains also some explicit effects of wave reflections. The smooth periodic orbits that result from the spring-block system in the regime of instability of steady sliding are "decorated" by an oscillation whose period is related to the travel time of the waves across the slabs. The approximating differential system reproduces this effect with reasonable accuracy when the mean sliding velocity is not too far from the critical velocity for the steady state. The differential system also displays a period-doubling bifurcation as the mean sliding velocity is increased, corresponding to similar behaviour of the exact differential-difference system.