Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction

The stability of quasi-static frictional slip of a single degree of freedom elastic system is studied for a Dieterich-Ruina rate and state dependent f riction law, showing steady-slate velocity weakening, and following the age ing (or slowness) version of the stale evolution law. Previous studies have been done for the slip version. Analytically determined phase plane trajectories and Liapunov function meth ods are used in this work. The stability results have an extremely simple f orm: (1) When a constant velocity is imposed at the load point, slip motion is always periodic when the elastic stiffness, K, has a critical value, K- cr. Slip is always stable when K > K-cr > 0, with rate approaching the load -point velocity, and unstable (slip rates within the quasi-static model bec ome unbounded) when K < K-cr. This is unlike results based on the slip vers ion of the state evolution law, in which instability occurs in response to sufficiently large perturbations from steady sliding when K > K-cr. An impl ication of this result for slip instabilities in continuum systems is that a critical nucleation size of coherent slip has to be attained before unsta ble slip can ensue. (2) When the load point is stationary, the system stabl y evolves towards slip at a monotonically decreasing rate whenever K greate r than or equal to K-cr > 0. However, when K < K-cr, initial conditions lea ding to stable and unstable slip motion exist. Hence self-driven creep mode s of instability exist, but only in the latter case. (C) 1999 Elsevier Scie nce Ltd. All rights reserved.