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 friction law, showing steady-slate velocity weakening, and following the ageing (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 methods are used in this work. The stability results have an extremely simple form: (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 become unbounded) when K < K-cr. This is unlike results based on the slip version of the state evolution law, in which instability occurs in response to sufficiently large perturbations from steady sliding when K > K-cr. An implication of this result for slip instabilities in continuum systems is that a critical nucleation size of coherent slip has to be attained before unstable slip can ensue. (2) When the load point is stationary, the system stably evolves towards slip at a monotonically decreasing rate whenever K greater than or equal to K-cr > 0. However, when K < K-cr, initial conditions leading to stable and unstable slip motion exist. Hence self-driven creep modes of instability exist, but only in the latter case. (C) 1999 Elsevier Science Ltd. All rights reserved.