Instabilities in the antiplane problem with slip dependent friction

Abstract
Quasi-static loading process and dynamic process modelling the interaction with slip displacement friction between an elastic body and a rigid body are considered. In these processes, stick-slip motions are related to the earthquake instabilities. On the contact interface we use the friction Coulomb law with a slip dependent friction coefficient in the case of a prescribed normal pressure. In quasi-static loading, the qualitative behaviour of the solution is decided by the competition between two parameters which involve the geometry, the normal stress, the elasticity properties and the dependence on the slip displacement of the friction coefficient. We explain how slow loads generate time discontinuities and space nonhomogeneities on the contact interface at a large time scale. In the dynamic process we provide a linear stability analysis: rapidly after the initial perturbation, the dominant exponential part governs the time and space evolution of the slip during the initiation phase. For large times, the slip velocity behaves qualitatively as it would be in the case of a propagating crack. Between these two phases a transition phase exists caracterised by an extremely high apparent velocity (supersonic) of the rupture front.