Existence results for a contact problem with varying friction coefficient and nonlinear forces

We consider the rate-independent problem of a particle moving in a three dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed in [MTL02, MT04]. We generalize the time-incremental minimization procedure of [MR06] for the present situation of a non-associative flow rule.