On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity

Rate-independent systems with nonconvex energies generate solutions with jumps. To resolve the full jump path we consider two different regularizations, namely (i) small viscosity and (ii) local minimization in the time discretized setting. After rescaling the solutions via arc-length parametrization we obtain a new limit problem, which is again rate independent. We establish convergence results for the viscously regularized solutions as well as for the time-discretized solutions. In general the limit function is no longer parametrized by arc length; however, another reparametrization leads to a solution. Using a Young-measure argument, we show that the latter reparamterization is not necessary if the dry-friction potential and the viscous potential satisfy a structural compatibility condition.