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Journal of Convex Analysis 13 (2006), No. 1, 151--167
Copyright Heldermann Verlag 2006

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

Messoud A. Efendiev
IBB, Forschungszentrum Umwelt Gesundheit, 85758 Neuherberg, Germany

Alexander Mielke
Present address: WIAS Berlin, Mohrenstr. 39, 10117 Berlin, Germany
Permanent Address: Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

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.

Keywords: Rate independence, differential inclusion, nonconvex energy, arc-length parametrization, dry friction, viscous slip.

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