Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Messoud A. Efendiev

Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, 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

mielke@wias-berlin.de

Keywords

  • Rate independence
  • differential inclusion
  • nonconvex energy
  • arc-length parametrization
  • dry friction
  • viscous slip

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Suggested citation

M. A. Efendiev, A. Mielke. “On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity.” Journal of Convex Analysis 13 (2006), No. 1, 151–167. https://doi.org/10.68381/jca13010

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press.