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Carnegie Mellon Center for Nonlinear Analysis


Vanishing-Viscosity Solutions for Rate-Independent Energetic System

Alexander Mielke
Humboldt-Universität Institut für Mathematik

Abstract: Rate-independent systems can be formulated in terms of a state space, a time-dependent energy functional, and a dissipation potential R. Applications occur in dry friction, elastoplasticity, magnetism, and phase-transforming systems like shape-memory alloys. The unique feature is that R is homogeneous of degree 1 in the velocity, while it is quadratic in viscous cases. This makes the relation between the velocity and the dissipation force homogeneous of degree 0 and hence discontinuous.

Since rate-independence occurs as a limit for systems under very slow loading, solutions may develop jumps when forced to leave a potential well. Such jumps may not be described correctly by the so-called energetic solutions, which were developed within the last decade and which allow for a general existence theory. To analyze the correct jump behavior, we propose to study the `limit of vanishing viscosity', which leads to new types of solutions for rate-independent systems, namely `parametrized solutions' and `BV solutions'. These solutions can switch between three distinct regimes: (i) sticking (no motion, since the forces are below the activation threshold), (ii) rate-independent sliding (the forces are at the activation threshold), and (iii) jump paths (the forces are above the activation threshold).

We discuss convergence results for the vanishing-viscosity limit as well as for suitable incremental minimization problems, where viscosity and the time-steps tend to 0 simultaneously.