BV SOLUTIONS AND VISCOSITY APPROXIMATIONS OF RATE-INDEPENDENT SYSTEMS ∗

By Alexander Mielke, Riccarda Rossi, Giuseppe Savar'e. 2009

149 citations recorded by the source corpus.

In the nonconvex case, solutions of rate-independent systems may develop jumps as a func- tion of time. To model such jumps, we adopt the philosophy that rate-independence should be consid- ered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dis- sipation potential. The resulting definition of "BV solutions" involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems. Mathematics Subject Classification. 49Q20, 58E99.

Subfields: Control and Systems Engineering, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/062472a7668c8f4dc5d247e208aacc14c264cc18

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In Control and Systems Engineering

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Citations in the corpus

Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling (2011, methodology): “space. This idea has already been used in [11,20, 21 ] for rate independent dissipative problems in finite dimension.”
Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems (2013, methodology): “n Ψ depends on the state of the system (as in [MRS13]), or it is replaced by a distance on D (as in [RMS08, MRS09]) and when the viscous correction Φ is a general convex superlinear functional (as in [MRS12a]). We have chosen the current simpler structure to focus on the main features and techniques”