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search.filters.applied.f.access: openAccess × Author: Mielke, Alexander × Version: publishedVersion × Subject: quasistatic evolution × WGL Institute: WIAS ×

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    Modeling solutions with jumps for rate-independent systems on metric spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.
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    Rate-independent elastoplasticity at finite strains and its numerical approximation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Roubíc̆ek, Tomáš
    Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.