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Global Existence Results for Viscoplasticity at Finite Strain

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Abstract

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate, and thus depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance and energy-dissipation-inequality solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Alexander Mielke

  3. DIMI, Università di Brescia, Via Valotti 9, 25133, Brescia, Italy

    Riccarda Rossi

  4. Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata, 27100, Pavia, Italy

    Giuseppe Savaré

  5. Institute for Applied Mathematics and Information Technologies “Enrico Magenes” (IMATI-CNR), Via Ferrata, 27100, Pavia, Italy

    Giuseppe Savaré

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  1. Alexander Mielke
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  2. Riccarda Rossi
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  3. Giuseppe Savaré
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Correspondence to Alexander Mielke.

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Communicated by C. Dafermos

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Mielke, A., Rossi, R. & Savaré, G. Global Existence Results for Viscoplasticity at Finite Strain. Arch Rational Mech Anal 227, 423–475 (2018). https://doi.org/10.1007/s00205-017-1164-6

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  • DOI: https://doi.org/10.1007/s00205-017-1164-6

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