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From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results

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Abstract

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.

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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. Laboratoire de Mécanique des Solides, Ecole Polytechnique, Route de Saclay, 91128, Palaiseau, France

    Lev Truskinovsky

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  1. Alexander Mielke
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Mielke, A., Truskinovsky, L. From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results. Arch Rational Mech Anal 203, 577–619 (2012). https://doi.org/10.1007/s00205-011-0460-9

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