WIAS Preprint No. 1541, (2010)

From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Truskinovsky, Lev

2010 Mathematics Subject Classification

  • 74C15 74N30 74D10 70K70

Keywords

  • Snap-spring potential, hysteresis, Gamma convergence for evolution, rate-independent plasticity, viscous gradient flow, wiggly energy

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 solid transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shape-memory alloys. The approach, however, is sufficiently general and can be used for similar reductions in the cases of more general plasticity and damage models.

Appeared in

  • Arch. Ration. Mech. Anal., 203 (2012) pp. 577--619.

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