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

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.