2011, Mielke, Alexander, Müller, Ingo

We consider the passage from viscous system to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R*) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

2010, Mielke, Alexander, Truskinovsky, Lev

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.