From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results
- Mielke, Alexander
- Truskinovsky, Lev
2010 Mathematics Subject Classification
- 74C15 74N30 74D10 70K70
- Snap-spring potential, hysteresis, Gamma convergence for evolution, rate-independent plasticity, viscous gradient flow, wiggly energy
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.
- Arch. Ration. Mech. Anal., 203 (2012) pp. 577--619.