Two-speed solutions to non-convex rate-independent systems

Abstract

We consider evolutionary PDE inclusions of the formwhere R 1 is a positively 1-homogeneous rate-independent dissipation potential and W 0 is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit λ ↓ 0. Contrary to existing approaches, our solutions have high regularity in space and time away from jumps. On the "slow" time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we resolve the jump transients at a "fast" time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This in particular yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent / rate-independent way. Our results are illustrated by examples, which highlight the effects that can occur.MSC (2010): 49J40 (PRIMARY); 47J20, 47J40, 74H30.

Separately Global Solutions to Rate-Independent Processes in Large-Strain Inelasticity

“…The list of possible notions of solutions presented here is by no means exhaustive and the reader is encouraged to consult for example [36,38,46,47] for a more detailed survey of both the mentioned and unmentioned solution strategies for RIS, as well as for recent developments.…”