On a Lack of Stability of Parametrized BV Solutions to Rate-Independent Systems with Non-Convex Energies and Discontinuous Loads

TL;DR

This paper examines the instability of parametrized BV solutions in rate-independent systems with non-convex energies under discontinuous loads, revealing that common solutions are not stable under load convergence and proposing a new, but physically questionable, solution concept.

Contribution

It demonstrates the instability of existing solution concepts under weak* load convergence and introduces a new solution framework with stability but physical limitations.

Findings

Common solution concepts lack stability under weak* load convergence.

Counterexamples show limits of existing solution stability.

A new stable solution concept admits physically meaningless solutions.

Abstract

We consider a rate-independent system with nonconvex energy under discontinuous external loading. The underlying space is finite dimensional and the loads are functions in $BV([0,T];\mathbb{R}^d)$. We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly$*$ in $BV([0,T];\mathbb{R}^d)$ with a particular emphasis on the so-called normalized, $\mathfrak{p}$-parametrized balanced viscosity solutions. By means of two counterexamples, it is shown that common solution concepts are not stable w.r.t. weak$*$ convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows "solutions" that are physically meaningless.