Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems

TL;DR

This paper establishes finite-time stabilization criteria for polyhedral sweeping processes and applies these results to elastoplastic systems, providing computational methods for practical verification.

Contribution

It introduces a new theorem on finite-time stabilization of polyhedral sweeping processes and applies it to elastoplastic systems with a practical computational guide.

Findings

Finite-time stabilization theorem for polyhedral sweeping processes.

Application of the theorem to elastoplastic spring systems.

A computational method for verifying stabilization conditions.

Abstract

We use the ideas of Adly-Attoych-Cabot [Adv. Mech. Math., 12, Springer, 2006] on finite-time stabilization of dry friction oscillators to establish a theorem on finite-time stabilization of differential inclusions with a moving polyhedral constraint (known as polyhedral sweeping processes) of the form $C+c(t).$ We then employ the ideas of Moreau [New variational techniques in mathematical physics, CIME, 1973] to apply our theorem to a system of elastoplastic springs with a displacement-controlled loading. We show that verifying the condition of the theorem ultimately leads to the following two problems: (i) identifying the active vertex ``A'' or the active face ``A'' of the polyhedron that the vector $c'(t)$ points at; (ii) computing the distance from $c'(t)$ to the normal cone to the polyhedron at ``A''. We provide a computational guide to implement steps (i)-(ii) in the case of an arbitrary elastoplastic system and apply the guide to a particular example. Due to the simplicity of the particular example, we can solve (i)-(ii) by the methods of linear algebra and minor combinatorics.