Finite-time stability of any polyhedral sweeping process with uni-directional loading and application to elastoplastic models

Abstract

A significant progress in the study of the dynamics of elastoplastic lattice spring models has been recently due to the development of a sweeping process framework where the dynamics of the stresses of springs is linked to a solution of the associated sweeping process. By using the sweeping process framework, a recent result by Gudoshnikov et al. (2022) ensures finite-time convergence of a lattice spring model under assumption that the vector g′(t) of the applied displacement controlled-loading lies strictly inside a suitable normal cone of the sweeping process. In this paper we drop the latter assumption by proving that g′(t) always belongs to a suitable normal cone when g′(t)=const (i.e. the loading is uni-directional) and when certain genericity condition holds. By using this result we establish that every elastoplastic lattice spring model with a generic associated polyhedron reaches a unique distribution of plastic deformations in finite time.