An implicit sweeping process approach to quasistatic evolution variational inequalities

TL;DR

This paper introduces a new implicit sweeping process approach with velocity constraints, establishing its well-posedness and equivalence to quasistatic evolution variational inequalities, with applications to mechanical contact problems.

Contribution

It develops a novel implicit sweeping process framework and proves its equivalence to quasistatic evolution variational inequalities, extending the mathematical tools for contact mechanics.

Findings

Proved well-posedness of the implicit sweeping process.

Established equivalence with quasistatic evolution variational inequalities.

Applied the framework to frictional contact problems in mechanics.

Abstract

In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with velocity constraint. The link between the implicit sweeping process and the quasistatic evolution variational inequality is possible thanks to some standard tools from convex analysis and is new in the literature.