A new class of history-dependent quasi variational-hemivariational inequalities with constraints

TL;DR

This paper introduces a new class of time-dependent quasi variational-hemivariational inequalities with history-dependent operators and constraints, establishing existence and uniqueness of solutions, and applying these results to a viscoelastic contact problem.

Contribution

It develops a novel theoretical framework for history-dependent inequalities and demonstrates their application to complex contact mechanics problems.

Findings

Proved existence and uniqueness of solutions for the new inequalities.

Applied the abstract results to a quasistatic viscoelastic contact problem.

Extended variational-hemivariational inequality theory to include history dependence.

Abstract

In this paper we consider an abstract class of time-dependent quasi variational-hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational-hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.