A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem

TL;DR

This paper introduces a new class of nonlinear systems combining differential equations and variational inequalities, and applies it to model complex quasistatic contact problems in viscoelastic materials with long-term effects.

Contribution

It develops a novel mathematical framework for these systems and proves existence and uniqueness of solutions, applying it to real-world viscoelastic contact problems.

Findings

Proved existence and uniqueness of solutions for the new system.

Applied the theory to solve a complex viscoelastic contact problem.

Established conditions for solvability in Banach spaces.

Abstract

The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality, and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.