Well-posedness of Constrained Evolutionary Differential Variational-Hemivariational Inequalities

TL;DR

This paper establishes the well-posedness of a complex system combining evolutionary variational-hemivariational inequalities with differential equations, with applications in contact mechanics demonstrating the theoretical results.

Contribution

It proves existence, uniqueness, and continuous dependence for a coupled system of inequalities and differential equations using a fixed point approach, with practical applications in contact mechanics.

Findings

Proved well-posedness of the coupled system.

Demonstrated solution regularity and dependence on data.

Applied results to contact mechanics problems.

Abstract

A system of a first order history-dependent evolutionary variational-hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.