Well-posedness of evolutionary differential variational-hemivariational inequalities and applications to frictional contact mechanics

TL;DR

This paper proves the well-posedness of a class of evolutionary variational-hemivariational inequalities coupled with differential equations, and applies these results to model frictional contact in viscoelastic materials with geophysical relevance.

Contribution

It establishes existence, uniqueness, and continuity of solutions for a new class of inequalities and introduces two novel friction models inspired by earth sciences.

Findings

Unique mild solutions exist for the inequalities.

Flow map is continuous with respect to initial data.

Applications to frictional contact in viscoelastic materials.

Abstract

In this paper, we study the well-posedness of a class of evolutionary variational-hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modelling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient $\mu$ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable $\alpha$ and the slip rate $|\dot{u}_\tau|$.