Existence results for variational-hemivariational inequality systems with nonlinear couplings

TL;DR

This paper establishes existence results for a fully nonlinear coupled system of variational and hemivariational inequalities in Banach spaces, with applications to contact mechanics involving multiple contact zones.

Contribution

It introduces a topological approach to prove existence for nonlinear coupled inequalities without linearity assumptions on the coupling functional.

Findings

Proved existence of solutions in both bounded and unbounded constraint sets.

Applied theoretical results to contact mechanics models with multivalued laws.

Demonstrated simultaneous determination of displacement and stress in contact zones.

Abstract

In this paper we investigate a system of coupled inequalities consisting of a variational-hemivariational inequality and a quasi-hemivariational inequality on Banach spaces. The approach is topological, and a wide variety of existence results is established for both bounded and unbounded constraint sets in real reflexive Banach spaces. The main point of interest is that no linearity condition is imposed on the coupling functional, therefore making the system fully nonlinear. Applications to Contact Mechanics are provided in the last section of the paper. More precisely, we consider a contact model with (possibly) multivalued constitutive law whose variational formulation leads to a coupled system of inequalities. The weak solvability of the problem is proved via employing the theoretical results obtained in the previous section. The novelty of our approach comes from the fact that we consider two potential contact zones and the variational formulation allows us to determine simultaneously the displacement field and the Cauchy stress tensor.