Global existence for a highly nonlinear temperature-dependent system modeling nonlocal adhesive contact

TL;DR

This paper establishes the global existence of solutions for a complex, highly nonlinear temperature-dependent PDE system modeling nonlocal adhesive contact with damage and heat flux effects.

Contribution

It introduces a novel analytical framework for proving global existence of solutions to a highly nonlinear, nonlocal PDE system in adhesive contact modeling.

Findings

Proved global-in-time existence of solutions.

Developed a careful time discretization and variational approach.

Addressed positivity of temperature in the model.

Abstract

In this paper we analyze a new temperature-dependent model for adhesive contact that encompasses nonlocal adhesive forces and damage effects, as well as nonlocal heat flux contributions on the contact surface. The related PDE system combines heat equations, in the bulk domain and on the contact surface, with mechanical force balances, including micro-forces, that result in the equation for the displacements and in the flow rule for the damage-type internal variable describing the state of the adhesive bonds. Nonlocal effects are accounted for by terms featuring integral operators on the contact surface. The analysis of this system poses several difficulties due to its overall highly nonlinear character, and in particular to the presence of quadratic terms, in the rates of the strain tensor and of the internal variable, that feature in the bulk and surface heat equations. Another major challenge is related to proving strict positivity for the bulk and surface temperatures. We tackle these issues by very careful estimates that enable us to prove the existence of global-in-time solutions and could be useful in other contexts. All calculations are rigorously rendered on an accurately devised time discretization scheme in which the limit passage is carried out via variational techniques.