Long-time behaviour of a thermomechanical model for adhesive contact

TL;DR

This paper analyzes the long-term behavior of a complex PDE system modeling adhesive contact with thermal effects, proving existence of solutions over infinite time and convergence to stationary states.

Contribution

It introduces a novel analysis of a highly nonlinear thermomechanical PDE system with internal constraints and surface coupling, establishing existence and asymptotic behavior results.

Findings

Solutions exist for all time

Solutions tend to stationary states

Dissipation vanishes asymptotically

Abstract

This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Fremond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.