Analysis of a quasi-variational contact problem arising in thermoelasticity

TL;DR

This paper develops and analyzes mathematical models of a thermoforming process involving a membrane and mould, focusing on contact problems governed by quasi-variational inequalities, with results on existence, regularity, and uniqueness of solutions.

Contribution

It introduces two coupled models for thermoforming contact problems, proving existence and regularity results, and extends these to evolutionary cases with uniqueness under certain conditions.

Findings

Existence of weak solutions for stationary models.

Regularity and uniqueness conditions for solutions.

Existence results for evolutionary (quasistatic) models.

Abstract

We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.