Duality Arguments in the Analysis of a Viscoelastic Contact Problem

TL;DR

This paper develops and analyzes dual variational formulations for a quasistatic viscoelastic contact problem, proving their equivalence, existence, uniqueness, and stability of solutions, supported by numerical simulations.

Contribution

It introduces three dual variational formulations for the viscoelastic contact problem and proves their equivalence and well-posedness, advancing mathematical understanding of such models.

Findings

Proved pairwise duality of three variational formulations.

Established unique weak solvability and Lipschitz stability.

Provided numerical simulations with mechanical interpretations.

Abstract

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.