Duality arguments in the analysis of a viscoelastic contact problem

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.

Automated summary

This article considers a mathematical model that describes the quasistatic frictionless contact between a viscoelastic body and a rigid-plastic foundation. It discusses the mechanical assumptions and hypotheses on the data, and provides three different variational formulations with different unknowns. Furthermore, it proves that these formulations are dual to each other and deduces the unique weak solvability of the contact problem and 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, numerical simulations are presented along with corresponding mechanical interpretations to study the contact problem.