Tykhonov well-posedness of a viscoplastic contact problem<sup>†</sup>

Abstract

We consider an initial and boundary value problem \begin{document}$ {{\mathcal{P}}} $\end{document} which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is \begin{document}$ \mathbb{R}_+ = [0,+\infty) $\end{document} . We list the assumptions on the data and derive a variational formulation \begin{document}${{\mathcal{P}}}_V $\end{document} of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple \begin{document}$ {{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}}) $\end{document} where \begin{document}$ I $\end{document} is set of parameters, \begin{document}$ \Omega $\end{document} represents a family of approximating sets and \begin{document}${{\mathcal{C} }} $\end{document} is a set of sequences, then we define the well-posedness of Problem \begin{document}${{\mathcal{P}}}_V $\end{document} with respect to \begin{document}$ {{\mathcal{T}}}$\end{document} . Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of \begin{document}$ {{\mathcal{P} }}_V $\end{document} with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem \begin{document}$ {{\mathcal{P}}} $\end{document} can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.