A history-dependent elastic contact problem

Abstract

We consider a mathematical model which describes the equilibrium of an elastic body in contact with a rigid-plastic foundation. The contact is frictionless and the memory effects of the foundations are taken into account. The weak formulation of the model is in a form of a history-dependent variational inequality for the displacement field which, under appropriate assumptions on the data, has a unique solution. Here, we are interested on the continuous dependence of the solution with respect to various data and parameters. To this end, we state and prove an abstract convergence criterion for a class of history-dependent variational inequalities in Hilbert spaces. We apply this criterion in the study of our contact problem and derive two convergence results that we present together with their mechanical interpretation. Finally, we use a finite element scheme to approximate the problem, implement it on the computer, and provide numerical simulations which validate the theoretical convergence results.