Analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation. The material’s behavior is modeled with a constitutive law with long memory. The contact is frictionless and is modeled with a multivalued normal compliance condition and unilateral constraint. We present the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. Then we prove its unique solvability. The proof is based on arguments of history-dependent quasivariational inequalities. We also study the dependence of the solution with respect to the data and prove a convergence result. Further, we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate. Finally, we provide numerical validations both for the convergence and the error estimate results, in the study of a two-dimensional test problem.