Modelling and analysis of a viscoelastic contact problem with unilateral constraints

We consider a mathematical model which describes the equilibrium of two viscoelastic membranes situated in parallel plans, submitted to the action of external forces. As a result, the membranes could arrive in contact with two obstacles and could arrive in contact each other, too. The contact is frictionless and is described both with normal compliance and the Signorini unilateral contact condition. We list the assumption on the data, then we derive a variational formulation of the model which is in a form of a history-dependent variational inequality for the displacement field. We prove the unique weak solvability of the contact model, then we consider an associated optimization problem, for which we prove the existence of the solution. We also provide some conclusions and list related problems for further research.