ANALYSIS OF A FRICTIONAL CONTACT PROBLEM FOR VISCOELASTIC MATERIALS WITH LONG MEMORY

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is time-dependent, the material behavior is described with a viscoelastic constitutive law with long memory and the contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of a hemivariational inequality with Volterra integral term for the displacement field. Then we prove existence and uniqueness results in the study of abstract inclusions as well as in the study of abstract hemivariational inequalities with Volterra integral term. The proofs are based on arguments on pseudomonotone operators, compactness and fixed point. We use the abstract results to prove the unique solvability of the frictional contact problem. Finally, we present examples of contact and frictional boundary conditions for which our results work.