Analysis of a general dynamic history-dependent variational-hemivariational inequality

This paper is devoted to the study of a general dynamic variational-hemivariational inequality with history-dependent operators. These operators appear in a convex potential and in a locally Lipschitz superpotential. The existence and uniqueness of a solution to the inequality problem is explored through a result on a class of nonlinear evolutionary abstract inclusions involving a nonmonotone multivalued term described by the Clarke generalized gradient. The result presented in this paper is new and general. It can be applied to study various dynamic contact problems. As an illustrative example, we apply the theory on a dynamic frictional viscoelastic contact problem in which the contact is modeled by a nonmonotone Clarke subdifferential boundary condition and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the total slip.