History-Dependent Inequalities for Contact Problems with Locking Materials

摘要

We consider a class of history-dependent variational-hemivariational inequalities with constraints. Besides the unique solvability of the inequalities, we study the behavior of the solution with respect to the set of constraints and prove a continuous dependence result. The proof is based on various estimates, monotonicity arguments and the properties of the Clarke subdifferential. Then, we consider a mathematical model which describes the equilibrium of a locking material with memory, in contact with an obstacle. We comment the model and state its weak formulation, which is in a form of a history-dependent variational-hemivariational inequality for the displacement field. We prove the unique weak solvability of the model, then we use our abstract result to prove the continuous dependence of the solution with respect to the set of constraints. We apply this convergence result in the study of an optimization problem associated to the contact model.