Analysis of Rothe Method for a Variational-Hemivariational Inequality in Adhesive Contact Problem for Locking Materials

Abstract

We study a system of differential variational–hemivariational inequality arising in the modelling of adhesive viscoelastic contact problems for locking materials. The system consists of a variational-hemivariational inequality for the displacement field and an ordinary differential equation for the adhesion field. The contact is described by the unilateral constraint and normal compliance contact condition in which adhesion is taken into account and the friction is modelled by the nonmonotone multivalued subdifferential condition with adhesion. The problem is governed by a linear viscoelastic operator, a nonconvex locally Lipschitz friction potential and the subdifferential of the indicator function of a convex set which describes the locking constraints. The existence and uniqueness of solution to the coupled system are proved. The proof is based on a time-discretization method, known as the Rothe method.