In this chapter we illustrate the use of the abstract results obtained in Chapters 2 and 3 in the study of three frictionless or frictional contact problems with piezoelectric bodies. We model the material's behavior with an electro-elastic, an electro-viscoelastic and an electro-viscoplastic constitutive law, respectively. The contact is either bilateral or modelled with the normal compliance condition, with or without unilateral constraint. The friction is modelled with versions of Coulomb's law. The foundation is assumed to be either an insulator or electrically conductive. For each problem we provide a variational formulation which is in the form of a nonlinear system in which the unknowns are the displacement field and the electric potential field. Then we use the abstract existence and uniqueness results presented in Chapters 2 and 3 to prove the unique weak solvability of the corresponding contact problems. For the electro-elastic problem we also provide a dual variational formulation in terms of the stress and electric displacement fields. Everywhere in this chapter we consider the physical setting and the notation presented in Section 4.5, as well as the function spaces introduced in Section 4.1.

An Electro-elastic frictional contact problem

In this section we consider a frictional contact problem for electro-elastic materials. The problem is static and, therefore, we investigate it by using the arguments of elliptic variational inequalities presented in Section 2.2.

Problem statement

We assume that the body is electro-elastic and the foundation is an insulator.