Elastic Problems

In this chapter, we study antiplane frictional contact models for elastic materials, both in the static and quasistatic case. We start by considering static problems in which friction is described with the Tresca law and its regularizations; we derive a variational formulation for each model that is in the form of an elliptic variational inequality for the displacement .eld; then we prove existence, uniqueness, and convergence results for the weak solution. Next, we consider a static slip dependent frictional contact problem that leads to an elliptic quasivariational inequality and, again, we prove the existence and uniqueness of the weak solution. We then extend part of the above results in the study of quasistatic versions of these models, which lead to evolutionary variational or quasivariational inequalities. Everywhere in this chapter, we use the space V (page 152) together with its inner product (·,·)V and the associated norm $||\cdot||\ X$, and we denote by [0, T] the time interval of interest, T > 0. Also, everywhere in this chapter, the use of the abstract results presented in Part II of this manuscript is made in the case X = V , (·,·)X = (·,·)V , without explicit specication.