Viscoelastic Problems with Long Memory

For the problems in this chapter, we assume that the behavior of the materials is described by a viscoelastic constitutive law with long memory. We consider both static and quasistatic antiplane problems in which the friction conditions are either the Tresca law or its regularization. For each model, we derive a variational formulation that is in the form of an elliptic or evolutionary variational inequality with a Volterra integral term for the displacement .eld. Then, by using the abstract results in Chapter 6, we derive existence, uniqueness, and convergence results for the weak solutions of the corresponding antiplane frictional contact problems. In particular, we study the behavior of the solutions as the relaxation coe.cient converges to zero and prove that they converge to the solution of the corresponding purely elastic problems. Again, everywhere in this chapter, we use the space V (page 152), endowed with its inner product (·,·)V and the associated norm $||\cdot||\ v$, 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 speci.cation.