In this talk, two contact problems will be presented. The first one concerns an elastic body in contact with a rigid or deformable obstacle, and the second one deals with the contact of a viscoelastic body with a deformable foundation. In both cases, piezoelectric effects are taken into account. Piezoelectricity is the ability of certain cristals, like the quartz (also ceramics (BaTiO3, KNbO3, LiNbO3, etc) and even the human bones), to produce a voltage when they are subjected to mechanical stress. Therefore, there is a coupling between the mechanical and electrical properties of the material.
The existence of a unique weak solution is stated in both problems using fixed point arguments and classical results on nonlinear variational equations. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and, if needed, an Euler scheme to discretize the time derivatives. Error estimates are derived from which, under suitable regularity assumptions, the linear convergence of the algorithm is deduced. Finally, numerical simulations which demonstrate the behaviour and accuracy of the method are shown.