Mathematical modelling of piezoelectric elastic materials

We consider a quasistatic contact modelled with the regularised friction law for electro-elastic and the foundation is assumed to be electrically conductive. This regularisation is obtained by replacing the function j(.) by the function jρ(.), where ρ is a strictly positive parameter. The classical formulation for the antiplane problem is formulated as a time dependent of corresponding variational formulation and is solved by the Banach fixed-point theorem and classical results for variational inequalities. We provide a weak formulation of the contact problem in the form variational system in which the unknowns are the displacement and the stress fields, then we establish the existence of a unique weak solution to the model. Finally, we have given a convergence criterion of the solution as the parameter of regularisation ρ converges to zero.