  In this paper, we study, from both variational and numerical points of view,
a dynamic contact problem between a viscoelastic-viscoplastic piezoelectric
body and a deformable obstacle. The contact is modelled using the classical
normal compliance contact condition. The variational formulation is written as
a nonlinear ordinary differential equation for the stress field, a nonlinear
hyperbolic variational equation for the displacement field and a linear
variational equation for the electric potential field. An existence and
uniqueness result is proved using Gronwall's lemma, adequate auxiliary problems
and fixed-point arguments. Then, fully discrete approximations are introduced
using an Euler scheme and the finite element method, for which some a priori
error estimates are derived, leading to the linear convergence of the algorithm
under suitable additional regularity conditions. Finally, some two-dimensional
numerical simulations are presented to show the accuracy of the algorithm and
the behaviour of the solution.
