Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem

¹ Centro Universitario de la Defensa, Escuela Naval Militar, Plaza de España s/n, 36920 Marín, Spain.
² Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain.
jose.fernandez@uvigo.es
³ Departamento de Métodos Matemáticos e Representación, Universidade de A Coruña, A Coruña, Spain.

Received: 11 January 2016
Accepted: 21 April 2016

Abstract

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.