Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem

TL;DR

This paper investigates a complex dynamic contact problem involving a viscoelastic-viscoplastic piezoelectric material, providing a mathematical formulation, proving existence and uniqueness, and demonstrating numerical methods with convergence analysis and simulations.

Contribution

It introduces a comprehensive variational and numerical framework for a nonlinear contact problem in piezoelectric materials, including error estimates and simulations.

Findings

Proved existence and uniqueness of the solution

Developed a convergent finite element and Euler scheme

Validated the approach with numerical simulations

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.