Analysis of a contact problem for a viscoelastic Bresse system

¹ LANA, Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS, Brazil
² Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Debbieh, Lebanon
³ Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
⁴ Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy
⁵ Department of Mathematics, Faculty of Sciences 1, Lebanese University, Hadath, Lebanon

^* Corresponding author: mimcopetti@ufsm.br

Received: 2 April 2020
Accepted: 22 March 2021

Abstract

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an adequate Liapunov function. Then, using the finite element method and the implicit Euler scheme, a finite element approximation is introduced. A discrete stability property and a priori error estimates are proved. Finally, some numerical experiments are performed to demonstrate the decay of the discrete energy and the numerical convergence.