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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 1562509, 15 pages
Research Article

Analysis of a Dynamic Viscoelastic Contact Problem with Normal Compliance, Normal Damped Response, and Nonmonotone Slip Rate Dependent Friction

Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 avenue Paul Alduy, 66860 Perpignan, France

Received 5 May 2016; Accepted 26 July 2016

Academic Editor: André Nicolet

Copyright © 2016 Mikaël Barboteu and David Danan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.