Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 36130, 22 pages

Viscoelastic frictionless contact problems with adhesion

Mohamed Selmani1 and Mircea Sofonea2

1Département de Mathématiques, Université Ferhat Abbas, Route de Scipion, Sétif 19 000, Algeria
2Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, Perpignan 66 860, France

Received 16 December 2005; Revised 8 March 2006; Accepted 9 March 2006

Copyright © 2006 Mohamed Selmani and Mircea Sofonea. 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 two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.