Journal of Applied Mathematics 
Volume 2 (2002), Issue 1, Pages 1-21
doi:10.1155/S1110757X02000219

A frictionless contact problem for viscoelastic materials

Mikäel Barboteu,1 Weimin Han,2 and Mircea Sofonea1

 1Laboratoire de Théorie des Systèmes, Université de Perpignan, 52 Avenue de Villeneuve, Perpignan 66860, France
2Department of Mathematics, University of Iowa, Iowa 52242, IA, USA
Received 12 March 2001; Revised 4 September 2001

Abstract

We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a viscoelastic behavior that we model with the Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the well-known Signorini condition in a form with a zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.