Journal of Applied Mathematics
Volume 2 (2002), Issue 1, Pages 1-21
doi:10.1155/S1110757X02000219
A frictionless contact problem for viscoelastic materials
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
Copyright © 2002 Mikäel Barboteu et al. 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.
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.