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Numerical analysis of history-dependent quasivariationalinequalities with applications in contact mechanics

Published online by Cambridge University Press:  24 April 2014

Kamran Kazmi ,
Mikael Barboteu ,
Weimin Han  and
Mircea Sofonea
Kamran Kazmi
Affiliation:
Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI 54901, USA. kazmis@uwosh.edu
Mikael Barboteu
Affiliation:
Laboratoire de Mathématiques et Physique, University of Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France; barboteu@univ-perp.fr; sofonea@univ-perp.fr
Weimin Han
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA; weimin-han@uiowa.edu
Mircea Sofonea
Affiliation:
Laboratoire de Mathématiques et Physique, University of Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France; barboteu@univ-perp.fr; sofonea@univ-perp.fr
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Abstract

A new class of history-dependent quasivariational inequalities was recently studied in[M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising incontact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491].Existence, uniqueness and regularity results were proved and used in the study of severalmathematical models which describe the contact between a deformable body and an obstacle.The aim of this paper is to provide numerical analysis of the quasivariationalinequalities introduced in the aforementioned paper. To this end we introduce temporallysemi-discrete and fully discrete schemes for the numerical approximation of theinequalities, show their unique solvability, and derive error estimates. We then applythese results to a quasistatic frictional contact problem in which the material’s behavioris modeled with a viscoelastic constitutive law, the contact is bilateral, and friction isdescribed with a slip-rate version of Coulomb’s law. We discuss implementation of thecorresponding fully-discrete scheme and present numerical simulation results on atwo-dimensional example.

Keywords

Quasivariational inequalitynumerical analysisfinite element methoderror estimatesquasistatic frictional contact problemviscoelastic constitutive lawCoulomb’s lawnumerical simulations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 919 - 942
Copyright
© EDP Sciences, SMAI, 2014

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