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History-dependent quasi-variational inequalities arising in contact mechanics

Published online by Cambridge University Press:  19 May 2011

MIRCEA SOFONEA  and
ANDALUZIA MATEI
MIRCEA SOFONEA
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France email: sofonea@univ-perp.fr
ANDALUZIA MATEI
Affiliation:
Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 Craiova, Romania
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Abstract

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

Keywords

Quasi-variational inequalitiesExistence and uniquenessRegularityContact problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 5 , October 2011 , pp. 471 - 491
Copyright
Copyright © Cambridge University Press 2011

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