# Penalization of history-dependent variational inequalities

## Abstract

The present paper represents a continuation of Sofonea and Matei’s paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math22, 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact.

In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material’s behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations.

We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

## Authors

Mircea Sofonea
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

history-dependent operator, variational inequality, penalization, viscoelastic material, frictionless contact, normal compliance, unilateral constraint, weak solution

## City this paper as

M. Sofonea, F. Pătrulescu, Penalization of history-dependent variational inequalities, European J. Appl. Math., vol. 25, no. 2 (2014), pp. 155-176
DOI: 10.1017/S0956792513000363

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##### Publisher Name

Cambridge University Press, Cambridge

0956-7925

1469-4425/e

3177272

1297.49016