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Penalization of history-dependent variational inequalities

Published online by Cambridge University Press:  09 October 2013

M. SOFONEA  and
F. PĂTRULESCU
M. SOFONEA
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan, 52 Avenue de Paul Alduy, 66 860 Perpignan, France email: sofonea@univ-perp.fr
F. PĂTRULESCU
Affiliation:
Tiberiu Popoviciu Institute of Numerical Analysis, P.O. Box 68-1, 400110 Cluj-Napoca, Romania email: flaviusolimpiu@yahoo.com
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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. Math. 22, 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.

Keywords

History-dependent operatorVariational inequalityPenalizationViscoelastic materialFrictionless contactNormal complianceUnilateral constraintWeak solution
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 2 , April 2014 , pp. 155 - 176
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Baiocchi, C. & Capelo, A. (1984) Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester, UK.Google Scholar
[2]Brézis, H. (1968) Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18, 115–175.CrossRefGoogle Scholar
[3]Corduneanu, C. (1965) Problèmes globaux dans la théorie des équations intégrales de Volterra. Ann. Math. Pure Appl. 67, 349–363.Google Scholar
[4]Duvaut, G. & Lions, J. L. (1976) Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, Germany.Google Scholar
[5]Eck, C., Jarušek, J. & Krbeč, M. (2005) Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics 270, Chapman/CRC Press, New York, NY.Google Scholar
[6]Glowinski, R. (1984) Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY.Google Scholar
[7]Han, W. & Reddy, B. D. (1995) Computational plasticity: The variational basis and numerical analysis. Comput. Mech. Adv. 2, 283–400.Google Scholar
[8]Han, W. & Reddy, B. D. (1999) Plasticity: Mathematical Theory and Numerical Analysis, Springer-Verlag, New York, NY.Google Scholar
[9]Han, W. & Sofonea, M. (2002) Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, American Mathematical Society–International Press, Providence, RI.CrossRefGoogle Scholar
[10]Hlaváček, I., Haslinger, J., Necǎs, J. & Lovíšek, J. (1988) Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, NY.Google Scholar
[11]Kikuchi, N. & Oden, J. T. (1980) Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Eng. Sci. 18, 1173–1284.Google Scholar
[12]Kikuchi, N. & Oden, J. T. (1988) Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, PA.Google Scholar
[13]Kinderlehrer, D. & Stampacchia, G. (2000) An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics 31, SIAM, Philadelphia, PA.Google Scholar
[14]Martins, J. A. C. & Monteiro Marques, M. D. P. (editors) (2002) Contact Mechanics, Kluwer, Dordrecht, Netherlands.Google Scholar
[15]Massera, J. J. & Schäffer, J. J. (1966) Linear Differential Equations and Function Spaces, Academic Press, New York, NY.Google Scholar
[16]Migórski, S., Ochal, A. & Sofonea, M. (2013) Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, NY.Google Scholar
[17]Panagiotopoulos, P. D. (1985) Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, MA.Google Scholar
[18]Raous, M., Jean, M. & Moreau, J. J. (editors) (1995) Contact Mechanics, Plenum Press, New York, NY.Google Scholar
[19]Shillor, M. (editor) (1998) Recent Advances in Contact Mechanics, Special issue of Mathematical and Computer Modelling 28 (4–8).Google Scholar
[20]Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact. Variational Methods, Lecture Notes in Physics 655, Springer, Berlin, Germany.Google Scholar
[21]Sofonea, M. & Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491.Google Scholar
[22]Sofonea, M. & Matei, A. (2012) Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Notes, 398, Cambridge University Press, Cambridge, UK.Google Scholar
[23]Sofonea, M. & Pătrulescu, F. (2013) Analysis of a history-dependent frictionless contact problem. Math. Mech. Solids 18, 409–430.Google Scholar
[24]Wriggers, P. & Nackenhorst, U. (editors) (2006) Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics 27, Springer, Berlin, Germany.Google Scholar