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Penalization of history-dependent variational inequalities

  • M. SOFONEA (a1) and F. PĂTRULESCU (a2)

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.

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[2] H. Brézis (1968) Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18, 115175.

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[16] S. Migórski , A. Ochal & M. Sofonea (2013) Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, NY.

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[23] M. Sofonea & F. Pătrulescu (2013) Analysis of a history-dependent frictionless contact problem. Math. Mech. Solids 18, 409430.

[24] P. Wriggers & U. Nackenhorst (editors) (2006) Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics 27, Springer, Berlin, Germany.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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