Articles

A VARIATIONAL-HEMIVARIATIONAL INEQUALITY IN CONTACT PROBLEM FOR LOCKING MATERIALS AND NONMONOTONE SLIP DEPENDENT FRICTION

  • Stanis ł ,
  • aw MIGÓ ,
  • RSKI ,
  • Justyna OGORZA LY
Expand
  • 1. Chair of Optimization and Control, Jagiellonian University in Kraków, ul. ?ojasiewicza 6, 30348 Kraków, Poland;
    2. Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. ?ojasiewicza 6, 30348 Kraków, Poland

Received date: 2016-09-14

  Revised date: 2017-06-13

  Online published: 2017-12-25

Supported by

Research supported by the National Science Center of Poland under the Maestro 3 Project No. DEC-2012/06/A/ST1/00262, and the project Polonium "Mathematical and Numerical Analysis for Contact Problems with Friction" 2014/15 between the Jagiellonian University and Université de Perpignan Via Domitia.

Abstract

We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modelled by the nonmonotone multivalued subdifferential condition which depends on the slip. The problem is governed by a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set which describes the locking constraints and a nonconvex locally Lipschitz friction potential. The result on existence and uniqueness of solution to the inequality is shown. The proof is based on a surjectivity result for maximal monotone and pseudomonotone operators combined with the application of the Banach contraction principle.

Cite this article

Stanis ł , aw MIGÓ , RSKI , Justyna OGORZA LY . A VARIATIONAL-HEMIVARIATIONAL INEQUALITY IN CONTACT PROBLEM FOR LOCKING MATERIALS AND NONMONOTONE SLIP DEPENDENT FRICTION[J]. Acta mathematica scientia, Series B, 2017 , 37(6) : 1639 -1652 . DOI: 10.1016/S0252-9602(17)30097-8

References

[1] Clarke F H. Optimization and Nonsmooth Analysis. New York:Wiley, Interscience, 1983[2] Demengel F, Suquet P. On locking materials. Acta Appl Math, 1986, 6:185-211[3] Denkowski Z, Migórski S, Papageorgiou N S. An Introduction to Nonlinear Analysis:Theory. Boston, Dordrecht, London, New York:Kluwer Academic/Plenum Publishers, 2003[4] Denkowski Z, Migórski S, Papageorgiou N S. An Introduction to Nonlinear Analysis:Applications. Boston, Dordrecht, London, New York:Kluwer Academic/Plenum Publishers, 2003[5] Eck C, Jarušek J, Krbeč M. Unilateral Contact Problems:Variational Methods and Existence Theorems. Pure and Applied Mathematics 270. New York:Chapman/CRC Press, 2005[6] Haines D W, Wilson W D. Strain-energy density function for rubber-like materials. J Mech Phys Solids, 1979, 27:345-360[7] Han W, Migórski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal, 2014, 46:3891-3912[8] Han W, Sofonea M. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics 30. Providence, RI:Amer Math Soc, Somerville, MA:International Press, 2002[9] Maier G. A quadratic programming approach for certain classes of nonlinear structural problems. Meccanica, 1968, 3:121-130[10] Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics 26. New York:Springer, 2013[11] Migórski S, Ochal A, Sofonea M. History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal Real World Appl, 2015, 22:604-618[12] Migórski S, Ochal A, Sofonea M. A class of variational-hemivariational inequalities in reflexive Banach spaces. J Elasticity, 2017, 127:151-178[13] Naniewicz Z, Panagiotopoulos P D. Mathematical Theory of Hemivariational Inequalities and Applications. New York, Basel, Hong Kong:Marcel Dekker, Inc, 1995[14] Panagiotopoulos P D. Inequality Problems in Mechanics and Applications. Boston:Birkhäuser, 1985[15] Panagiotopoulos P D. Hemivariational Inequalities, Applications in Mechanics and Engineering. Berlin:Springer-Verlag, 1993[16] Prager W. On ideal-locking materials. Trans Soc Rheol, 1957, 1:169-175[17] Prager W. Elastic solids of limited compressibility//Proceedings of the 9th International Congress of Applied Mechanics, vol 5. Brussels, 1958:205-211[18] Prager W. On elastic, perfectly locking materials//Görtler H, ed. Applied Mechanics, Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964. Berlin, Heidelberg:Springer-Verlag, 1966:538-544[19] Shillor M, Sofonea M, Telega J J. Models and Analysis of Quasistatic Contact. Lect Notes Phys 655. Berlin, Heidelberg:Springer, 2004[20] Sofonea M, Matei A. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series 398. Cambridge University Press, 2012
Outlines

/