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Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

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Central European Journal of Mathematics

Abstract

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

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References

  1. Clarke F.H., Optimization and Nonsmooth Analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1983

    Google Scholar 

  2. Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston, 2003

    Google Scholar 

  3. Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Applications, Kluwer, Boston, 2003

    Google Scholar 

  4. Duvaut G., Lions J.-L., Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., 219, Springer, Berlin-New York, 1976

    Google Scholar 

  5. Eck C., Jarušek J., Krbec M., Unilateral Contact Problems, Pure Appl. Math. (Boca Raton), 270, Chapman Hall/CRC, Boca Raton, 2005

    Google Scholar 

  6. Han W., Sofonea M., Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Stud. Adv. Math., 30, American Mathematical Society, Providence, 2002

    Google Scholar 

  7. Jarušek J., Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., 1996, 46(121)(3), 475–487

    MATH  MathSciNet  Google Scholar 

  8. Jarušek J., Eck C., Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Math. Models Methods Appl. Sci., 1999, 9(1), 11–34

    Article  MATH  MathSciNet  Google Scholar 

  9. Migórski S., Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 2005, 84(7), 669–699

    Article  MATH  MathSciNet  Google Scholar 

  10. Migórski S., Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Comput. Math. Appl., 2006, 52(5), 677–698

    Article  MATH  MathSciNet  Google Scholar 

  11. Migórski S., Ochal A., Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Anal., 2005, 61(1–2), 135–161

    Article  MATH  MathSciNet  Google Scholar 

  12. Migórski S., Ochal A., A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 2006, 83(3), 247–275

    Article  MATH  MathSciNet  Google Scholar 

  13. Migórski S., Ochal A., Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 2009, 41(4), 1415–1435

    Article  MATH  MathSciNet  Google Scholar 

  14. Migórski S., Ochal A., Sofonea M., History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 2011, 12(6), 3384–3396

    Article  MATH  MathSciNet  Google Scholar 

  15. Migórski S., Ochal A., Sofonea M., Nonlinear Inclusions and Hemivariational Inequalities, Adv. Mech. Math., 26, Springer, New York, 2012

    Google Scholar 

  16. Naniewicz Z., Panagiotopoulos P.D., Mathematical Theory of Hemivariational Inequalities and Applications, Monogr. Textbooks Pure Appl. Math., 188, Marcel Dekker, New York, 1995

    Google Scholar 

  17. Panagiotopoulos P.D., Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985

    Book  MATH  Google Scholar 

  18. Panagiotopoulos P.D., Hemivariational Inequalities, Springer, Berlin, 1993

    Book  MATH  Google Scholar 

  19. Shillor M., Sofonea M., Telega J.J., Models and Analysis of Quasistatic Contact, Lecture Notes in Phys., 655, Springer, Berlin, 2004

    Book  MATH  Google Scholar 

  20. Sofonea M., Rodríguez-Arós A., Viaño J.M., A class of integro-differential variational inequalities with applications to viscoelastic contact, Math. Comput. Modelling, 2005, 41(11–12), 1355–1369

    Article  MATH  MathSciNet  Google Scholar 

  21. Zeidler E., Nonlinear Functional Analysis and its Applications, II/B, Springer, New York, 1990

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    Stanisław Migórski

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  1. Stanisław Migórski
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Correspondence to Stanisław Migórski.

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Migórski, S. Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems. centr.eur.j.math. 10, 1953–1968 (2012). https://doi.org/10.2478/s11533-012-0123-6

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  • DOI: https://doi.org/10.2478/s11533-012-0123-6

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