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2012 | 10 | 6 | 1953-1968
Tytuł artykułu

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

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EN
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EN
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.
Twórcy
  • Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland, stanislaw.migorski@ii.uj.edu.pl
Bibliografia
  • [1] Clarke F.H., Optimization and Nonsmooth Analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1983
  • [2] Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston, 2003
  • [3] Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Applications, Kluwer, Boston, 2003
  • [4] Duvaut G., Lions J.-L., Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., 219, Springer, Berlin-New York, 1976
  • [5] Eck C., Jarušek J., Krbec M., Unilateral Contact Problems, Pure Appl. Math. (Boca Raton), 270, Chapman Hall/CRC, Boca Raton, 2005
  • [6] Han W., Sofonea M., Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Stud. Adv. Math., 30, American Mathematical Society, Providence, 2002
  • [7] Jarušek J., Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., 1996, 46(121)(3), 475–487
  • [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 http://dx.doi.org/10.1142/S0218202599000038[Crossref]
  • [9] Migórski S., Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 2005, 84(7), 669–699 http://dx.doi.org/10.1080/00036810500048129[Crossref]
  • [10] Migórski S., Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Comput. Math. Appl., 2006, 52(5), 677–698 http://dx.doi.org/10.1016/j.camwa.2006.10.007[Crossref]
  • [11] Migórski S., Ochal A., Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Anal., 2005, 61(1–2), 135–161 http://dx.doi.org/10.1016/j.na.2004.11.018[Crossref]
  • [12] Migórski S., Ochal A., A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 2006, 83(3), 247–275 http://dx.doi.org/10.1007/s10659-005-9034-0[Crossref]
  • [13] Migórski S., Ochal A., Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 2009, 41(4), 1415–1435 http://dx.doi.org/10.1137/080733231[WoS][Crossref]
  • [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 http://dx.doi.org/10.1016/j.nonrwa.2011.06.002[Crossref]
  • [15] Migórski S., Ochal A., Sofonea M., Nonlinear Inclusions and Hemivariational Inequalities, Adv. Mech. Math., 26, Springer, New York, 2012
  • [16] Naniewicz Z., Panagiotopoulos P.D., Mathematical Theory of Hemivariational Inequalities and Applications, Monogr. Textbooks Pure Appl. Math., 188, Marcel Dekker, New York, 1995
  • [17] Panagiotopoulos P.D., Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985 http://dx.doi.org/10.1007/978-1-4612-5152-1[Crossref]
  • [18] Panagiotopoulos P.D., Hemivariational Inequalities, Springer, Berlin, 1993 http://dx.doi.org/10.1007/978-3-642-51677-1[Crossref]
  • [19] Shillor M., Sofonea M., Telega J.J., Models and Analysis of Quasistatic Contact, Lecture Notes in Phys., 655, Springer, Berlin, 2004 http://dx.doi.org/10.1007/b99799[Crossref]
  • [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 http://dx.doi.org/10.1016/j.mcm.2004.01.011[Crossref]
  • [21] Zeidler E., Nonlinear Functional Analysis and its Applications, II/B, Springer, New York, 1990 http://dx.doi.org/10.1007/978-1-4612-0985-0[Crossref]
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0123-6
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