Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 12, 2012

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

  • Stanisław Migórski EMAIL logo
From the journal Open Mathematics


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.

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

[2] Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston, 2003 10.1007/978-1-4419-9158-4Search in Google Scholar

[3] Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Applications, Kluwer, Boston, 2003 10.1007/978-1-4419-9156-0Search in Google Scholar

[4] Duvaut G., Lions J.-L., Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., 219, Springer, Berlin-New York, 1976 10.1007/978-3-642-66165-5Search in Google Scholar

[5] Eck C., Jarušek J., Krbec M., Unilateral Contact Problems, Pure Appl. Math. (Boca Raton), 270, Chapman Hall/CRC, Boca Raton, 2005 10.1201/9781420027365Search in 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 10.1090/amsip/030Search in Google Scholar

[7] Jarušek J., Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Math. J., 1996, 46(121)(3), 475–487 10.21136/CMJ.1996.127309Search in 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 in 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 in 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 in 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 in Google Scholar

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

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

[15] Migórski S., Ochal A., Sofonea M., Nonlinear Inclusions and Hemivariational Inequalities, Adv. Mech. Math., 26, Springer, New York, 2012 10.1007/978-1-4614-4232-5Search in 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 Search in Google Scholar

[17] Panagiotopoulos P.D., Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985 in Google Scholar

[18] Panagiotopoulos P.D., Hemivariational Inequalities, Springer, Berlin, 1993 in Google Scholar

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

[21] Zeidler E., Nonlinear Functional Analysis and its Applications, II/B, Springer, New York, 1990 in Google Scholar

Published Online: 2012-10-12
Published in Print: 2012-12-1

© 2012 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 1.12.2023 from
Scroll to top button