Analysis of Quasistatic Frictional Contact Problem with Subdifferential Form, Unilateral Condition and Long-Term Memory

Main Article Content

A. Ourahmoun, B. Bouderah, T. Serrar


We consider a quasistatic problem which models the contact between a deformable body and an obstacle called foundation. The material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. In Contact Mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions. The mathematical analysis of contact models leads to the study of variational and hemivariational inequalities. For this reason a large number of contact problems lead to inequalities which involve history dependent operators, called history dependent inequalities. Such inequalities could be variational or hemivariational and variational hemivariational.

In this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses, we stablish an existence and uniqueness result. The proof of the result is based on arguments of variational inequalities monotone operators and Banach fixed point theorem.

Article Details


  1. Adly, S. Ernst, E., Thera, M.: Stability of the solution set of non-coercive variational inequalities. Commun. Contemp. Math. 4 (2002), 145-160.
  2. Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces; Springer: Heidelberg, Germany, 1976.
  3. Br ´ezis, H. Probl`emes unilat ´eraux. J. Math. Pures Appl. 51 (1972), 1-168.
  4. Carstensen, C., Gwinner, J.: A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. Ann. Mat. Pura Appl. 177 (1999), 363-394.
  5. Chau, O. Analyse variationnelle et num ´erique en m ´ecanique du contact. Thesis, Perpignan, June 2000.
  6. Ciarlet, P.G.: Mathematical Elasticity, vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam, 1988.
  7. Duvaut, G., Lions, J.L.: Les In ´equations en M ´ecanique et en Physique. Dunod, Paris, 1972.
  8. Eck, Ch., Jarusek, J., Krbec, M.: Unilateral Contact Problems, Variational Methods and ExistenceTheorems. Monographs & Texbooks in Pure & Applied Mathematics, vol. 270, Chapman and Hall, London, 2005.
  9. Glowinski, R Numerical Methods for Nonlinear Variational Problems. Springer, Berlin, 1984.
  10. Goeleven, D, Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities,Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics. Kluwer, Dordrecht, 2003.
  11. Han, W, Sofonea, M.: Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM J. Numer. Anal. 38 (2000), 556-579. .
  12. Lions, J.L. Quelques m ´ethodes de r ´esolution des probl`emes aux limites non lin ´eaires. Dunod et Gauthier-Villars, Paris, 1969.
  13. Lions, J., Magenes, E.: Probl`emes aux limites non homog`enes et applications, vol. 1. Dunod, Paris, 1968.
  14. Matei, A, Sofonea, M.: Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Advances in Mechanics and Mathematics, vol. 18. Springer, Berlin, 2009.
  15. Kikuchi, N., Oden, J.T. Contact Problems in Elasticity. SIAM, Philadelphia, 1988.
  16. Necas, J., Hlavacek, I. Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction.Elsevier, Amsterdam, 1981.
  17. Panagiotopoulos, P.D. Inequality Problems in Meechanics and Applications. Birkh ¨auser, Basel, 1985.
  18. Panagiotopoulos, P.D. Inequality Problems in Meechanics and Applications. Birkh ¨auser, Basel, 1985.
  19. Sofonea, M., Han, W., Mig ´orski, S. Numerical analysis of history-dependent variational inequalities with applications to contact problems. Eur. J. Appl. Math. 26 (2015), 427-452.
  20. Sofonea, M., Matei, A. History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011), 471-491.
  21. Sofonea, M., Matei, A.Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge, 2012.
  22. Sofonea, M., Migorski, S. A class of history-dependent variational-hemivariational inequalities. Nonlinear Differ. Equ. Appl. 23 (2016), 38.
  23. Sofonea, M., Xiao, Y. Fully history-dependent quasivariational inequalities in contact mechanics. Appl. Anal. 95 (2016), 2464-2484.
  24. Zeidler, E.Nonlinear Functional Analysis and Its Applications, II/A, Linear Monotone Operators. Springer, Berlin, 1997.