Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T11:17:29.163Z Has data issue: false hasContentIssue false
  • English
  • Français

History-dependent quasi-variational inequalities arising in contact mechanics

Published online by Cambridge University Press:  19 May 2011

MIRCEA SOFONEA  and
ANDALUZIA MATEI
MIRCEA SOFONEA
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France email: sofonea@univ-perp.fr
ANDALUZIA MATEI
Affiliation:
Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 Craiova, Romania
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

Keywords

Quasi-variational inequalitiesExistence and uniquenessRegularityContact problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 5 , October 2011 , pp. 471 - 491
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baiocchi, C. & Capelo, A. (1984) Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester.Google Scholar
[2]Brezis, H. (1968) Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18, 115–175.CrossRefGoogle Scholar
[3]Duvaut, G. & Lions, J. L. (1976) Inequalities in Mechanics and Physics, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[4]Eck, C., Jarušek, J. & Krbeč, M. (2005) Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, Vol. 270, Chapman/CRC Press, New York.CrossRefGoogle Scholar
[5]Glowinski, R. (1984) Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York.CrossRefGoogle Scholar
[6]Han, W. & Reddy, B. D. (1995) Computational plasticity: The variational basis and numerical analysis. Comput. Mech. Adv. 2, 283–400.Google Scholar
[7]Han, W. & Reddy, B. D. (1999) Plasticity: Mathematical Theory and Numerical Analysis, Springer-Verlag, New York.Google Scholar
[8]Han, W. & Sofonea, M. (2002) Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, Vol. 30, American Mathematical Society, Providence, RI-International Press, Sommerville, MA.CrossRefGoogle Scholar
[9]Hlaváček, I., Haslinger, J., Necas, J. & Lovíšek, J. (1988) Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York.CrossRefGoogle Scholar
[10]Jarušek, J. & Sofonea, M. (2008) On the solvability of dynamic elastic-visco-plastic contact problems. Z. für Angew. Mate. Mech. 88, 3–22.CrossRefGoogle Scholar
[11]Kikuchi, N. & Oden, J. T. (1980) Theory of variational inequalities with applications to problems of flow through porous media. Int. J. Engng. Sci. 18, 1173–1284.Google Scholar
[12]Kikuchi, N. & Oden, T. J. (1988) Contact Problems in Elasticity, SIAM, Philadelphia.CrossRefGoogle Scholar
[13]Kinderlehrer, D. & Stampacchia, G. (2000) An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics, Vol. 31, SIAM, Philadelphia.CrossRefGoogle Scholar
[14]Martins, J. A. C. & Monteiro Marques, M. D. P. (editors) (2002) Contact Mechanics, Kluwer, Dordrecht.CrossRefGoogle Scholar
[15]Panagiotopoulos, P. D. (1985) Inequality Problems in Mechanics and Applications, Birkhäuser, Boston.CrossRefGoogle Scholar
[16]Raous, M., Jean, M. & Moreau, J. J. (1995) Contact Mechanics, Plenum Press, New York.CrossRefGoogle Scholar
[17]Rochdi, M., Shillor, M. & Sofonea, M. (1998) A quasistatic viscoelastic contact problem with normal compliance and friction. J. Elast 51, 105–126.CrossRefGoogle Scholar
[18]Rochdi, M., Shillor, M. & Sofonea, M. (1998) A quasistatic contact problem with directional friction and damped response. Appl. Anal. 68, 409–422.CrossRefGoogle Scholar
[19]Rodríguez-Aros, A. D., Sofonea, M. & Viaño, J. M. (2004) A class of evolutionary variational inequalities with volterra-type integral rerm. Math. Models Methods Appl. Sci. 14, 555–577.CrossRefGoogle Scholar
[20]Shillor, M. (editor) (1998) Recent advances in contact mechanics. Special issue of Math. Comput. Modelling 28 (4–8).Google Scholar
[21]Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact. Variational Methods, Lecture Notes in Physics, Vol. 655, Springer, Berlin.CrossRefGoogle Scholar
[22]Sofonea, M., Avramescu, C. & Matei, A. (2008) A Fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Anal. 7, 645–658.CrossRefGoogle Scholar
[23]Sofonea, M. & Matei, A. (2009) Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems, Advances in Mechanics and Mathematics, Vol. 18, Springer, New York.Google Scholar
[24]Wriggers, P. & Nackenhorst, U. (editors). (2006) Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, Vol. 27, Springer, Berlin.CrossRefGoogle Scholar