Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:25:21.198Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of history-dependent variational–hemivariational inequalities with applications to contact problems†

Published online by Cambridge University Press:  20 April 2015

MIRCEA SOFONEA ,
WEIMIN HAN  and
STANISŁAW MIGÓRSKI
MIRCEA SOFONEA
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France email: [email protected]
WEIMIN HAN
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China email: [email protected]
STANISŁAW MIGÓRSKI
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, ul. Stanisława Łojasiewicza 6, 30348 Krakow, Poland email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

Keywords

Variational–hemivariational inequalityClarke subdifferentialhistory-dependent operatorContinuous dependenceFinite element methodConvergenceError estimatesViscoelastic materialFrictional contactWeak solution
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 427 - 452
Copyright
Copyright © Cambridge University Press 2015 

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.)

Footnotes

†

Research supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under the Maestro Advanced Project no. DEC-2012/06/A/ST1/00262, and the Polonium Project No. 31155YH/2014 between the University of Perpignan and the Jagiellonian University. The second author is also partially supported by grants from the Simons Foundation. The third author is also partially supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. W111/7.PR/2012.

References

[1]Atkinson, K. & Han, W. (2009) Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd ed., Springer-Verlag, New York.Google Scholar
[2]Baiocchi, C. & Capelo, A. (1984) Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester.Google Scholar
[3]Brézis, H. (1968) Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18, 115–175.CrossRefGoogle Scholar
[4]Brézis, H. (1972) Problèmes unilatéraux, J. Math. Pures Appl. 51, 1–168.Google Scholar
[5]Ciarlet, P. G. (1978) The Finite Element Method for Elliptic Problems, North Holland, Amsterdam.Google Scholar
[6]Clarke, F. H. (1975) Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262.CrossRefGoogle Scholar
[7]Clarke, F. H. (1981) Generalized gradients of Lipschitz functionals. Adv. Math. 40, 52–67.CrossRefGoogle Scholar
[8]Clarke, F. H. (1983) Optimization and Nonsmooth Analysis, Wiley Interscience, New York.Google Scholar
[9]Denkowski, Z., Migórski, S. & Papageorgiou, N. S. (2003) An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York.CrossRefGoogle Scholar
[10]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
[11]Friedman, A. (1982) Variational Principles and Free-Boundary Problems, John Wiley, New York.Google Scholar
[12]Glowinski, R. (1984) Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York.CrossRefGoogle Scholar
[13]Han, W., Migórski, S. & Sofonea, M. (2014) A class of variational-hemivariational inequalities with applications to elastic contact problems. SIAM J. Math. Anal. 46, 3891–3912.CrossRefGoogle Scholar
[14]Han, W. & Reddy, B. D. (2013) Plasticity: Mathematical Theory and Numerical Analysis, 2nd ed., Springer-Verlag, New York.CrossRefGoogle Scholar
[15]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, Somerville, MA.CrossRefGoogle Scholar
[16]Haslinger, J., Hlaváček, I. & Nečas, J. (1996) Numerical methods for unilateral problems in solid mechanics. In: Ciarlet, P. G. & Lions, J.-L. (editors), Handbook of Numerical Analysis, Vol. IV, North-Holland, Amsterdam, pp. 313–485.Google Scholar
[17]Haslinger, J., Miettinen, M. & Panagiotopoulos, P. D. (1999) Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Boston, Dordrecht, London.CrossRefGoogle Scholar
[18]Hlaváček, I., Haslinger, J., Nečas, J. & Lovíšek, J. (1988) Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York.CrossRefGoogle Scholar
[19]Kazmi, K., Barboteu, M., Han, W. & Sofonea, M. (2014) Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics. Math. Modelling Numer. Anal. (M2AN) 48, 919–942.CrossRefGoogle Scholar
[20]Kikuchi, N. & Oden, J. T. (1988) Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia.CrossRefGoogle Scholar
[21]Kinderlehrer, D. & Stampacchia, G. (2000) An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics, Vol. 31, SIAM, Philadelphia.CrossRefGoogle Scholar
[22]Lions, J.-L. (1969) Quelques méthodes de resolution des problémes aux limites non linéaires, Dunod, Paris.Google Scholar
[23]Martins, J. A. C. & Monteiro Marques, M. D. P. eds. (2002) Contact Mechanics, Kluwer, Dordrecht.CrossRefGoogle Scholar
[24]Migórski, S., Ochal, A. & Sofonea, M. (2013) Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, Vol. 26, Springer, New York.CrossRefGoogle Scholar
[25]Migórski, S., Ochal, A. & Sofonea, M. (2015) History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Ser. B: Real World Appl. 22, 604–618.CrossRefGoogle Scholar
[26]Naniewicz, Z. & Panagiotopoulos, P. D. (1995) Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong.Google Scholar
[27]Panagiotopoulos, P. D. (1985) Nonconvex problems of semipermeable media and related topics, ZAMM Z. Angew. Math. Mech. 65, 29–36.CrossRefGoogle Scholar
[28]Panagiotopoulos, P. D. (1985) Inequality Problems in Mechanics and Applications, Birkhäuser, Boston.CrossRefGoogle Scholar
[29]Panagiotopoulos, P. D. (1993) Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[30]Raous, M., Jean, M. & Moreau, J. J. (1995) Contact Mechanics, Plenum Press, New York.CrossRefGoogle Scholar
[31]Shillor, M. ed. (1998) Recent advances in contact mechanics. Special issue of Math. Comput. Modelling 28 (4–8), 1–531.Google Scholar
[32]Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact, Lect. Notes Phys., Vol. 655, Springer, Berlin Heidelberg.CrossRefGoogle Scholar
[33]Sofonea, M. & Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics, Eur. J. Appl. Math. 22, 471–491.CrossRefGoogle Scholar