Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T21:39:12.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence results for primal and dual history-dependent quasivariational inequalities

Part of: Equilibrium (steady-state) problems Existence theories Equations and inequalities involving nonlinear operators Special kinds of problems Thin bodies, structures

Published online by Cambridge University Press:  27 December 2018

Mircea Sofonea  and
Ahlem Benraouda
Mircea Sofonea*
Affiliation:
Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia 52 Avenue Paul Alduy, 66860 Perpignan, France (sofonea@univ-perp.fr)
Ahlem Benraouda
Affiliation:
Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia 52 Avenue Paul Alduy, 66860 Perpignan, France (sofonea@univ-perp.fr)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of history-dependent quasivariational inequalities for which we prove the continuous dependence of the solution with respect to the set of constraints. Then, under additional assumptions, we associate with each inequality in the class a new inequality, the so-called dual variational inequality, for which we state and prove existence, uniqueness, equivalence and convergence results. The proofs are based on various estimates, monotonicity and fixed-point arguments for history-dependent operators. Our abstract results are useful in the study of various mathematical models of contact. To provide an example, we consider a boundary value problem which describes the equilibrium of a viscoelastic body in contact with an elastic-rigid foundation. We list the assumptions on the data and derive both the primal and the dual variational formulation of the problem. Then, we state and prove existence, uniqueness and convergence results. We also provide the link between the two formulations, together with their mechanical interpretation.

Keywords

history-dependent quasivariational inequalityprimal and dual variational formulationconvergence resultunilateral constraintfrictionless contactnormal complianceweak solution

MSC classification

Primary: 47J20: Variational and other types of inequalities involving nonlinear operators (general)
Secondary: 49J40: Variational methods including variational inequalities 49J45: Methods involving semicontinuity and convergence; relaxation 74G25: Global existence of solutions 74G30: Uniqueness of solutions 74K10: Rods (beams, columns, shafts, arches, rings, etc.) 74M15: Contact
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 471 - 494
Copyright
Copyright © Royal Society of Edinburgh 2018 

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

1Awbi, B., Shillor, M. and Sofonea, M.. Dual formulation of a quasistatic viscoelastic contact problem with Tresca's friction law. Appl. Anal. 79 (2001), 1–20.Google Scholar
2Baiocchi, C. and Capelo, A.. Variational and quasivariational inequalities: applications to free-boundary problems (Chichester: John Wiley, 1984).Google Scholar
3Benraouda, A. and Sofonea, M.. A convergence result for history-dependent quasivariational inequalities. Appl. Anal. 96 (2017), 2635–2651.Google Scholar
4Duvaut, G. and Lions, J.-L.. Inequalities in Mechanics and Physics (Berlin: Springer-Verlag, 1976).Google Scholar
5Glowinski, R.. Numerical Methods for Nonlinear Variational Problems (New York: Springer-Verlag, 1984).Google Scholar
6Glowinski, R., Lions, J.-L. and Trémolières, R.. Numerical Analysis of Variational Inequalities (Amsterdam: North-Holland, 1981).Google Scholar
7Han, W. and Sofonea, M.. Quasistatic contact problems in viscoelasticity and viscoplasticity. Studies in Advanced Mathematics, vol. 30 (Providence, Somerville, MA: Americal Mathematical Society, RI–International Press, 2002).Google Scholar
8Haslinger, J., Hlaváček, I. and Nečas, J.. Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis (ed. Ciarlet, P.G. and Lions, J.-L.) vol. IV, pp. 313–485 (Amsterdam: North-Holland, 1996).Google Scholar
9Hlaváček, I., Haslinger, J., Necǎs, J. and Lovíšek, J.. Solution of Variational Inequalities in Mechanics (New York: Springer-Verlag, 1988).Google Scholar
10Kalita, P., Migorski, S. and Sofonea, M.. A class of subdifferential inclusions for elastic unilateral contact problems. Set-Valued and Variational Analysis 24 (2016), 355–379.Google Scholar
11Kikuchi, N. and Oden, J.T.. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods (Philadelphia: SIAM, 1988).Google Scholar
12Kinderlehrer, D. and Stampacchia, G.. An introduction to variational inequalities and their applications. In Classics in Applied Mathematics, vol. 31 (Philadelphia: SIAM, 2000).Google Scholar
13Migórski, S., Ochal, A. and Sofonea, M.. History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 22 (2015), 604–618.Google Scholar
14Panagiotopoulos, P.D.. Inequality Problems in Mechanics and Applications (Boston: Birkhäuser, 1985).Google Scholar
15Pipkin, A.C.. Lectures in Viscoelasticity Theory. Applied Mathematical Sciences, vol. 7 (London, New York: George Allen & Unwin Ltd., Springer-Verlag, 1972).Google Scholar
16Shillor, M., Sofonea, M. and Telega, J.J.. Models and Analysis of Quasistatic Contact. Lect. Notes Phys., vol. 655 (Berlin, Heidelberg: Springer, 2004).Google Scholar
17Sofonea, M. and Matei, A.. History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011), 471–491.Google Scholar
18Sofonea, M. and Matei, A.. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398 (Cambridge: Cambridge University Press, 2012).Google Scholar
19Sofonea, M. and Migorski, S.. A class of history-dependent variational-hemivariational inequalities. Nonlin. Differ. Equ. Appl. 23 (2016), Art. 38, 23.Google Scholar
20Sofonea, M. and Pătrulescu, F.. Penalization of history-dependent variational inequalities. Eur. J. Appl. Math. 25 (2014), 155–176.Google Scholar
21Sofonea, M. and Xiao, Y.. Fully history-dependent quasivariational inequalities in contact mechanics. Appl. Anal. 95 (2016), 2464–2484.Google Scholar
22Sofonea, M., Renon, N. and Shillor, M.. Stress formulation for frictionless contact of an elastic-perfectly-plastic body. Appl. Anal. 83 (2004), 1157–1170.Google Scholar
23Sofonea, M., Avramescu, C. and Matei, A.. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Math. 7 (2008), 645–658.Google Scholar
24Sofonea, M., Danan, D. and Zheng, C.. Primal and dual variational formulation of a frictional contact problem. Mediterr. J. Math. 13 (2016), 857–872.Google Scholar
25Sofonea, M., Migorski, S. and Han, W.. A penalty method for history-dependent variational-hemivariational inequalities. Comput. Math. Appl. 75 (2018), 2561–2573.Google Scholar