Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Annals of West University of Timisoara - Mathematics and Computer Science

The Journal of West University of Timisoara

Editor-in-Chief: Sasu, Bogdan

2 Issues per year

Mathematical Citation Quotient (MCQ) 2016: 0.01

Open Access
See all formats and pricing
More options …

Analysis of a Unilateral Contact Problem with Normal Compliance

Arezki Touzaline / Rachid Guettaf
Published Online: 2014-12-11 | DOI: https://doi.org/10.2478/awutm-2014-0010


The paper deals with the study of a quasistatic unilateral contact problem between a nonlinear elastic body and a foundation. The contact is modelled with a normal compliance condition associated to unilateral constraint and the Coulomb's friction law. The adhesion between contact surfaces is taken into account and is modelled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We establish a variational formulation of the mechanical problem and prove an existence and uniqueness result in the case where the coefficient of friction is bounded by a certain constant. The technique of the proof is based on arguments of time-dependent variational inequalities, differential equations and fixed-point theorem.

Keywords: elastic; normal compliance; adhesion,friction; unilateral; weak solution


  • [1] L.-E. Andersson, Existence result for quasistatic contact problem with Coulomb friction, Appl. Math. Optimiz., 42, (2000), 169-202Google Scholar

  • [2] H. T. Banks, S. Hu, and Z. R Kenz, A brief review of elasticity and viscoelasticity for solids, Adv. Appl. Math. Mech., 3, (2011), 1-51Web of ScienceGoogle Scholar

  • [3] L. Cangémi, Frottement et adhérence: modèle, traitement numérique et application à l'interface fibre/matrice, Ph.D. Thesis, Univ. Méditerranée, Aix Marseille I, 1997Google Scholar

  • [4] O. Chau, J. R. Fernandez, M. Shillor, and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion, Journal of Computational and Applied Mathematics, 159, (2003), 431-465Google Scholar

  • [5] O. Chau, M. Shillor, and M. Sofonea, J. Appl. Math. Phys. (ZAMP), 55, (2004), 32-47Google Scholar

  • [6] M.Cocou, E. Pratt, and M. Raous, Formulation and approximation of quasistatic frictional contact, Int.J.Engng Sc., 34, (1996), 783-798Google Scholar

  • [7] M. Cocou and R. Rocca, Existence results for unilateral quasistatic contact problems with friction and adhesion, 34, (2000), 981-1001Google Scholar

  • [8] M. Cocou, M. Schyvre, and M. Raous, A dynamic unilateral contact problem with adhesion and friction in viscoelasticity, Z. Angew. Math. Phys., 61, (2010), 721-743Web of ScienceGoogle Scholar

  • [9] S. Drabla and Z. Zellagui, Analysis of a electro-elastic contact problem with friction and adhesion, Studia Univ. "Babes-Bolyai", Mathematica, LIV, (2009)Google Scholar

  • [10] G. Duvaut, C. R. Acad. Sc. Paris Série A, 290, (1980), 263Google Scholar

  • [11] G. Duvaut and J-L Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972Google Scholar

  • [12] C. Eck, J. Jarušek, and M. Krbec, Unilateral Contact Problems. Variational Methods and Existence Theorems, Chapman & Hall / CRC (Taylor & Francis Group), Boca Raton-London-New York-Singapore, 2005Google Scholar

  • [13] J. R. Fernandez, M. Shillor, and M. Sofonea, Analysis and numerical simulations of a dynamic contact problem with adhesion, 37, (2003), 1317-1333Google Scholar

  • [14] M. Frémond, Adhérence des solides, J. Mécanique Théorique et Appliquée, 6, (1987), 383-407Google Scholar

  • [15] M. Frémond, Equilibre des structures qui adhèrent à leur support, C. R. Acad. Sci.Paris, 295, (1982), 913-916Google Scholar

  • [16] M. Frémond, Non smooth Thermomechanics, Springer, Berlin, 2002Google Scholar

  • [17] J. Jarušek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift fur Angewandte Mathematik and Mechanik, (ZAMM), 88, (2008), 3-22Google Scholar

  • [18] J. Jarušek and M. Sofonea, On the of dynamic elastic-visco-plastic contact problems with adhesion, Annals of AOSR, Series on Mathematics and its applications, 1, (2009), 191-214Google Scholar

  • [19] S. Migorski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, (2013)Google Scholar

  • [20] M. Migorski, A. Ochal, and M. Sofonea, Analysis of a quasistatic contact problem for piezoelectric materials, J. Math. Anal. Appl., 382, (2011), 701-713Web of ScienceGoogle Scholar

  • [21] S. Migorski, A. Ochal, and M. Sofonea, A dynamic frictional contact problem for piezoelectric materials, J. Math. Anal. Appl., 361, (2010), 161-176Web of ScienceGoogle Scholar

  • [22] S.A. Nassar, T. Andrews, S. Kruk, and M. Shillor, Modelling and Simulations of a bonded rod, Math. Comput. Modelling, 42, (2005), 553-572Google Scholar

  • [23] M. Raous, L. Cangémi, and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact, Comput.Meth. Appl. Mech. Engng., 177, (1999), 383-399Google Scholar

  • [24] R. Rocca, Analyse et numérique de problèmes quasistatiques de contact avec frottement local de Coulomb en élasticité, Thèse, Aix Marseille 1, 2005Google Scholar

  • [25] J. Rojek and J. J. Telega, Contact problems with friction, adhesion and wear in orthopeadic biomechanics. I: General developements, J. Theor. Appl. Mech., 39, (2001), 655-677Google Scholar

  • [26] M. Shillor, M. Sofonea, and J. J. Telega, Models and Variational Analysis of Quasistatic Contact, Lecture Notes Physics, 655, (2004)Google Scholar

  • [27] M. Sofonea, W. Han, and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, 276, (2006)Google Scholar

  • [28] M. Sofonea and T.V. Hoarau -Mantel, Elastic frictionless contact problems with adhesion, Adv. Math. Sci. Appl, 15, (2005), 49-68Google Scholar

  • [29] M. Sofonea and A. Matei, Variational inequalities with applications, Advances in Mathematics and Mechanics, 18, (2009)Google Scholar

  • [30] M. Sofonea and A. Matei, An elastic contact problem with adhesion and normal compliance, Journal of Applied Analysis, 12, (2006), 19-36Google Scholar

  • [31] M. Sofonea, F. Patrelescu, and A. Farcas, A viscoplastic contact problem with normal compliance, unilateral constraint and memory term, Applied Mathematics & Optimization, 69, (2014), 175-198Google Scholar

  • [32] M. Sofonea and F. Patrelescu, Analysis of a history-dependent frictionless contact problem, Mathematics and Mechanics of solids, 18, (2012), 409-430Google Scholar

  • [33] M. Sofonea and A. Matei, Mathematical models in Contact Mechanics, London, Mathematical Society, Lecture Notes, Cambridge University, Press 398, Cambridge, 2012Google Scholar

  • [34] A. Touzaline, Frictionless contact problem with finite penetration for elastic mate- rials, Ann. Pol. Math., 98, (2010), 23-38Google Scholar

  • [35] A. Touzaline, Analysis of a contact adhesive problem with normal compliance, Ann. Pol. Math., 104, (2012), 175-188 Google Scholar

About the article

Received: 2014-04-12

Revised: 2014-07-18

Accepted: 2014-07-26

Published Online: 2014-12-11

Published in Print: 2014-06-01

Citation Information: Annals of West University of Timisoara - Mathematics, Volume 52, Issue 1, Pages 157–171, ISSN (Online) 1841-3307, DOI: https://doi.org/10.2478/awutm-2014-0010.

Export Citation

© Annals of West University of Timisoara - Mathematics. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in