Skip to main content
SpringerLink
Log in

Analysis of a quasistatic contact problem with adhesion and nonlocal friction for viscoelastic materials

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A mathematical model is established to describe a contact problem between a deformable body and a foundation. The contact is bilateral and modelled with a nonlocal friction law, in which adhesion is taken into account. Evolution of the bonding field is described by a first-order differential equation. The materials behavior is modelled with a nonlinear viscoelastic constitutive law. A variational formulation of the mechanical problem is derived, and the existence and uniqueness of the weak solution can be proven if the coefficient of friction is sufficiently small. The proof is based on arguments of time-dependent variational inequalities, differential equations, and the Banach fixed-point theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Duvaut, G. and Lions, J. L. Les Inéquations en Mécanique et en Physique, Dunod, Paris (1972)

    MATH  Google Scholar 

  2. Sofonea, M., Han, W., and Shillor, M. Analysis and approximations of contact problems with adhesion or damage. Pure and Applied Mathematics 276, Chapman & Hall/CRC Press, Boca Raton, Florida (2006)

    Google Scholar 

  3. Awbi, B. Analyse Variationnelle de Quelques Problèmes Viscoélastiques et Viscoplastiques Avec Frottement, Ph. D. disserdation, Université de Perpigan (2001)

  4. Chau, O., Fernandez, J. R., Shillor, M., and Sofonea, M. Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. Journal of Computational and Applied Mathematics 159(2), 431–465 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chau, O., Shillor, M., and Sofonea, M. Dynamic frictionless contact with adhesion. Zeitschrift für Angewandte Mathematik und Physik, ZAMP 55(1), 32–47 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fernandez, J. R., Shillor, M., and Sofonea, M. Analysis and numerical simulations of a dynamic contact problem with adhesion. Mathematical and Computer Modelling 37, 1317–1333 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Sofonea, M. and Hoarau-Mantel, T. V. Elastic frictionless contact problems with adhesion. Advances in Mathematical Sciences and Applications 15(1), 49–68 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Cangémi, L. Frottement et Adhérence: Modèle, Traitement Numérique et Application à l’interface Fibre/Matrice, Ph. D. disserdation, Université de la Méditerranée, Marseille, France (1997)

    Google Scholar 

  9. Frémond, M. Adhérence des solides. Journal of Mécanique Théorique et Appliquée 6, 383–407 (1987)

    MATH  Google Scholar 

  10. Frémond, M. Equilibre des structures qui adhèrent à leur support. C. R. Acad. Sci. Paris, Ser. II 295, 913–916 (1982)

    Google Scholar 

  11. Raous, M., Cangémi, L., and Cocu, M. A consistent model couplingadhesion, friction, and unilateral contact. Computer Methods in Applied Mechanics and Engineering 177, 383–399 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rojek, J. and Telega, J. J. Contact problems with friction, adhesion and wear in orthopeadic biomechanics. I: general developements. Journal of Theoretical and Applied Mechanics 39, 655–677 (2001)

    MATH  Google Scholar 

  13. Shillor, M., Sofonea, M., and Telega, J. J. Models and variational analysis of quasistatic contact. Lecture Notes in Physics, Vol. 655, Springer, Berlin (2004)

    Google Scholar 

  14. Sofonea, M., Arhab, R., and Tarraf, R. Analysis of electroelastic frictionless contact problems with adhesion. Journal of Applied Mathematics, Article ID 64217, 1–25 (2006) DOI 10.1155/JAM/2006/64217

    Google Scholar 

  15. Nassar, S. A., Andrews, T., Kruk, S., and Shillor, M. Modelling and simulations of a bonded rod. Mathematical and Computer Modelling 42, 553–572 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Frémond, M. Non-smooth Thermomechanics, Springer, Berlin (2002)

    MATH  Google Scholar 

  17. Brezis, H. Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Annales de l’institut Fourier 18(1), 115–175 (1968)

    MATH  MathSciNet  Google Scholar 

  18. Cocou, M. and Rocca, R. Existence results for unilateral quasistatic contact problems with friction and adhesion. Modélisation Mathématique et Analyse Numérique 34, 981–1001 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Systèmes Dynamiques, Faculté de Mathématiques, Université des Sciences et de la Technologie Houari Boumediene, BP 32 EL ALIA, Bab-Ezzouar, 16111, Algérie

    Arezki Touzaline

Authors
  1. Arezki Touzaline
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arezki Touzaline.

Additional information

Communicated by Xing-ming GUO

Rights and permissions

Reprints and permissions

About this article

Cite this article

Touzaline, A. Analysis of a quasistatic contact problem with adhesion and nonlocal friction for viscoelastic materials. Appl. Math. Mech.-Engl. Ed. 31, 623–634 (2010). https://doi.org/10.1007/s10483-010-0510-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-0510-z

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation