Skip to main content
SpringerLink
Log in

Mixed finite element analysis of semi-coercive unilateral contact problems with given friction

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “ bolted” problem and the algorithm of Uzawa.

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. J. P. Aubin: Approximation of Elliptic Boundary Value Problems. Wiley, New York, 1972.

    MATH  Google Scholar 

  2. L. Baillet, T. Sassi: Méthodes d’éléments finis avec hybridisation frontière pour les problèmes de contact avec frottement. C. R. Acad. Sci. Paris, Ser. I 334 (2002), 917–922.

    MATH  MathSciNet  Google Scholar 

  3. P. Coorevits, P. Hild, K. Lhalouani, and T. Sassi: Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies. Math. Comput. 71 (2002), 1–25.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Daněk, I. Hlaváček, and J. Nedoma: Domain decomposition for generalized unilateral semi-coercive contact problem with friction in elasticity. Math. Comput. Simul. 68 (2005), 271–300.

    Article  Google Scholar 

  5. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels. Dunod, Paris, 1974.

    MATH  Google Scholar 

  6. M. Fiedler: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ. (member of Kluwer), Dordrecht-Boston, 1986.

    MATH  Google Scholar 

  7. M. Frémond: Dual formulations for potentials and complementary energies. In: MAFELAP (J. R. Whiteman, ed.). Academic Press, London, 1973.

    Google Scholar 

  8. J. Haslinger, I. Hlaváček: Approximation of the Signorini problem with friction by a mixed finite element method. J. Math. Anal. Appl. 86 (1982), 99–122.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Haslinger, I. Hlaváček, and J. Nečas: Numerical methods for unilateral problems in solid mechanics. In: Handbook of Numerical Analysis, vol. IV (P. G. Ciarlet, J.-L. Lions, eds.). North Holland, Amsterdam, 1996, pp. 313–485.

    Google Scholar 

  10. J. Haslinger, T. Sassi: Mixed finite element approximation of 3D contact problems with given friction: error analysis and numerical realization. M2AN, Math. Model. Numer. Anal. 38 (2004), 563–578.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Haslinger, R. Kučera, and Z. Dostál: An algorithm for numerical realization of 3D contact problems with Coulomb friction. J. Comput. Appl. Math. 164–165 (2004), 387–408.

    Article  Google Scholar 

  12. I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek: Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York, 1988.

    Google Scholar 

  13. I. Hlaváček: Finite element analysis of a static contact problem with Coulomb friction. Appl. Math. 45 (2000), 357–379.

    Article  MathSciNet  Google Scholar 

  14. I. Hlaváček, J. Nedoma: On a solution of a generalized semi-coercive contact problem in thermo-elasticity. Math. Comput. Simul. 60 (2002), 1–17.

    Article  Google Scholar 

  15. V. Janovský, P. Procházka: Contact problem for two elastic bodies, Parts I–III. Apl. Mat. 25 (1980), 87–109, 110–136, 137–146.

    MATH  MathSciNet  Google Scholar 

  16. J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam, 1981.

    Google Scholar 

  17. T. Sassi: Nonconforming mixed variational formulation of the Signorini problem with a given friction. Preprint of MAPLY No. 365.-2003. http://maply,univ-lyon1.fr/publis.

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Ivan Hlaváček

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07, Praha 8, Czech Republic

    Ivan Hlaváček

Authors
  1. Ivan Hlaváček
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hlaváček, I. Mixed finite element analysis of semi-coercive unilateral contact problems with given friction. Appl Math 52, 25–58 (2007). https://doi.org/10.1007/s10492-007-0002-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-007-0002-9

Keywords

Search

Navigation