Implicit Variational Inequalities Arising in Frictional Unilateral Contact Mechanics: Analysis and Numerical Solution of Quasistatic Problems

  • Marius Cocu
  • Michel Raous
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 55)

Abstract

This paper is a survey on implicit variational inequalities arising in the study of unilateral contact problems with friction. Recent works on mathematical and numerical approaches of quasistatic problems are presented. The coupling of unilateral contact, friction, and adhesion is considered and previous results are generalized to this case.

Keywords

Cocu 

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References

  1. Andersson, L. E. (1995). A global existence result for a quasistatic contact problem with friction. Advances in Mathematical Sciences and Applications, 5:249–286.MathSciNetMATHGoogle Scholar
  2. Cangémi, L. (1997). Frottement et adhérence: modèle, traitement numérique et application à l’interface fibre/matrice. Thèse, Université d’Aix-Marseille II, Marseille.Google Scholar
  3. Capatina, A., and Cocu, M. (1991). Internal approximation of quasi-variational inequalities. Numer. Math., 59:385–398.MathSciNetMATHCrossRefGoogle Scholar
  4. Chabrand, P., Dubois, F., and Raous, M. (1998). Various numerical methods for solving unilateral contact problems with friction. Mathematical and Computer Modelling, 28:97–108.MATHCrossRefGoogle Scholar
  5. Cocu, M. (1984). Existence of solutions of Signorini problems with friction. Int. J. Engrg. Sci., 22:567–575.MathSciNetMATHCrossRefGoogle Scholar
  6. Cocu, M., Cangémi, L., and Raous, M. (1999). Approximation results for a class of quasistatic contact problems including adhesion and friction. In Proceedings of the IUTAM Symposium on the Variations of Domains and Free-Boundary Problems in Solid Mechanics-1997, pages 211–218. Kluwer Academic Publishers.CrossRefGoogle Scholar
  7. Cocu, M., Pratt, E., and Raous, M. (1996). Formulation and approximation of quasistatic frictional contact. Int. J. Engrg. Sci., 34:783–798.MathSciNetMATHCrossRefGoogle Scholar
  8. Cottle, R.W., Giannessi, F., and Lions, J.L., editors (1979). Variational Inequalities and Complementarity Problems in Mathematical Physics and Economics. John Wiley, New York.Google Scholar
  9. Demkowicz, L., and Oden, J. T. (1982). On some existence and uniqueness results in contact problems with nonlocal friction. Nonlinear Analysis, Theory, Methods and Applications, 6:1075–1093.MathSciNetMATHGoogle Scholar
  10. Duvaut, G. (1980). Equilibre d’un solide élastique avec contact unilatéral et frottement de Coulomb. C. R. Acad. Sci. Paris série A, 290:263–265.MathSciNetMATHGoogle Scholar
  11. Duvaut, G., and Lions, J.L. (1972). Les inéquations en mécanique et en physique. Dunod, Paris.MATHGoogle Scholar
  12. Fichera, G. (1964). Problemi elastostatici con vincoli unilaterali: i1 problema di Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei Ser. VIII,7:91–140.MathSciNetGoogle Scholar
  13. Fichera, G. (1972). Boundary value problems of elasticity with unilateral constraints. In Flügge, S., editor, Encyclopedia of Physics, Vol. VI a/2, pages 391–424. Springer, Berlin.Google Scholar
  14. Frémond, M. (1987). Adhérence des solides. J. Méc. Théor. et Appl., 6:383–407.MATHGoogle Scholar
  15. Glowinski, R., Lions, J.L., and Trémolières, R. (1976). Analyse numérique des inéquations variationnelles. Dunod, Paris.Google Scholar
  16. Klarbring, A. (1986). A mathematical programming approach to three dimensional contact problems with friction. Cornp. Meth. Appl. Mech. Engrg., 58:175–200.MathSciNetMATHCrossRefGoogle Scholar
  17. Klarbring, A. (1990). Derivation and analysis of rate boundary problems of frictional contact. European Journal of Mechanics A/Solids, 9:53–85.MathSciNetMATHGoogle Scholar
  18. Klarbring, A., Mikelić, A., and Shillor, M. (1991). A global existence result for the quasistatic frictional contact problem with normal compliance. In Inter-national Series of Numerical Mathematics,101:85–111.Birkhäuser Verlag, Basel.Google Scholar
  19. Lebon, F., and Raous, M. (1992). Multibody contact problems including friction in structure assembly. Computers and Structures, 42:925–934.CrossRefGoogle Scholar
  20. Licht, C., Pratt, E., and Raous, M. (1991). Remarks on a numerical method for unilateral contact including friction. In International Series of Numerical Mathematics, 101:129–144. Birkhäuser Verlag, Basel.MathSciNetGoogle Scholar
  21. Lions, J.L., and Stampacchia, G. (1967). Variational inequalities. Cornm. Pure Appl. Math.,20:493–519.MathSciNetMATHCrossRefGoogle Scholar
  22. Martins, J.A.C., Barbarin, S., Raous, M., and Pinto da Costa, A. (1999). Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction. Comp. Meth. Appl. Mech. Engrg.,177:289–328.MathSciNetMATHCrossRefGoogle Scholar
  23. Mosco, U. (1975). Implicit variational problems and quasi variational inequalities. In Lecture Notes in Mathematics 543: Nonlinear Operators and the Calculus of Variations, Bruxelles, pages 83–156. Springer, Berlin.Google Scholar
  24. Oden, J. T., and Pires, E. B. (1983). Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. ASME Journal of Applied Mechanics,50:67–76.MathSciNetMATHCrossRefGoogle Scholar
  25. Panagiotopoulos, P., Wriggers, P., Frémond, M., Curnier, A., Klarbring, A., and Raous, M. (1998). Contact problems: theory, methods, applications, CISM Course. Springer, Wien. to appear.Google Scholar
  26. Raous, M., and Barbarin, S. (1992). Preconditioned conjugate gradient method for a unilateral problem with friction. In Curnier, A., editor, Contact Mechanics, pages 423–432. Presses Polytechniques et Universitaires Romandes, Lausanne.Google Scholar
  27. Raous, M., Chabrand, P., and Lebon, F. (1998). Numerical methods for frictional contact problems and applications. Journal de Mécanique Théorique et Appliquée, special issue, supplement n°1 to 7:111–128.MathSciNetGoogle Scholar
  28. Raous, M., Cocu, M., and Cangémi, L. (1997). Un modele couplant adhérence et frottement pour le contact entre deux solides déformables. C. R. Acad. Sci. Paris, Série II b325:503–509.Google Scholar
  29. Raous, M., Cocu, M., and Cangémi, L. (1999). A consistent model coupling adhesion, friction, and unilateral contact. Comp. Meth. Appl. Mech. Engng., 177:383–399.MATHCrossRefGoogle Scholar
  30. Signorini, A. (1959). Questioni di elasticità nonlinearizzata et semilinearizzata. Rend. di Matem. e delle sue appl.,18:1–45.MathSciNetGoogle Scholar
  31. Telega, J.J. (1991). Quasi-static Signorini’s contact problem with friction and duality. In International Series of Numerical Mathematics, 101:199–214. Birkhäuser Verlag, Basel.MathSciNetGoogle Scholar
  32. Vola, D., Pratt, E., Jean, M., and Raous, M. (1998). Consistent time discretization for a dynamical frictional contact problem and complementarity techniques. Revue Européenne des Eléments Finis,7:149–162.MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Marius Cocu
    • 1
  • Michel Raous
    • 2
  1. 1.Laboratoire de Mécanique et d’Acoustique - C.N.R.S.Université de ProvenceMarseilleFrance
  2. 2.Laboratoire de Mécanique et d’Acoustique - C.N.R.S.Marseille Cedex 20France

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