Contact Problems with Friction by Linear Complementarity

  • A. Klarbring
Part of the International Centre for Mechanical Sciences book series (CISM, volume 304)


This study presents a new approach to the quasi-static contact problem with friction. It is assumed that the contacting bodies (structures) are linear elastic and that their displacement fields can be described by a finite number of variables. The approach is based on concepts from the discipline of Mathematical Programming, particularly those related to the Linear Complementarity Problem. It is pointed out that the problem studied may in some cases have multiple solutions, and that in other cases a solution may not even exist.


Contact Problem Linear Complementary Problem Full Column Rank Complementarity Condition Contact Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Grierson, D.E. et al: Mathematical programming and nonlinear finite element analysis, Comp. Meth. in Appl. Mech. and Eng., 17/18(1979), 497–518.ADSCrossRefGoogle Scholar
  2. 2.
    Cohn, M.Z., G. Maier and D Grierson (eds.): Engineering Plasticity by Mathematical Programming, Pergamon Press, New York 1979.MATHGoogle Scholar
  3. 3.
    Panagiotopoulos, P.D.: A nonlinear programming approach to the unilateral contact- and friction-boundary value problem in the theory of elasticity, Ingenieur-Archiv, 44(1975), 421–432.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Fredriksson, B., G. Rydholm and P. Sjöblom: Variational Inequalities in Structural Mechanics with Emphasis on Contact Problems, in: Proc. International Conference on Finite Elements in Nonlinear Solid and Structural Mechanics, Geilo 1977.Google Scholar
  5. 5.
    Oden, J.T. and E.B. Pires: Numerical analysis of certain contact problems in elasticity with non-classical friction laws, Comp. & Struct., 16, 1–4(1983), 481–485.CrossRefMATHGoogle Scholar
  6. 6.
    Klarbring, A.: Contact Problems With Friction — Using a Finite Dimensional Description and the Theory of Linear Complementarity, Linköping Studies in Science and Technology, Thesis No. 20, Linköping Institute of Technology, Linköping, Sweden 1984.Google Scholar
  7. 7.
    Klarbring, A.: The Influence of Slip Hardening and Interface Compli-ance on Contact Stress Distributions. A Mathematical Programming Approach, to appear in: Mechanics of Material Interfaces (Eds. A.P.S. Selvadurai, G. Voyiadjis), Elsevier 1985.Google Scholar
  8. 8.
    Klarbring, A.: A mathematical programming approach to threedimensio-nal contact problems with friction, to appear in Comp. Meth. in App. Mech. and Eng.Google Scholar
  9. 9.
    Cottle, R.W.: Monotone solutions of the parametric linear complementary problem, Mathem. Program.,3(1972),210–224.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kaneko, I.: A parametric linear complementarity problem involving derivatives, Mathematical Programming, 15(1978),146–154.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Klarbring, A.: General contact boundary conditions and the analysis of frictional systems, to appear in Int. J. of Solids and Struct.Google Scholar
  12. 12.
    Klarbring, A.: Quadratic programming in frictionless contact problem, to appear in Int. J. of Eng. Sci.Google Scholar
  13. 13.
    Eaves, B.C.: The linear complementarity problem, Management Sci, 17(1971), 612–634.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Murty, K.G.: On the number of solutions of the complementarity problem and spanning properties of complementary cones, Linear Algebra and Its Appl., 5(1972), 65–108.CrossRefMATHGoogle Scholar
  15. 15.
    Kalker,J.J.: A minimum principle for the law of dry friction, with application to elastic cylinders in rolling contact, J. of Appl. Mech., Dec. 1971, 875–887.Google Scholar
  16. 16.
    Barber, J.: Private Communication, October 12, 1984.Google Scholar
  17. 17.
    Lötstedt, P.: Coulomb friction in two-dimensional rigid body systems, ZAMM, 61(1981), 605–615.CrossRefMATHGoogle Scholar
  18. 18.
    Janovsky V.: Catastrophic Features of Coulomb Friction Model, in: Proc. Mathematics of Finite Elements and Applications, Brunei University 1981.Google Scholar

Copyright information

© Springer-Verlag Wien 1987

Authors and Affiliations

  • A. Klarbring
    • 1
  1. 1.Department of Mechanical EngineeringInstitute of Technology of LinkopingSweden

Personalised recommendations