Archive for Rational Mechanics and Analysis

, Volume 208, Issue 1, pp 25–57 | Cite as

Normal Compliance Contact Models with Finite Interpenetration

  • Christof Eck
  • Jiří Jarušek
  • Jana StaráEmail author


We study contact problems with contact models of normal compliance type, where the compliance function tends to infinity for a given finite interpenetration. Such models are physically more realistic than standard normal compliance models, where the interpenetration is not restricted, because the interpenetration is typically justified by deformations of microscopic asperities of the surface; therefore it should not exceed a certain value that corresponds to a complete flattening of the asperities. The model can be interpreted as intermediate between the usual normal compliance models and the unilateral contact condition of Signorini type. For the static problem without friction, we prove the existence and uniqueness of solutions and establish the equivalence to an optimization problem. For the static problem with Coulomb friction, we show the existence of a solution. The analysis is based on an approximation of the problems by standard normal compliance models, the derivation of regularity results for these auxiliary problems in Sobolev spaces of fractional order by a special translation technique, and suitable limit procedures.


Variational Inequality Contact Problem Coulomb Friction Normal Compliance Friction Term 
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.
    Andersson L.-E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. 16(4), 347–369 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andersson L.-E.: Existence result for quasistatic contact problem with Coulomb friction. Appl. Math. Optim. 42, 169–202 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ballard P., Jarušek J.: Indentation of an elastic half-space by a rigid flat punch as a model problem for analysing contact problems with Coulomb friction. J. Elasticity 103(1), 15–52 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Eck C., Jarušek J.: Solvability of static contact problems with Coulomb friction for orthotropic material. J. Elasticity 93(1), 93–104 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Eck, C., Jarušek, J., Krbec, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, Series of Monographs and Textbooks, vol. 270. Chapman/CRC Press, New York, 2005Google Scholar
  6. 6.
    Eck C., Jarušek J., Sofonea M.: Dynamic elastic-visco-plastic contact problems with normal damped response and Coulomb friction. Eur. J. Appl. Math. 21(3), 229–251 (2010)zbMATHCrossRefGoogle Scholar
  7. 7.
    Hild P.: Two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance. Nonlinear Anal. 71(11), 5560–5571 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Jarušek J., Sofonea M.: On the solvability of dynamic elastic-visco-plastic contact problems. Z. Angew. Math. Mech. 88(1), 3–22 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kato Y.: Signorini’s problem with friction in linear elasticity. Japan J. Appl. Math. 4, 237–268 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Klarbring A., Mikelić A., Shillor M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)zbMATHCrossRefGoogle Scholar
  11. 11.
    Klarbring A.: Derivation and analysis of rate boundary value problems of frictional contact. Eur. J. Mech. A Solids 9(1), 53–85 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Martins J.A.C., Oden J.T.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634 (1985)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Maz’ja V.G.: Sobolev Spaces. Springer, Berlin (1985)zbMATHGoogle Scholar
  14. 14.
    Nečas J., Jarušek J., Haslinger J.: On the solution of the variational inequality to the Signorini problem with small friction. Bolletino Unione Mat. Ital. 5(17B), 796–811 (1980)Google Scholar
  15. 15.
    Renard Y.: A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM J. Math. Anal. 38(2), 452–467 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany
  2. 2.Mathematical InstituteAcademy of Sciences of the Czech RepublicPraha 1Czech Republic
  3. 3.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

Personalised recommendations