Skip to main content
SpringerLink
Log in

A Mixed Variational Formulation of a Contact Problem with Wear

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014).

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. Archard, J.F., Hirst, W.: Wear of metals under unlubricated conditions. Proc. R. Soc. Lond. Ser. A 236, 3–55 (1956)

    Article  Google Scholar 

  2. Barboteu, M., Matei, A., Sofonea, M.: On the behavior of the solution of a viscoplastic contact problem. Q. Appl. Math. 72, 625–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Céa, J.: Optimization. Théorie et Algorithmes. Dunod, Paris (1971)

    MATH  Google Scholar 

  4. Chudzikiewicz, A., Myśliński, A.: Thermoelastic wheel-rail contact problem with elastic graded-materials. Wear 271, 417–425 (2011)

    Article  Google Scholar 

  5. Cristescu, N., Suliciu, I.: Viscoplasticity. Nijhoff/Editura Tehnică, Bucharest (1982)

    MATH  Google Scholar 

  6. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  7. Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, vol. 270. Chapman/CRC Press, New York (2005)

    MATH  Google Scholar 

  8. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  9. Glodež, S., Ren, Z., Flašker, J.: Simulation of surface pitting due to contact loading. Int. J. Numer. Methods Eng. 43, 33–50 (1998)

    Article  MATH  Google Scholar 

  10. Goryacheva, I.G.: Contact Mechanics in Tribology. Kluwer, Dordrecht (1988)

    MATH  Google Scholar 

  11. Gu, R.J., Shillor, M.: Thermal and wear analysis of an elastic beam in sliding contact. Int. J. Solids Struct. 38, 2323–2333 (2001)

    Article  MATH  Google Scholar 

  12. Gu, R.J., Kuttler, K.L., Shillor, M.: Frictional wear of a thermoelastic beam. J. Math. Anal. Appl. 242, 212–236 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics, vol. 30. Am. Math. Soc./International Press, Providence/Somerville (2002)

    MATH  Google Scholar 

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

    Google Scholar 

  15. Hild, P., Renard, Y.: A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 115, 101–129 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hüeber, S., Matei, A., Wohlmuth, B.: Efficient algorithms for problems with friction. SIAM J. Sci. Comput. 29, 70–92 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ionescu, I.R., Sofonea, M.: Functional and Numerical Methods in Viscoplasticity. Oxford Univ. Press, Oxford (1993)

    MATH  Google Scholar 

  18. Lions, J.-L., Glowinski, R., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  19. Matei, A.: On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers. Proc. R. Soc. Edinb., Sect. A, Math. 143, 1047–1059 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matei, A.: An evolutionary mixed variational problem arising from frictional contact mechanics. Math. Mech. Solids 19, 223–239 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Matei, A., Ciurcea, R.: Contact problems for nonlinearity elastic materials: weak solvability involving dual Lagrange multipliers. ANZIAM J. 52, 160–178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Matei, A., Ciurcea, R.: Weak solutions for contact problems involving viscoelastic materials with long memory. Math. Mech. Solids 16, 393–405 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhauser, Boston (1985)

    Book  MATH  Google Scholar 

  24. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  25. Rabinowiz, E.: Friction and Wear of Materials, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  26. Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rodriguez-Arós, A., Viaño, J.M., Sofonea, M.: Asymptotic derivation of quasistatic frictional contact models with wear for elastic rods. J. Math. Anal. Appl. 401, 641–653 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rojek, J., Telega, J.J., Stupkiewicz, S.: Contact problems with friction, adhesion and wear in orthopaedic biomechanics, II: numerical implementation and application to implanted knee joints. J. Theor. Appl. Mech. 39, 679–706 (2001)

    MATH  Google Scholar 

  29. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)

    MATH  Google Scholar 

  30. Sofonea, M., Matei, A.: History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491 (2011)

    Article  MATH  Google Scholar 

  31. Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge Univ. Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  32. Sofonea, M., Matei, A.: History-dependent mixed variational problems in contact mechanics. J. Glob. Optim. 61, 591–614 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sofonea, M., Avramescu, C., Matei, A.: A fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Anal. 7, 645–658 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sofonea, M., Pătrulescu, F., Souleiman, Y.: Analysis of a contact problem with wear and unilateral constraint. Appl. Anal. 95, 2602–2619 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Strömberg, N.: A Newton method for three-dimensional fretting problems. Int. J. Solids Struct. 36, 2075–2090 (1999)

    Article  MATH  Google Scholar 

  36. Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact, friction and wear. Int. J. Solids Struct. 33, 1817–1836 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zmitrowicz, A.: Variational descriptions of wearing out solids and wear particles in contact mechanics. J. Theor. Appl. Mech. 39, 791–808 (2001)

    MATH  Google Scholar 

  38. Zmitrowicz, A.: Wear patterns and laws of wear-a review. J. Theor. Appl. Mech. 44, 219–253 (2006)

    Google Scholar 

Download references

Acknowledgements

The work of the authors has been partially supported by project LEA Math Mode 2014/2015. The work of the second author has been partially supported by project POSDRU/159/1.5/S/132400: Young successful researchers-professional development in an international and interdisciplinary environment at Babeş-Bolyai University.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique, Université de Perpignan, 52 Avenue de Paul Alduy, 66 860, Perpignan, France

    Mircea Sofonea & Ahmad Ramadan

  2. Tiberiu Popoviciu Institute of Numerical Analysis, P.O. Box 68-1, 400110, Cluj-Napoca, Romania

    Flavius Pătrulescu

Authors
  1. Mircea Sofonea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Flavius Pătrulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmad Ramadan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mircea Sofonea.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sofonea, M., Pătrulescu, F. & Ramadan, A. A Mixed Variational Formulation of a Contact Problem with Wear. Acta Appl Math 153, 125–146 (2018). https://doi.org/10.1007/s10440-017-0123-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-017-0123-4

Keywords

Mathematics Subject Classification

Search

Navigation