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Viscoelastic frictionless contact problems with adhesion

Abstract

We consider two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.

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References

  1. 1.

    Andrews KT, Chapman L, Fernández JR, Fisackerly M, Shillor M, Vanerian L, Van Houten T: A membrane in adhesive contact. SIAM Journal on Applied Mathematics 2003,64(1):152–169. 10.1137/S0036139902406206

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Andrews KT, Shillor M: Dynamic adhesive contact of a membrane. Advances in Mathematical Sciences and Applications 2003,13(1):343–356.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Chau O, Fernández JR, Shillor M, Sofonea M: Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. Journal of Computational and Applied Mathematics 2003,159(2):431–465. 10.1016/S0377-0427(03)00547-8

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Chau O, Shillor M, Sofonea M: Dynamic frictionless contact with adhesion. Journal of Applied Mathematics and Physics (ZAMP) 2004,55(1):32–47. 10.1007/s00033-003-1089-9

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Cocu M, Rocca R: Existence results for unilateral quasistatic contact problems with friction and adhesion. Mathematical Modelling and Numerical Analysis 2000,34(5):981–1001. 10.1051/m2an:2000112

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Fernández JR, Shillor M, Sofonea M: Analysis and numerical simulations of a dynamic contact problem with adhesion. Mathematical and Computer Modelling 2003,37(12–13):1317–1333. 10.1016/S0895-7177(03)90043-4

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Frémond M: Équilibre de structures qui adhèrent à leur support. Comptes Rendus des Séances de l'Académie des Sciences. Série II 1982,295(11):913–916.

    Google Scholar 

  8. 8.

    Frémond M: Adhérence des solides. Journal de Mécanique Théorique et Appliquée 1987,6(3):383–407.

    MATH  Google Scholar 

  9. 9.

    Frémond M: Non-Smooth Thermomechanics. Springer, Berlin; 2002:xvi+480.

    Google Scholar 

  10. 10.

    Han W, Kuttler KL, Shillor M, Sofonea M: Elastic beam in adhesive contact. International Journal of Solids and Structures 2002,39(5):1145–1164. 10.1016/S0020-7683(01)00250-5

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Nečas J, Hlavaček I: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Studies in Applied Mechanics. Volume 3. Elsevier Scientific, Amsterdam; 1981.

    Google Scholar 

  12. 12.

    Raous M, Cangémi L, Cocu M: A consistent model coupling adhesion, friction, and unilateral contact. Computer Methods in Applied Mechanics and Engineering 1999,177(3–4):383–399. 10.1016/S0045-7825(98)00389-2

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Rojek J, Telega JJ: Contact problems with friction, adhesion and wear in orthopaedic biomechanics. I: general developments. Journal of Theoretical and Applied Mechanics 2001,39(3):655–677.

    MATH  Google Scholar 

  14. 14.

    Rojek J, Telega JJ, Stupkiewicz S: Contact problems with friction, adhesion and wear in orthopaedic biomechanics. II: numerical implementation and application to implanted knee joints. Journal of Theoretical and Applied Mechanics 2001,39(3):679–706.

    MATH  Google Scholar 

  15. 15.

    Shillor M, Sofonea M, Telega JJ: Models and Analysis of Quasistatic Contact. Variational Methods, Lect. Notes Phys.. Volume 655. Springer, Berlin; 2004.

    Google Scholar 

  16. 16.

    Sofonea M, Han W, Shillor M: Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton). Volume 276. Chapman & Hall/CRC Press, Florida; 2006:xviii+220.

    Google Scholar 

  17. 17.

    Talon C, Curnier A: A model of adhesion added to contact with friction. In Contact Mechanics (Praia da Consolação, 2001), Solid Mech. Appl.. Volume 103. Edited by: Martins JAC, Monteiro Marques MDP. Kluwer Academic, Dordrecht; 2002:161–168.

    Google Scholar 

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Correspondence to Mircea Sofonea.

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Selmani, M., Sofonea, M. Viscoelastic frictionless contact problems with adhesion. J Inequal Appl 2006, 36130 (2006). https://doi.org/10.1155/JIA/2006/36130

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Keywords

  • Differential Equation
  • Weak Solution
  • Variational Inequality
  • Contact Surface
  • Point Theorem
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