Journal of Applied Mathematics

Analysis of electroelastic frictionless contact problems with adhesion

Mircea Sofonea, Rachid Arhab, and Raafat Tarraf

Full-text: Open access

Abstract

We consider two quasistatic frictionless contact problems for piezoelectric bodies. For the first problem the contact is modelled with Signorini's conditions and for the second one is modelled with normal compliance. In both problems the material's behavior is electroelastic and 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 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 fixed point. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solution of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.

Article information

Source
J. Appl. Math., Volume 2006 (2006), Article ID 64217, 25 pages.

Dates
First available in Project Euclid: 16 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.jam/1168975521

Digital Object Identifier
doi:10.1155/JAM/2006/64217

Mathematical Reviews number (MathSciNet)
MR2251808

Zentralblatt MATH identifier
1143.74042

Citation

Sofonea, Mircea; Arhab, Rachid; Tarraf, Raafat. Analysis of electroelastic frictionless contact problems with adhesion. J. Appl. Math. 2006 (2006), Article ID 64217, 25 pages. doi:10.1155/JAM/2006/64217. https://projecteuclid.org/euclid.jam/1168975521


Export citation

References

  • K. T. Andrews, L. Chapman, J. R. Fernández, M. Fisackerly, M. Shillor, L. Vanerian, and T. VanHouten, A membrane in adhesive contact, SIAM Journal on Applied Mathematics 64 (2003), no. 1, 152--169.
  • K. T. Andrews and M. Shillor, Dynamic adhesive contact of a membrane, Advances in Mathematical Sciences and Applications 13 (2003), no. 1, 343--356.
  • R. C. Batra and J. S. Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity 38 (1995), no. 2, 209--218.
  • P. Bisegna, F. Lebon, and F. Maceri, The unilateral frictional contact of a piezoelectric body with a rigid support, Contact Mechanics (Praia da Consolação, 2001) (J. A. C. Martins and M. D. P. Monteiro Marques, eds.), Solid Mech. Appl., vol. 103, Kluwer Academic, Dordrecht, 2002, pp. 347--354.
  • T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electroelasticity, Memoirs on Differential Equations and Mathematical Physics 10 (1997), 1--53.
  • O. Chau, J. R. Fernández, M. Shillor, and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion, Journal of Computational and Applied Mathematics 159 (2003), no. 2, 431--465.
  • O. Chau, M. Shillor, and M. Sofonea, Dynamic frictionless contact with adhesion, Zeitschrift für Angewandte Mathematik und Physik 55 (2004), no. 1, 32--47.
  • M. Cocu and R. Rocca, Existence results for unilateral quasistatic contact problems with friction and adhesion, Mathematical Modelling and Numerical Analysis 34 (2000), no. 5, 981--1001.
  • J. R. Fernández, M. Shillor, and M. Sofonea, Analysis and numerical simulations of a dynamic contact problem with adhesion, Mathematical and Computer Modelling 37 (2003), no. 12-13, 1317--1333.
  • M. Frémond, Équilibre de structures qui adhèrent à leur support, Comptes Rendus des Séances de l'Académie des Sciences. Série II 295 (1982), no. 11, 913--916.
  • --------, Adhérence des solides, Journal de Mécanique Théorique et Appliquée 6 (1987), no. 3, 383--407.
  • --------, Non-Smooth Thermomechanics, Springer, Berlin, 2002.
  • W. Han, K. L. Kuttler, M. Shillor, and M. Sofonea, Elastic beam in adhesive contact, International Journal of Solids and Structures 39 (2002), no. 5, 1145--1164.
  • W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Rhode Island; International Press, Massachusetts, 2002.
  • T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1990.
  • F. Maceri and P. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Mathematical and Computer Modelling 28 (1998), no. 4--8, 19--28.
  • M. Raous, L. Cangémi, and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact, Computer Methods in Applied Mechanics and Engineering 177 (1999), no. 3-4, 383--399.
  • J. Rojek and J. J. Telega, Contact problems with friction, adhesion and wear in orthopaedic biomechanics. I: general developments, Journal of Theoretical and Applied Mechanics 39 (2001), 655--677.
  • J. Rojek, J. J. Telega, and S. Stupkiewicz, 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 39 (2001), 679--706.
  • M. Shillor, M. Sofonea, and J. J. Telega, Models and Variational Analysis of Quasistatic Contact, Lecture Notes in Physics, vol. 655, Springer, Berlin, 2004.
  • M. Sofonea and El-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Mathematical Modelling and Analysis 9 (2004), no. 3, 229--242.
  • --------, Quasistatic frictional contact of a viscoelastic piezoelectric body, Advances in Mathematical Sciences and Applications 14 (2004), no. 2, 613--631.
  • M. Sofonea, W. Han, and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), vol. 276, Chapman & Hall/CRC, Florida, 2006.
  • M. Sofonea and T.-V. Hoarau-Mantel, Elastic frictionless contact problems with adhesion, Advances in Mathematical Sciences and Applications 15 (2005), no. 1, 49--68.
  • C. Talon and A. Curnier, A model of adhesion added to contact with friction, Contact Mechanics (Praia da Consolação, 2001) (J. A. C. Martins and M. D. P. Monteiro Marques, eds.), Solid Mech. Appl., vol. 103, Kluwer Academic, Dordrecht, 2002, pp. 161--168.