Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

MATHEMATICAL ANALYSIS OF CONTACT PROBLEM WITH DAMPED RESPONSE OF AN ELECTRO-VISCOELASTIC ROD

  • LAHCEN OUMOUACHA (Laboratory LS2ME, Polydisciplary Faculty of Khouribga) ;
  • YOUSSEF MANDYLY (Laboratory LMAI, ENS of Casablanca, Hassan II University of Casablanca) ;
  • RACHID FAKHAR (Laboratory LS2ME, Polydisciplary Faculty of Khouribga) ;
  • EL HASSAN BENKHIRA (Laboratory MACS, Faculty of Sciences, University Moulay Ismail)
  • Received : 2022.12.27
  • Accepted : 2023.11.17
  • Published : 2024.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

We consider a mathematical model which describes the quasistatic contact of electro-viscoelastic rod with an obstacle. We use a modified Kelvin-Voigt viscoelastic constitutive law in which the elasticity operator is nonlinear and locally Lipschitz continuous, taking into account the piezoelectric effect of the material. We model the contact with a general damped response condition. We establish a local existence and uniqueness result of the solution by using arguments of time-dependent nonlinear equations and Schauder's fixed-point theorem and obtain a global existence for small enough data.

Keywords

References

  1. K.T. Andrews, M. Shillor and S. Wright, A hyperbolic-parabolic system modelling the thermoelastic impact of two rods, Math. Methods Appl. Sci. 17 (1994), 901-908.  https://doi.org/10.1002/mma.1670171105
  2. K.T. Andrews, M. Shillor and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle, J. Elasticity 42 (1996), 1-30.  https://doi.org/10.1007/BF00041221
  3. K.T. Andrews, P. Shi, M. Shillor and S. Wright, Thermoelastic contact with Barber's heat exchange condition, Appl. Math. Optim. 28 (1993), 11-48.  https://doi.org/10.1007/BF01188756
  4. K.L. Kuttler and M. Shillor, A one-dimensional thermoviscoelastic contact problem, Adv. Math. Sci. Appl. 4 (1994), 141-159. 
  5. EL-H. Essoufi, A global existence and uniqueness result of a non linear quadratic Kelvin-Voigt model, Conference MTNS2000, Perpignan, France, June, 2000. 
  6. K. Bartosz and M. Sofonea, Modeling and analysis of a contact problem for a viscoelastic rod, Zeitschrift fur Angewandte Mathematik und Physik 67 (2016), 127. 
  7. M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity 51 (1998), 105-126.  https://doi.org/10.1023/A:1007413119583
  8. M. Rochdi, M. Shillor and M. Sofonea, A quasistatic contact problem with directional friction and damped response, Applic. Anal. 68 (2006), 409-422.  https://doi.org/10.1080/00036819808840639
  9. W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30 (2002), AMS/IP. 
  10. R.C. Batra and J.S. Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity 38 (1995), 209-218.  https://doi.org/10.1007/BF00042498
  11. El. H. Benkhira, R. Fakhar and Y. Mandyly, Analysis and Numerical Approximation of a Contact Problem Involving Nonlinear Hencky-Type Materials with Nonlocal Coulomb's Friction Law, Numerical Functional Analysis and Optimization 40 (2019), 1291-1314.  https://doi.org/10.1080/01630563.2019.1600546
  12. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
  13. J.L. Lions and E. Magenes, Nonhomogenuous Boundary Value Problems and Applications, Vol II, Springer-Verlag, Berlin, 1972. 
  14. H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst. Fourier 18 (1968), 115-175.  https://doi.org/10.5802/aif.280
  15. J. Simon, Compact sets in the space Lp(0, T; B), Ann. Math. Pura. Appli. 4 (1987), 65-96. 
  16. R.E. Edwards, Functional Analysis, Theory and Applications, Reinhart and Winston, Holt, 1965. 
  17. K. Kameyama, T. Inoue, I. Yu. Demin, K. Kobayashi, T. Sato, Acoustical tissue nonlinearity characterization using bispectral analysis, Signal processing 53 (1996), 117-13. 
  18. G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. 
  19. I.R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford, 1993.