Advertisement

Antisymmetric dark solitary SH waves in a nonlinear heterogeneous plate

  • Dilek DemirkuşEmail author
Article
  • 35 Downloads

Abstract

In the present work, we investigate the propagation of the nonlinear shear horizontal (SH) waves in a plate, which is composed of isotropic, hyperelastic, heterogeneous, and generalized neo-Hookean materials. Using the method of multiple scales, we strike a balance between the nonlinearity and the dispersion, and by then, we see that the nonlinear modulation of these waves can express in terms of a nonlinear Schrödinger equation. We know that this equation has been derived from many areas of physics and has some solitary wave solutions. Therefore, we claim that the antisymmetric dark solitary SH waves exist and propagate in this plate, in addition to considering both the heterogeneous effect and the nonlinear effect on the deformation field for these waves.

Keywords

Nonlinear SH waves Heterogeneous plate Dark solitary SH waves 

Mathematics Subject Classification

00A69 35G30 35B20 35Q55 74B20 65Z99 74E05 74L05 74J35 74J30 

Notes

Acknowledgements

We thank Editor Prof. David Steigmann and the referees for their useful suggestions.

References

  1. 1.
    Demirkuş, D.: Antisymmetric bright solitary SH waves in a nonlinear heterogeneous plate. Z. Angew. Math. Phys. 69(5), 128 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Demirkuş, D.: Symmetric bright solitary SH waves in a nonlinear heterogeneous plate. Z. Angew. Math. Phys. 70(2), 63 (2019)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Demirkuş, D.: Symmetric dark solitary SH waves in a nonlinear heterogeneous plate. Z. Angew. Math. Phys. 70(4), 108 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ahmetolan, S., Teymur, M.: Nonlinear modulation of SH waves in an incompressible hyperelastic plate. Z. Angew. Math. Phys. 58, 457–474 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fu, Y.: On the propagation of nonlinear traveling waves in an incompressible elastic plate. Wave Motion 19, 271–292 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fu, Y., Zeng, Q.: Nonlinear traveling waves in a neo-Hookean plate subjected to a simple shear. Math. Mech. Solids 2, 27–48 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Prikazchikova, L., Aydın, Y.E., Erbaṣ, B., Kaplunov, J.: Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate. Math. Mech. Solids (2018).  https://doi.org/10.1177/1081286518790804 CrossRefGoogle Scholar
  8. 8.
    Craster, R., Joseph, L., Kaplunov, J.: Long-wave asymptotic theories: the connection between functionally graded waveguides and periodic media. Wave Motion 51(4), 581–588 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaplunov, J., Prikazchikov, D.A., Prikazchikova, L.A.: Dispersion of elastic waves in a strongly inhomogeneous three-layered plate. Int. J. Solids Struct. 113–114, 169–179 (2017)CrossRefGoogle Scholar
  10. 10.
    Demirkuş, D., Teymur, M.: Shear horizontal waves in a nonlinear elastic layer overlying a rigid substratum. Hacet. J. Math. Stat. 46(5), 801–815 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Graff, K.F.: Wave Motion in Elastic Solids. Dever Publishing Inc., New York (1975)zbMATHGoogle Scholar
  12. 12.
    Jeffrey, A., Kawahara, T.: Asymptotic Methods in Nonlinear Wave Theory. Pitman, Boston (1981)zbMATHGoogle Scholar
  13. 13.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B 25, 16–43 (1983)CrossRefGoogle Scholar
  14. 14.
    Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic Press, London (1982)zbMATHGoogle Scholar
  15. 15.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Non-linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and LettersBeykent UniversityBuyukcekmece, IstanbulTurkey

Personalised recommendations