Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Zemlyanukhin A. I., Bochkarev A. V., Ratushny A. V., Chernenko A. V. Generalized model of nonlinear elastic foundation and longitudinal waves in cylindrical shells. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 2, pp. 196-204. DOI: 10.18500/1816-9791-2022-22-2-196-204

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 488)
Article type: 

Generalized model of nonlinear elastic foundation and longitudinal waves in cylindrical shells

Zemlyanukhin Alexandr Isaevich, Yuri Gagarin State Technical University of Saratov
Bochkarev Andrey Vladimirovich, Yuri Gagarin State Technical University of Saratov
Ratushny Aleksandr Vasilyevich, Yuri Gagarin State Technical University of Saratov
Chernenko Aleksandr V., Yuri Gagarin State Technical University of Saratov

A non-integrable quasi-hyperbolic sixth-order equation is derived that simulates the axisymmetric propagation of longitudinal waves along the generatrix of a cylindrical Kirchhoff – Love shell interacting with a nonlinear elastic medium. A six-parameter generalized model of a nonlinear elastic medium, which is reduced in particular cases to the models of Winkler, Pasternak, and Hetenyi, is introduced into consideration. The equation was derived by the asymptotic multiscale expansions method under the assumption that the dimensionless parameters of nonlinearity, dispersion, and thinness have the same order of smallness. The use of the introduced model made it possible to reveal additional high-frequency and low-frequency dispersions characterizing the response of the external environment to bending and shear. It is shown that non-classical shell theories should be used to reveal nonlinear effects that compensate for dispersion. It was found that the Pasternak model admits a  “dispersionless” state when the dispersion due to the inertia of normal displacement is compensated by the dispersion generated by the reaction of the nonlinear elastic foundation to shear.

This work was supported by the Russian Foundation for Basic Research (project No. 20-01-00123).
  1. Winkler E. Die Lehre von der Elastizitat und Festigkeit. Prague, Verlag von H. Dominicus, 1867. 388 S. (in German).
  2. Biot M. A. Bending of an infinite beam on an elastic foundation. Journal of Applied Mechanics, 1937, vol. 4, no. 1, pp. A1–A7. https://doi.org/10.1115/1.4008739
  3. Filonenko-Borodich M. Some approximate theories of elastic foundation. Uchenyie Zapiski Moskovkogo Gosudarstuennogo Universiteta. Mekhanika, 1940, vol. 46, pp. 3–18 (in Russian).
  4. Pasternak P. L. On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants. Moscow, Gosstrojizdat, 1954. 56 p. (in Russian).
  5. Hetenyi M. Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering. Ann Arbor, University of Michigan Press, 1958. 255 p.
  6. Vlasov V. Z., Leont’ev N. N. Beams, Plates and Shells on Elastic Foundations. Jerusalem, Israel, Israel Program for Scientific Translations, 1966. 357 p.
  7. Thompson J. M. T. Advances in shell buckling: Theory and experiments. International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 1, Art. 1530001. https://doi.org/10.1142/S0218127415300013
  8. Hunt G. Buckling in space and time. Nonlinear Dynamics, 2006, vol. 43, pp. 29–46. https://doi.org/10.1007/s11071-006-0748-8
  9. Champneys A. R., Hunt G. W., Thompson J. M. T. Localization and solitary waves in solid mechanics. Philosophical Transactions of the Royal Society A, 1997, vol. 355, pp. 2077–2081. https://doi.org/10.1098/rsta.1997.0110
  10. Kerr A. D. On the formal development of elastic foundation models. Ingenieur-Archiv, 1984, vol. 54, no. 6, pp. 455–464. https://doi.org/10.1007/BF00537376
  11. Younesian D., Hosseinkhani A., Askari H., Esmailzadeh E. Elastic and viscoelastic foundations: A review on linear and nonlinear vibration modeling and applications. Nonlinear Dynamics, 2019, vol. 97, pp. 853–895. https://doi.org/10.1007/s11071-019-04977-9
  12. Dillard D., Mukherjee B., Karnal P., Batra R. C., Frechette J. A review of Winkler’s foundation and its profound influence on adhesion and soft matter applications. Soft Matter, 2018, vol. 14, pp. 3669–3683. https://doi.org/10.1039/C7SM02062G
  13. Kaplunov J., Prikazchikov D. A., Rogerson G. A. Edge bending wave on a thin elastic plate resting on a Winkler foundation. Proceedings of the Royal Society A, 2016, vol. 472, Art. 20160178. https://doi.org/10.1098/rspa.2016.0178
  14. Kaplunov J., Nobili A. The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation. Journal of Vibration and Control, 2017, vol. 23, no. 12, pp. 2014–2022. https://doi.org/10.1177/1077546315606838
  15. Indeitsev D. A., Kuklin T. S., Mochalova Yu. A. Localization in a Bernoulli – Euler beam on an inhomogeneous elastic foundation. Vestnik St. Petersburg University: Mathematics, 2015, vol. 48, no. 1, pp. 41–48. https://doi.org/10.3103/S1063454115010069
  16. Indeitsev D. A., Osipova E. V. Localization of nonlinear waves in elastic bodies with inclusions. Acoustical Physics, 2004, vol. 50, pp. 420–426. https://doi.org/10.1134/1.1776219
  17. Erofeev V. I., Leontieva A. V. Dispersion and spatial localization of bending waves propagating in a Timoshenko beam laying on a nonlinear elastic base. Mechanics of Solids, 2021, vol. 56, no. 4, pp. 443–454. https://doi.org/10.3103/S0025654421040051
  18. Erofeev V. I., Leonteva A. V. Localized bending and longitudinal waves in rods interacting with external nonlinear elastic medium. Journal of Physics: Conference Series, 2019, vol. 1348, Art. 012004. https://doi.org/10.1088/1742-6596/1348/1/012004
  19. Zemlyanukhin A. I., Bochkarev A. V. Axisymmetric nonlinear modulated waves in a cylindrical shell. Acoustical Physics, 2018, vol. 64, pp. 408–414. https://doi.org/10.1134/S1063771018040139
  20. Zemlyanukhin A. I., Bochkarev A. V., Andrianov I. V., Erofeev V. I. The Schamel – Ostrovsky equation in nonlinear wave dynamics of cylindrical shells. Journal of Sound and Vibration, 2021, vol. 491, 115752. https://doi.org/10.1016/j.jsv.2020.115752
  21. Stepanyants Y. A. On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons. Chaos, Soliton and Fractals, 2006, vol. 28, no. 1, pp. 193–204. https://doi.org/10.1016/j.chaos.2005.05.020
  22. Volmir A. The Nonlinear Dynamics of Plates and Shells. Foreign Tech. Div., Wright-Patterson AFB, 1974. 450 p.
  23. Bochkarev A. V., Zemlyanukhin A. I., Mogilevich L. I. Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium. Acoustical Physics, 2017, vol. 63, pp. 148–153. https://doi.org/10.1134/S1063771017020026
  24. Ostrovsky L. A. Nonlinear internal waves in a rotating ocean. Okeanologia, 1978, vol. 18, no. 2, pp. 181–191.
  25. Conte R., Musette M. The Painleve Handbook. Springer, Berlin, 2008. https://doi.org/10.1007/978-1-4020-8491-1
  26. Pelinovsky E. N., Didenkulova (Shurgalina) E. G., Talipova T. G., Tobish E., Orlov Yu. F., Zen’kovich A. V. Korteweg – de Vries type equations in applications. Transactions of NNSTU n.a. R. E. Alekseev, 2018, no. 4, pp. 41–47 (in Russian). https://doi.org/10.46960/1816-210X_2018_4_41
  27. Obregon M. A., Stepanyants Yu. A. On numerical solution of the Gardner – Ostrovsky equation. Mathematical Modelling of Natural Phenomena, 2012, vol. 7, no. 2, pp. 113–130. https://doi.org/10.1051/mmnp/20127210
  28. Stepanyants Yu. A. Nonlinear waves in a rotating ocean (The Ostrovsky equation and its generalizations and applications). Izvestiya, Atmospheric and Oceanic Physics, 2020, vol. 56, pp. 16–32. https://doi.org/10.1134/S0001433820010077
  29. Grimshaw R. H. J., Helfrich K., Johnson E. R. The reduced Ostrovsky equation: Integrability and breaking. Studies in Applied Mathematics, 2012, vol. 129, no. 4, pp. 414–436. https://doi.org/10.1111/j.1467-9590.2012.00560.x
  30. Galkin V. N., Stepanyants Yu. A. On the existence of stationary solitary waves in a rotating fluid. Journal of Applied Mathematics and Mechanics, 1991, vol. 55, iss. 6, pp. 939–943. https://doi.org/10.1016/0021-8928(91)90148-N