Skip to main content

Relaxation oscillations and buckling prognosis for shallow thin shells

Abstract

This paper reports the possibilities of predicting buckling of thin shells with nondestructive techniques during operation. It examines shallow shells fabricated from high-strength materials. Such structures are known to exhibit surface displacements exceeding the thickness of the elements. In the explored shells, relaxation oscillations of significant amplitude could be generated even under relatively low internal stresses. The problem of cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, works by many authors who studied the geometry of the surface formed after buckling (postbuckling behavior) were used. The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. This circumstance enables a detailed analysis of the obtained equation to be performed and a description to be given of the physical phenomena—relaxation oscillations—that are unique to thin high-strength shells. It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence at which stable periodic motions are converted into dynamic chaos. The amplitude and period of the relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure. It is emphasized that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Because the amplitude and the frequency of the oscillations are known, this fact enables an experimental facility to be proposed for prediction of shell buckling with nondestructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding the specified limits. Based on the results of the experimental measurements, a formula is obtained to estimate the safety against buckling (safety factor) of the structure.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. 1.

    Wu, J., Wheatley, J., Putterman, S., Rudnick, I.: Observation of envelope solitons in solids. Phys. Rev. Lett. 59(24), 2744–2747 (1987)

    Article  Google Scholar 

  2. 2.

    Maugin, G.A.: Solitons in elastic solids (1938–2010). Mech. Res. Commun. 38(5), 341–349 (2011)

    Article  Google Scholar 

  3. 3.

    Guo, B., Pang, X.F., Wang, Y.F., Liu, N.: Solitons. Walter de Gruyter GmbH, Berlin (2018)

    Book  Google Scholar 

  4. 4.

    Timoshenko, S.P., Gere, G.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill Int. book corp, London (1985)

    Google Scholar 

  5. 5.

    Hutchinson, J.W.: Buckling of spherical shells revisited. Proc. R. Soc. A 472(2195), 20160577 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Virot, E., Kreilos, T., Schneider, T.M., Rubinstein, S.M.: Stability landscape of shell buckling. Phys. Rev. Lett. 119, 224101 (2017)

    Article  Google Scholar 

  7. 7.

    Neustadt, Y.S., Grachev, V.A.: Buckling prognosis for thin elastic shallow shells. Z. für Angew. Math. und Phys. 70(4), 113 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Volmir, A.S.: Stability of Elastic Systems, Moscow, Nauka. English Translation: Foreign Tech-nology, Division, Air Force Systems Command. Wright-Patterson Air Force Base, Ohio (1965)

  9. 9.

    Timoshenko, S.P., Voinovsky-Krieger, I.: Theory of Plates and Shells. McGraw-Hill, NewYork (1989)

    Google Scholar 

  10. 10.

    Donnell, L.H.: Beams, Plates and Shells. McGraw-Hili Book Company, New York (1976)

    MATH  Google Scholar 

  11. 11.

    Moss, R.D., Basic, M.: Pressure Vessel Design Manual, 4th edn. Butter-worth-Heinemann, Amsterdam (2012)

    Google Scholar 

  12. 12.

    Kabritz, S.A., Michailovskii, E.I., Tovstik, P.E., Chernyh, K.F., Shamina, V.F.: General Nonlinear Theory of Elastic Shells. SPb.: Sankt-Petersburg University Publishing (2002). (in Russian)

  13. 13.

    Doerich, C., Rotter, J.M.: Behavior of cylindrical steel shells supported on local brackets. J. Struct. Eng. 134(8), 1269–1277 (2008)

    Article  Google Scholar 

  14. 14.

    Kovacic, I., Brennan, M.J. (eds.): The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley, Chichester (2011)

  15. 15.

    Mishchenko E.F., Rozov N.H.: Differential Equations with Small Parameters and Relaxation Oscillations, Springer, New York. Reprint of the original 1st ed. (1980)

  16. 16.

    Feigenbaum, M.J.: Universal behavior in nonlinear systems. Los Alamos Sci. 1(1), 4–27 (1980)

    MathSciNet  Google Scholar 

  17. 17.

    Kanamaru, T.: Duffing\_oscillator. [Electronic resource]: http://scholarpedia.org/article/ Duffing_oscillator https://doi.org/10.4249/scholarpedia.6327. Accessed 22 Oct 2014

  18. 18.

    Lukasiewicz, S.: Local Loads in Plates and Shells. Noordhoff Int. Publ, Leyden (1979)

    Book  Google Scholar 

  19. 19.

    Darevskii, V.M.: Determination of displacements and stresses in a cylindrical shell at local loads in book “Durability and Dynamics of Aircraft Engines”, v. 1, Moscow, Mashinery, pp. 23–83 (1964) (in Russian)

  20. 20.

    Darevskii, V.M.: Shells under local loading. In handbook “Durability, buckling, oscillations” (Editors: Birger I.A. and Panovko Ya. G), v.2. Moscow, Mashinery, P. 49–96 (1968) (in Russian)

  21. 21.

    Novozhilov, V.V.: Foundations of the Nonlinear Theory of Elasticity. Graylock Press, Rochester (1953)

    MATH  Google Scholar 

  22. 22.

    Koiter W.T.: Elastic stability and post buckling behavior in nonlinear problems. In: Proceedings of the Symposium on Nonlinear Problems. University of Wisconsin Press, Madison, pp. 257–275 (1963)

  23. 23.

    Fermi E., Pasta J., Ulam S.: Studies of nonlinear problems. Los-Alamos scientific report, LA-(1940, 1955). Collected works of Enrico Fermi, The University of Chicago Press. v. 2, pp. 977–988 (Copyright 1965)

  24. 24.

    Thompson, J.M.T.: Advances in Shell Buckling: Theory and Experiments. p. 31. arxiv:1409.3156 (2014)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yu. S. Neustadt.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Neustadt, Y.S., Grachev, V.A. Relaxation oscillations and buckling prognosis for shallow thin shells. Z. Angew. Math. Phys. 71, 142 (2020). https://doi.org/10.1007/s00033-020-01369-7

Download citation

Mathematics Subject Classification

  • 74B20
  • 74K25
  • 74J35
  • 62K99

Keywords

  • Elastic shells
  • Buckling
  • Relaxation oscillations
  • Duffing oscillator
  • Safety factor
  • Experimental buckling prediction