Skip to main content
Log in

Long-wave oscillations and waves in anisotropic beams

  • Mechanics
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

The asymptotic integrating method is used to investigate long-wave oscillations and waves in an infinite heterogeneous (with respect to width) anisotropic beam-belt. A dispersion equation of the second-order accuracy with respect to the relative width of the beam-belt is constructed and additional qualitative effects related to the anisotropy are found.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Kirchhoff, Vorlesungen über Matematische Physik. Mechanik (B.G. Teubner, Leipzig, 1876).

    MATH  Google Scholar 

  2. A. E. H. Love, A Treatise on the Mathematical Theory Elasticity (Dover, New York. 1927; ONTI NKTP, Moscow, 1935).

    Google Scholar 

  3. S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars,” Philos. Mag. Ser. 6 41, 744–746 (1921).

    Article  Google Scholar 

  4. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” J. Appl. Mech. 12, 68–77 (1945).

    MathSciNet  MATH  Google Scholar 

  5. V. A. Rodionova, B. F. Titaev, and K. F. Chernykh, Applied theory of Anisotropic Plates and Shells (St.-Petersb. Gos. Univ., St. Petersburg, 1996) [in Russian].

    Google Scholar 

  6. I. N. Vekua, “On one method of calculating prismatic shells,” Tr. Tbilis. Mat. Inst. 21, 191–259 (1955) [in Russian].

    MathSciNet  Google Scholar 

  7. A. L. Gol’denveizer, Theory of Elastic Thin Shells (Nauka, Moscow, 1976; Pergamon, Oxford, 1961).

    Google Scholar 

  8. L. A. Agalovyan, Asymptotic Theory of Anisotropic Plates and Shells (Nauka, Moscow, 1997; World Sci., Singapore, 2005).

    Google Scholar 

  9. S. A. Nazarov, Asymptotic Analysis of Thin Plates and Rods (Nauchn. Kn., Novosibirsk, 2002), Vol. 1 [in Russian].

    Google Scholar 

  10. B. A. Zimin, and I. S. Zorin, “On flat equilibrium shapes of inhomogeneous anisotropic elastic plates and rods,” Vestn. St. Petersburg Univ.: Math. 41, 271–274 (2015).

    Article  MathSciNet  Google Scholar 

  11. Y. Vetyukov, A. Kuzin, and M. Krommer, “Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates,” Int. J. Solids Struct. 48, 12–23 (2011).

    Article  MATH  Google Scholar 

  12. P. E. Tovstik, “Models of plates made of an anisotropic material,” Dokl. Phys. 54, 205–209 (2009).

    Article  Google Scholar 

  13. P. E. Tovstik and T. P. Tovstik, “Two-dimensional linear model of elastic shell accounting for general anisotropy of material,” Acta Mech. 225, 647–661 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. P. E. Tovstik and T. P. Tovstik, “Two-dimensional model of plate made of anisotropic inhomogeneous material,” in Proc. Int. Conf. on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014), Rhodes, Greece, Sept. 22–28, 2014, AIP Conf. Proc. 1648, 300011 (2015).

    Google Scholar 

  15. K. Vijayakumar, “Poisson’s theory for analysis of bending of isotropic and anisotropic plates,” ISRN Civ. Eng. 2013, 562482 (2013). doi doi 10.1155/2013/562482

    Google Scholar 

  16. K. L. Verma, “Wave propagation in laminated composite plates,” Int. J. Adv. Struct. Eng. 5, 10 (2013). doi doi 10.1186/2008-6695-5-10

    Article  Google Scholar 

  17. S. V. Kuznetsov, “Lamb waves in anisotropic plates (review),” Acoust. Phys. 60, 95–103 (2014).

    Article  Google Scholar 

  18. R. Kienzler and P. Schneider, “Comparison of various linear plate theories in the light of a consistent secondorder approximation,” in Proc. 10th Shell Structures: Theory and Applications Conf. (SSTA 2013), Gdańsk, Poland, Oct. 16–18, 2013, Ed. by W. Pietraszkiewicz and J. Górski (CRC, London, 2014), Vol. 3, pp. 109–112.

    Google Scholar 

  19. P. E. Tovstik and T. P. Tovstik, “A thin-plate bending equation of second-order accuracy,” Dokl. Phys. 59, 389–392 (2014).

    Article  Google Scholar 

  20. N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, “Generalized Timoshenko–Reissner model for a multilayer plate,” Mech. Solids (Engl. Transl.) 51, 527–537 (2016).

    Article  Google Scholar 

  21. P. E. Tovstik and T. P. Tovstik, “Generalized Timoshenko–Reissner models for beams and plates, strongly heterogeneous in the thickness direction,” Z. Angew. Math. Mech. (in press).

  22. V. M. Babich and A. P. Kiselev, Elastic Waves. High-Frequency Theory (BKhV-Petersburg, St. Petersburg, 2014) [in Russian].

    Google Scholar 

  23. L. Pochhammer, “Biegung des Kreiscylinders-Fortpflanzungs-Geschwindigkeit kleiner Schwingungen in einem Kreiscylinder,” J. Reine Angew. Math. 81 (1876).

    Google Scholar 

  24. C. Chree, “Longitudinal vibrations in solid and hollow cylinders,” Phil. Mag. 47, 333–349 (1899).

    Article  MATH  Google Scholar 

  25. P. E. Tovstik and T. P. Tovstik, “Two-dimensional models of anisotropic plates,” in Proc. Semin. on Computer Methods in Continuum Mechanics (S.-Petersb. Gos. Univ., St. Petersburg., 2008), pp. 4–16 [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. E. Tovstik.

Additional information

Original Russian Text © P.E. Tovstik, T.P. Tovstik, N.V. Naumova, 2017, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2017, Vol. 62, No. 2, pp. 149–161.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tovstik, P.E., Tovstik, T.P. & Naumova, N.V. Long-wave oscillations and waves in anisotropic beams. Vestnik St.Petersb. Univ.Math. 50, 198–207 (2017). https://doi.org/10.3103/S1063454117020121

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1063454117020121

Keywords

Navigation