and Computational Methods

doi: 10.18698/2309-3684-2015-1-5066

The dispersion relation for a symmetric 3-layered elastic plate is derived and analysed, both numerically and asymptotically. Each layer is assumed to be composed of a linear isotropic elastic material. Numerical solutions of the relation are first presented. After presentation of these numerical solutions, particular focus is applied to the short wave regime, within which appropriate asymptotic approximations are established. These are shown to provide excellent agreement with the numerical solution over a surprisingly larger than might be expected wave number regime. It is envisaged that these solutions might offer some potential for estimation of truncation error for wave number integrals and thereby enable the development of hybrid numerical-asymptotic methods to determine transient structural response to impact.

[1] Kaplunov J.D., Kossovich L.Yu. and Nolde E.V. Dynamics of thin walled elastic bodies. San-Diego, Academic Press, 1998, 226 p.

[2] Khanh C.L. Vibrations of shells and rods. Berlin, Springer, 1999, 423 p.

[3] Kossovitch L.Yu. and Rogerson G.A. Approximations for the dispersion relation for a plate composed of transversely isotropic elastic material. Journal of Sound and Vibration, 1999, vol. 225, pp. 283–305.

[4] Kaplunov J.D., Nolde E.V., and Rogerson G.A. A low frequency model for dynamic motion in a pre-stressed incompressible elastic plate. Proceedings of the Royal Society of London, Series A, 2000, vol. 456, pp. 2589–2610.

[5] Kaplunov J.D. Long-wave vibrations of a thin walled body with fixed faces. Quarterly Journal of Mechanics and Applied Mathematics, 1995, vol. 48, pp. 311–327.

[6] Kaplunov J.D., Nolde E.V., and Rogerson G.A. An asymptotically consistent model for long wave high frequency motion in a pre-stressed elastic plate. Mathematics and Mechanics of Solids, 2002, vol. 7, pp. 581–606.

[7] Kaplunov J.D., Nolde E.V., and Rogerson G.A. An asymptotic short wave approximation for waves for an elastic layer. IMA Journal of Applied Mathematics, 2002, vol. 67, pp. 383–399.

[8] Ryazantseva M.Yu. Vysokochastotnye kolebaniya trekhsloynykh plastin simmetrichnogo stroeniya [High frequency vibration of three layer symmetric plates]. Izvestiya Akademii nauk SSSR. Mekhanika tverdogo tela – Proceedings of the USSR Academy of Sciences. Mechanics of Solids, 1989, no. 5, pp. 175–181.

[9] Ryazantseva M.Yu. O dispersii voln v beskonechnoy uprugoy nrekhsloynoy plastine [Оn wave dispersion in infinite elastic three layer plates]. Izvestiya Rossiyskoy Akademii nauk. Mekhanika tverdogo tela – Proceedings of the Russian Academy of Sciences. Mechanics of Solids, 1998, no. 1, pp. 166–172.

[10] Berdichevski, V.L. An asymptotic theory of sandwich plates. International Journal of Engineering Science, 2010, vol. 48, pp. 383–404.

[11] Dowaikh M.A. and Ogden R.W. On surface waves and deformation in a prestressed incompressible elastic solid. IMA Journal of Applied Mathematics, 1990, vol. 44, pp. 261–284.

[12] Lord Rayleigh. On waves propagated along the plane surface of an elastic solid. Proceedings of the London Mathematical Society, 1885, vol. 17, pp. 4–11.

[13] Stoneley R. Elastic waves at the surface of separation of two solids. Proceedings of the Royal Society London, Series A, 1924, vol. 106, pp. 416–428.

[14] Kiselev A.P., Parker D.F. Omni-directional Rayleigh, Stoneley and Scholte waves with general time dependence. Proceedings of the Royal Society London, Series A, 2010, vol. 466, pp. 2241–2258.

Lashab M., Rogerson G., Sandiford K. A short wave asymptotic analysis of the dispersion relation for a symmetric three-layered elastic plate. Маthematical Modeling and Coтputational Methods, 2015, №1 (5), pp. 50-66

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