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On 3D theory of acoustic metamaterials with a triple-periodic system of interior obstacles

  • M. A. SumbatyanEmail author
  • M. Y. Remizov
Original Article
  • 11 Downloads

Abstract

The paper is devoted to a semi-analytical method, to develop analytical expressions for the wave field and the scattering parameters—the reflection and transmission coefficients, when a longitudinal plane wave falls on the system of a finite number of identical doubly periodic gratings parallel to each other, each of them consisting of a periodic array of rectangular cracks in the elastic isotropic medium. In the one-mode frequency range, the problem is reduced to a system of hypersingular integral equations whose solution gives the physical parameters as well as an explicit representation of the wave field inside the structure. The present work continues to study the 3-D problem for arbitrary finite number of such plane arrays which form a triple-periodic system of cracks. The principal aim is to generalize the results obtained previously for elastic 2-D and 3-D problems. Some presented diagrams emphasize new physical properties of the cascading structure.

Keywords

Triple-periodic array of cracks One-mode regime Hypersingular integral equations Semi-analytical method Reflection and transmission coefficients Acoustic filters 

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Notes

Acknowledgments

The first author is thankful to the Russian Science Foundation (RSCF) for the support by Project 15-19-10008-P.

References

  1. 1.
    Shenderov, E.L.: Propagation of sound through a screen of arbitrary wave thickness with gaps. Sov. Phys. Acoust. 16(1), 115–131 (1970)Google Scholar
  2. 2.
    Angel, Y.C., Achenbach, J.D.: Harmonic waves in an elastic solid containing a doubly periodic array of cracks. Wave Motion 9, 377–385 (1987)CrossRefzbMATHGoogle Scholar
  3. 3.
    Angel, Y.C., Bolshakov, A.: In-plane waves in an elastic solid containing a cracked slab region. Wave Motion 31, 297–315 (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Achenbach, J.D., Li, Z.L.: Reflection and transmission of scalar waves by a periodic array of screens. Wave Motion 8, 225–234 (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Miles, J.W.: On Rayleigh scattering by a grating. Wave Motion 4, 285–292 (1982)CrossRefGoogle Scholar
  6. 6.
    Yang, C., Achenbach, J.D.: Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces. Int. J. Eng. Sci. 115, 43–50 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Scarpetta, E., Sumbatyan, M.A.: On the oblique wave penetration in elastic solids with a doubly periodic array of cracks. Q. Appl. Math. 58, 239–250 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sumbatyan, M.A.: Low-frequency penetration of acoustic waves through a periodic arbitrary shaped grating: the three-dimensional problem. Wave Motion 22, 133–144 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Scarpetta, E., Tibullo, V.: Explicit results for scattering parameters in three-dimensional wave propagation through a doubly periodic system of arbitrary openings. Acta Mech. 185, 1–9 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Scarpetta, E., Tibullo, V.: On the three-dimensionl wave propagation through cascading screens having a periodic system of arbitrary openings. Int. J. Eng. Sci. 46, 105–111 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Scarpetta, E.: In-plane problem for wave propagation through elastic solids with a periodic array of cracks. Acta Mech. 154, 179–187 (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    Remizov, M.Y., Sumbatyan, M.A.: One-mode penetration of elastic waves through a doubly periodic array of cracks. Mech. Solids 3, 67–80 (2018)zbMATHGoogle Scholar
  13. 13.
    Remizov, M.Y., Sumbatyan, M.A.: 3-D one-mode penetration of elastic waves through a doubly periodic array of cracks. Math. Mech. Solids 23(4), 636–650 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sumbatyan, M.A., Remizov, M.Y.: On the theory of acoustic metamaterials with a triple-periodic system of interior obstacles. Springer Proc. Phys. 175, 459–474 (2016)CrossRefGoogle Scholar
  15. 15.
    Homentcovschi, D., Miles, R.N.: Influence of viscosity on the diffraction of sound by a periodic array of screens. The general 3-D problem. J. Acoust. Soc. Am. 117(5), 2761–2771 (2005)ADSCrossRefGoogle Scholar
  16. 16.
    Sotiropoulos, D.A., Achenbach, J.D.: Ultrasonic reflection by a planar distribution of cracks. J. NDE 7, 123–129 (1988)Google Scholar
  17. 17.
    Mykhaskiv, V.V., Zhbadynskyi, I.Y., Zhang, C.: Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks. J. Math. Sci. 203, 114–122 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289(5485), 1734–1736 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    Craster, R.V., Guenneau, S.: Acoustic Metamaterials. Springer Series in Materials Science, vol. 166. Springer, Dordrecht (2013)Google Scholar
  20. 20.
    Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47, 610–617 (2009)CrossRefGoogle Scholar
  21. 21.
    Han, Z., Mogilevskaya, S.G., Schillinger, D.: Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann–Ogden interfaces. Int. J. Solids Struct. 147, 166–182 (2018)CrossRefGoogle Scholar
  22. 22.
    dellIsola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D., Giorgio, I., Andreaus, U., Turco, E., Golaszewski, M., Rizzi, N., Boutin, C., Eremeyev, V.A., Misra, A.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Therm. 1–34 (2018)Google Scholar
  23. 23.
    Sumbatyan, M.A., Brigante, M.: Analysis of strength and wave velocity for micro-damaged elastic media. Eng. Fract. Mech. 145, 43–53 (2015)CrossRefGoogle Scholar
  24. 24.
    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  25. 25.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 1. Gordon & Breach, Amsterdam (1986)zbMATHGoogle Scholar
  26. 26.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1965)zbMATHGoogle Scholar
  27. 27.
    Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. Wiley, London (1969)zbMATHGoogle Scholar
  28. 28.
    Belotserkovsky, S.M., Lifanov, I.K.: Method of Discrete Vortices. CRC Press, Boca Raton (1992)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Mechanics and Computer ScienceSouthern Federal UniversityRostov-on-DonRussia

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