On the 3D elastic wave propagation through a cascading system of three doubly-periodic arrays of co-planar cracks

Authors

  • Sumbatyan M.A. South Federal University, Rostov-on-Don, Russian Federation
  • Remizov M.Yu. South Federal University, Rostov-on-Don, Russian Federation

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-15-4-40-53

Abstract

The paper is devoted to the calculation of the reflection and transmission coefficients, when a plane wave is incident on a three-dimensional system of three parallel doubly-periodic gratings of rectangular cracks in the elastic material. In the one-mode frequency range the problem is reduced to a system of integral equations holding over the single chosen crack. The semi-analytical method previously introduced for three-dimensional scalar and two-dimensional elastic problems gives an explicit representations for the wave field and the scattering parameters.

Keywords:

double-periodic crack array, low-frequency mode, integral equation, transformation of hypersingular integral equation kernel, semi-analytical method, reflection and transmission coefficient, acoustic filter

Acknowledgement

Исследование выполнено при финансовой поддержке Российского научного фонда (проект № 15-19-10008-П).

Author Infos

Mezhlum A. Sumbatyan

д-р физ.-мат. наук, профессор, главный научный сотрудник Южного федерального университета, профессор кафедры теоретической и компьютерной гидроаэродинамики Института математики, механики и компьютерных наук им. Воровича И.И.

e-mail: masumbatyan@sfedu.ru

Michael Yu. Remizov

канд. физ.-мат. наук, доцент, старший научный сотрудник Института математики, механики и компьютерных наук им. Воровича И.И. Южного федерального университета

e-mail: remizov72@mail.ru

References

  1. Angel Y.C., Achenbach J.D. Harmonic waves in an elastic solid containing a doubly periodic array of cracks. Wave Motion, 1987, vol. 9, pp. 377–385.
  2. Scarpetta E., Sumbatyan M.A. On wave propagation in elastic solids with a doubly periodic array of cracks. Wave Motion, 1997, vol. 25, pp. 61–72.
  3. Zarrillo G., Aguiar K. Closed-form low frequency solutions for electromagnetic waves through a frequency selective surface. IEEE Trans. Anten., 1998, vol. AP-35, pp. 1406–1417.
  4. Angel Y.C., Bolshakov A. In-plane waves in an elastic solid containing a cracked slab region. Wave Motion, 2000, vol. 31, pp. 297–315.
  5. Mykhas’kiv V.V., Zhbadynskyi I.Ya., Zhang Ch. Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks. J. Math. Sci., 2014, vol. 203, pp. 114–122.
  6. Liu J., Li L., Xia B., Man X. Fractal labyrinthine acoustic metamaterial in planar lattices. Int. J. Solids Struct., 2018, vol. 132–133, pp. 20–30.
  7. Scarpetta E., Tibullo V. On the three-dimensionl wave propagation through cascading screens having a periodic system of arbitrary openings. Int. J. Eng. Sci., 2008, vol. 46, pp. 105–111.
  8. Remizov M.Yu., Sumbatyan M.A. On 3D theory of acoustic metamaterials with a triple-periodic system of interior obstacles. Proc. of National Academy of Sciences of Armenia. Mechanics, 2017, vol. 70, iss. 4, pp. 35–49.
  9. Liu Z.,Zhang X., Mao Y. Locally resonant sonic materials. Science, 2000, no. 289(5485), pp. 1734–1736.
  10. Babeshko V.A., Babeshko O.M., Evdokimova O.V. O treshchinakh v pokrytiyakh v staticheskikh zadachakh seysmologii i nanomaterialav [About the cracks in the coatings in static problems of seismology and nanomaterialov]. Doklady of Russian Academy of Science, 2013, no 453, iss. 2, pp. 162–166. (In Russian)
  11. Glushkov Ye.V., Glushkova N.V., Golub M.V., Bostrom A.E. Natural resonance frequencies, wave blocking, and energy localization in an elastic half-space and waveguide with a crack. J. Acoust. Soc. Am., 2006, vol. 119, iss. 6, pp. 3589–3598.
  12. Craster R.V. Guenneau S. Acoustic metamaterials. Springer series in materials science. Dordrecht: Springer, 2013, 166 p.
  13. Golub M.V., Doroshenko O.V., Bostrom A.E. Transmission of elastic waves through an interface between dissimilar media with random and periodic distributions of strip-like micro-cracks. Materials Physics and Mechanics, 2018, vol. 37, pp. 52–59.
  14. Babeshko V.A., Ratner S.V., Syromyatnikov P.V. Anizotropnye tela s neodnorodnostyami; sluchay sovokupnosti treshchin [Anisotropic bodies with inhomogeneities, the case of a set of cracks]. News of wounds. Solid mechanics, 2007, no. 5, pp. 49–59. (In Russian)
  15. Belotserkovsky S.M. Lifanov I.K. Chislennye metody v singulyarnykh integral'nykh uravneniyakh i ikh primenenie v aerodinamike, teorii uprugosti, elektrodinamike [Numerical methods in singular integral equations and their application in aerodynamics, elasticity theory, electrodynamics]. Nauka, Moscow, 1985. (In Russian)

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Issue

2018. Vol. 15. No. 4

Section

Mechanics

Pages

40-53

Submitted

2018-09-22

Published

2018-12-21

How to Cite

Sumbatyan M.A., Remizov M.Yu. On the 3D elastic wave propagation through a cascading system of three doubly-periodic arrays of co-planar cracks. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 4, pp. 40-53. DOI: https://doi.org/10.31429/vestnik-15-4-40-53 (In Russian)

Copyright (c) 2018 Sumbatyan M.A., Remizov M.Yu.

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