Advertisement

Mechanics of Solids

, Volume 53, Issue 3, pp 295–306 | Cite as

One-Mode Propagation of Elastic Waves through a Doubly Periodic Array of Cracks

  • M. Yu. Remizov
  • M. A. SumbatyanEmail author
Article
  • 5 Downloads

Abstract

The article is devoted to the derivation of analytical expressions for the reflection and propagation coefficients, when a plane longitudinal wave falls on a system of a finite number of consecutively located identical flat gratings, each of which consists of a periodic array of rectilinear cracks in an elastic isotropic medium. The problem is solved in a flat statement. In the mode of single-mode frequency range, the problem is reduced to a system of hypersingular integral equations, the solution of which gives the reflection and propagation coefficients, as well as the representation of the wave field inside the medium.

Keywords

reflection and propagation coefficients doubly periodic lattice hypersingular integral equation crack system acoustic filter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. D. Achenbach and Z. L. Li, “Reflection and Propagation of ScalarWaves by a Periodic Array of Screens,” Wave Motion 8, 225–234 (1986).CrossRefzbMATHGoogle Scholar
  2. 2.
    J.W. Miles, “On Rayleigh Scattering by a Grating,” Wave Motion 4, 285–292 (1982).CrossRefGoogle Scholar
  3. 3.
    E. L. Shenderov, “Sound Propagation through a Hard Screen of Finite Thickness with Holes,” Akust. Zh. 16 (2), 295–304 (1970).Google Scholar
  4. 4.
    Z. Liu, X. Zhang, Y. Mao, et al. “Locally Resonant SonicMaterials,” Science 289 (5485), 1734–1736 (2000).ADSCrossRefGoogle Scholar
  5. 5.
    M. A. Sumbatyan, “Low-Frequency Propagation of Acoustic Waves through a Periodic Arbitrary-Shaped Grating: the Three-Dimensional Problem,” Wave Motion 22, 133–144 (1995).CrossRefzbMATHGoogle Scholar
  6. 6.
    E. Scarpetta and M. A. Sumbatyan, “On Wave Propagation in Elastic Solids with a Doubly Periodic Array of Cracks,” Wave Motion 25, 61–72 (1997).CrossRefzbMATHGoogle Scholar
  7. 7.
    E. Scarpetta and M. A. Sumbatyan, “On the Oblique Wave Propagation in Elastic Solids with a Doubly Periodic Array of Cracks,” Quart. Appl.Math. 58, 239–250 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. Scarpetta, “In-Plane Problem for Wave Propagation through Elastic Solids with a Periodic Array of Cracks,” Acta Mech. 154, 179–187 (2002).CrossRefzbMATHGoogle Scholar
  9. 9.
    E. Scarpetta and V. Tibullo, “On the Three-Dimensional Wave Propagation through Cascading Screens Having a Periodic System of Arbitrary Openings,” Int. J. Engng Sci. 46, 105–118 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. A. Sumbatyan and M. Yu. Remizov, “On the Theory of Acoustic Metamaterials with a Triple-Periodic System of Interior Obstacles,” in M. Sumbatyan (Editor), Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials. Advanced Structured Materials, Vol 59 (Springer, Singapore, 2017), pp. 19–33.Google Scholar
  11. 11.
    M. A. Sumbatyan and M. Yu. Remizov, “Asymptotic Analysis in the Anti-Plane High-Frequency Diffraction by Interface Cracks,” Appl.Math. Lett. 34, 72–75 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Yu. Remizov and M. A. Sumbatyan, “Semi-AnalyticalMethod for Solving Problems of High-Frequency Diffraction of ElasticWaves on Cracks,” Appl.Math. Mech. 77 (4), 629–635 (2013).MathSciNetGoogle Scholar
  13. 13.
    M. A. Sumbatyan, M. Yu. Remizov, and V. Zampoli, “A Semi-Analytical Approach in the High-Frequency Diffraction by Cracks,”Mech. Res. Comm. 38, 607–609 (2011).CrossRefzbMATHGoogle Scholar
  14. 14.
    R. V. Craster and S. Guenneau, Acoustic Metamaterials (Springer, Dordrecht, 2013).CrossRefGoogle Scholar
  15. 15.
    R. C. McPhedran, A. B. Movchan, and N. V. Movchan, “Platonic Crystals: Bloch Bands, Neutrality and Defects,” Mech.Mater. 41, 356–363 (2009).CrossRefGoogle Scholar
  16. 16.
    N. V. Movchan, R. C. McPhedran, A. B. Movchan, and C.G. Poulton, “Wave Scattering by PlatonicGrating Stacks,” Proc. Royal Soc. A 465, 3383–3400 (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. V. Mykhaskiv, I. Ya. Zhbadynskyi, and Ch. Zhang, “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.
    H. H. Huang, C. T. Sun, and G. L. Huang, “On the Negative Effective Mass Density in Acoustic Metamaterials,” Int. J. Engng Sci. 47, 610–617 (2009).CrossRefGoogle Scholar
  19. 19.
    Ch. Yang and J. D. Achenbach, “Time Domain Scattering of Elastic Waves by a Cavity, Represented by Radiation from Equivalent Body Forces,” Int. J. Engng Sci. 115, 43–50 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    W. Nowacki, Theory of Elasticity (PWN,Warsaw, 1970;Mir,Moscow, 1975).zbMATHGoogle Scholar
  21. 21.
    I. N. Sneddon and M. Lowengrub, Crack Problems in the Classical Theory of Elasticity (Wiley, London, 1969).zbMATHGoogle Scholar
  22. 22.
    S. M. Belotserkovsky and I. K. Lifanov, Numerical Methods in Singular Integral Equations and Their Application in Aerodynamics, Elasticity Theory, Electrodynamics (Nauka, Moscow, 1985) [in Russian].Google Scholar
  23. 23.
    E. Scarpetta and V. Tibullo, “P-Wave Propagation through Elastic Solids with a Doubly Periodic Array of Cracks,”Quart. J.Mech. Appl. Math. 58, 535–550 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    G. A. Kriegsmann, “Scattering Matrix Analysis of a Photonic Fabry–Perot Resonator,” Wave Motion 37, 43–61 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    S. K. Datta, “Diffraction of Plane Elastic Waves by Ellipsoidal Inclusions,” J. Acoust. Soc. Am. 61, 1432–1437 (1977).ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    J. R. Willis, “A Polarization Approach to the Scattering of Elastic Waves–II. Multiple Scattering from Inclusions,” J. Mech. Phys. Solids 28, 307–327 (1980).Google Scholar
  27. 27.
    S. V. Kuznetsov, “Elastic Wave Scattering in Porous Media,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 81–86 (1995) [Mech. Solids (Engl. Transl.) 30 (3), 71–76 (1995)].Google Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.South Federal UniversityRostov-on-DonRussia

Personalised recommendations