Abstract
Locally resonant phononic crystals (LRPC) are a new type of sound insulating material. Using the plane wave expansion method based on the Bloch theorem, we compute the band structure of two dimensional (2D) phononic crystals (PC) with square and triangular lattices. Such PC typically consists of infinitely long carbon rods coated with silicon rubber and embedded in an elastic background. Computational results show that gaps appear at the lower frequency range, which are lower than those expected from the Bragg mechanism. Those gaps are generated due to local resonances; the optimum gap is obtained by tuning the thickness ratio of the coating layer. The gap created by the LRPC depends on the filling fraction of the coating cylinders.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achaoui, Y., Khelif, A., Benchabane, S., Robert, L., Laude, V.: Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars. Phys. Rev. B 83(10), 104201 (2011)
Bria, D., Assouar, M.B., Oudich, M., Pennec, Y., Vasseur, J., Djafari-Rouhani, B.: Opening of simultaneous photonic and phononic band gap in two-dimensional square lattice periodic structure. J. Appl. Phys. 109(1), 014507 (2011)
De Espinosa, F.M., Jimenez, E., Torres, M.: Ultrasonic band gap in a periodic two-dimensional composite. Phys. Rev. Lett. 80(6), 1208 (1998)
Herring, C.: A new method for calculating wave functions in crystals. Phys. Rev. 57(12), 1169 (1940)
Hou, Z., Assouar, B.M.: Numerical investigation of the propagation of elastic wave modes in a one-dimensional phononic crystal plate coated on a uniform substrate. J. Phys. D Appl. Phys. 42(8), 085103 (2009)
Huang, Z.G., Wu, T.T.: Analysis of wave propagation in phononic crystals with channel using the plane-wave expansion and supercell techniques. In: IEEE Ultrasonics Symposium, 2005. vol. 1, pp. 77–80. IEEE, New York (2005)
Jiang, S., Hu, H., Laude, V.: Low-frequency band gap in cross-like holey phononic crystal strip. J. Phys. D Appl. Phys. 51(4), 045601 (2018)
Khelif, A., Adibi, A.: Phononic Crystals. Springer, Berlin (2015)
Khelif, A., Achaoui, Y., Benchabane, S., Laude, V., Aoubiza, B.: Locally resonant surface acoustic wave band gaps in a two-dimensional phononic crystal of pillars on a surface. Phys. Rev. B 81(21), 214303 (2010)
Kushwaha, M.S.: Band gap engineering in phononic crystals. Recent Res. Dev. Appl. Phys. 2, 743–855 (1999)
Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71(13), 2022 (1993)
Larabi, H., Pennec, Y., Djafari-Rouhani, B., Vasseur, J.O.: Locally resonant phononic crystals with multilayers cylindrical inclusions. J. Phys: Conf. Ser. 92(1), 012112 (2007)
Li, Y.F., Meng, F., Li, S., Jia, B., Zhou, S., Huang, X.: Designing broad phononic band gaps for in-plane modes. Phys. Lett. A 382(10), 679–684 (2018)
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)
Meade, R.D., Brommer, K.D., Rappe, A.M., Joannopoulos, J.D.: Existence of a photonic band gap in two dimensions. Appl. Phys. Lett. 61(4), 495–497 (1992)
Pennec, Y., Vasseur, J.O., Djafari-Rouhani, B., Dobrzyński, L., Deymier, P.A.: Two-dimensional phononic crystals: examples and applications. Surf. Sci. Rep. 65(8), 229–291 (2010)
Plihal, M., Maradudin, A.A.: Photonic band structure of two-dimensional systems: the triangular lattice. Phys. Rev. B 44(16), 8565 (1991)
Sigalas, M.M.: Elastic wave band gaps and defect states in two-dimensional composites. J. Acoust. Soc. Am. 101(3), 1256–1261 (1997)
Sigalas, M.M., Economou, E.N.: Solid State Commun. 86, 141 (1993)
Sigalas, M.M., Economou, E.N.: Elastic waves in plates with periodically placed inclusions. J. Appl. Phys. 75(6), 2845–2850 (1994)
Sigalas, M.M., Economou, E.N.: Attenuation of multiple-scattered sound. EPL (Europhys. Lett.) 36(4), 241 (1996)
Vasseur, J.O., Djafari-Rouhani, B., Dobrzynski, L., Kushwaha, M.S., Halevi, P.: Complete acoustic band gaps in periodic fibre reinforced composite materials: the carbon/epoxy composite and some metallic systems. J. Phys.: Condens. Matter 6(42), 8759 (1994)
Vasseur, J.O., Deymier, P.A., Frantziskonis, G., Hong, G., Djafari-Rouhani, B., Dobrzynski, L.: Experimental evidence for the existence of absolute acoustic band gaps in two-dimensional periodic composite media. J. Phys.: Condens. Matter 10(27), 6051 (1998)
Vasseur, J.O., Deymier, P.A., Djafari-Rouhani, B., Pennec, Y., Hladky-Hennion, A.C.: Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates. Phys. Rev. B 77(8), 085415 (2008)
Zhang, X., Liu, Y., Wu, F., Liu, Z.: Large two-dimensional band gaps in three-component phononic crystals. Phys. Lett. A 317(1–2), 144–149 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sellami, K., Ketata, H. & Ben Ghozlen, M.H. Locally resonant phononic crystals band-gap analysis on a two dimensional phononic crystal with a square and a triangular lattice. Opt Quant Electron 51, 311 (2019). https://doi.org/10.1007/s11082-019-2028-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-019-2028-0