Skip to main content
Log in

Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This paper presents a metamaterial plate (metaplate) consisting of a periodic array of holes on a homogeneous thin plate with slot-embedded resonators. The study numerically proves that the proposed model can generate a complete vibrational bandgap in the low-frequency range. A simplified analytical model was proposed for feasibly and accurately capturing the dispersion behavior and first bandgap characteristics in the low-frequency range, which can be used for initial design and bandgap study of the metaplate. A realistic and practical unit metaplate was subsequently designed to verify the analytical model through finite element simulations. The metaplate not only generated a complete vibrational bandgap but also exhibited excellent agreement in both analytical and finite element models for predicting the bandgap characteristics. This study facilitates the design of opening and tuning bandgaps for potential applications such as low-frequency vibration isolation and stress wave mitigation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289, 1734–1736 (2000). https://doi.org/10.1126/science.289.5485.1734

    Article  Google Scholar 

  2. Liu, Z., Chan, C.T., Sheng, P.: Analytic model of phononic crystals with local resonances. Phys. Rev. B 71, 014103 (2005). https://doi.org/10.1103/PhysRevB.71.014103

    Article  Google Scholar 

  3. Yao, S., Zhou, X., Hu, G.: Experimental study on negative effective mass in a 1D mass-spring system. New J. Phys. 10, 043020 (2008). https://doi.org/10.1088/1367-2630/10/4/043020

    Article  Google Scholar 

  4. 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). https://doi.org/10.1016/j.ijengsci.2008.12.007

    Article  Google Scholar 

  5. Yu, D., Liu, Y., Zhao, H., Wang, G., Qiu, J.: Flexural vibration band gaps in Euler–Bernoulli beams with locally resonant structures with two degrees of freedom. Phys. Rev. B 73, 064301 (2006). https://doi.org/10.1103/PhysRevB.73.064301

    Article  Google Scholar 

  6. Yu, D., Liu, Y., Wang, G., Zhao, H., Qiu, J.: Flexural vibration band gaps in Timoshenko beams with locally resonant structures. J. Appl. Phys. 100, 124901 (2006). https://doi.org/10.1063/1.2400803

    Article  Google Scholar 

  7. Huang, H.H., Lin, C.K., Tan, K.T.: Attenuation of transverse waves by using a metamaterial beam with lateral local resonators. Smart Mater. Struct. 25, 085027 (2016). https://doi.org/10.1088/0964-1726/25/8/085027

    Article  Google Scholar 

  8. Huang, G.L., Sun, C.T.: Band gaps in a multiresonator acoustic metamaterial. J. Vib. Acoust. 132, 031003 (2010). https://doi.org/10.1115/1.4000784

    Article  Google Scholar 

  9. Tan, K.T., Huang, H.H., Sun, C.T.: Blast-wave impact mitigation using negative effective mass density concept of elastic metamaterials. Int. J. Impact Eng. 64, 20–29 (2014). https://doi.org/10.1016/j.ijimpeng.2013.09.003

    Article  Google Scholar 

  10. Zhu, R., Liu, X.N., Hu, G.K., Sun, C.T., Huang, G.L.: A chiral elastic metamaterial beam for broadband vibration suppression. J. Sound Vib. 333, 2759–2773 (2014). https://doi.org/10.1016/j.jsv.2014.01.009

    Article  Google Scholar 

  11. Huang, H.H., Sun, C.T.: Locally resonant acoustic metamaterials with 2D anisotropic effective mass density. Philos. Mag. 91, 981–996 (2011). https://doi.org/10.1080/14786435.2010.536174

    Article  Google Scholar 

  12. Zhu, R., Huang, H.H., Huang, G.L., Sun, C.T.: Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. 49, 1477–1485 (2011). https://doi.org/10.1016/j.ijengsci.2011.04.005

    Article  Google Scholar 

  13. Wu, T.T., Huang, Z.G., Tsai, T.C., Wu, T.C.: Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett. 93, 111902 (2008). https://doi.org/10.1063/1.2970992

    Article  Google Scholar 

  14. Oudich, M., Li, Y., Assouar, B.M., Hou, Z.: A sonic band gap based on the locally resonant phononic plates with stubs. New J. Phys. 12, 083049 (2010). https://doi.org/10.1088/1367-2630/12/8/083049

    Article  Google Scholar 

  15. Oudich, M., Senesi, M., Assouar, M.B., Ruzenne, M., Sun, J.H., Vincent, B., Hou, Z., Wu, T.T.: Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates. Phys. Rev. B 84, 165136 (2011). https://doi.org/10.1103/PhysRevB.84.165136

    Article  Google Scholar 

  16. Xiao, Y., Wen, J., Wen, X.: Flexural wave band gaps in locally resonant thin plates with periodically attached spring-mass resonators. J. Phys. D Appl. Phys. 45, 195401 (2012). https://doi.org/10.1088/0022-3727/45/19/195401

    Article  Google Scholar 

  17. Zhu, R., Huang, G.L., Huang, H.H., Sun, C.T.: Experimental and numerical study of guided wave propagation in a thin metamaterial plate. Phys. Lett. A 375, 2863–2867 (2011). https://doi.org/10.1016/j.physleta.2011.06.006

    Article  Google Scholar 

  18. Huang, T.Y., Shen, C., Jing, Y.: Membrane-and plate-type acoustic metamaterials. J. Acoust. Soc. Am. 139(6), 3240–3250 (2016). https://doi.org/10.1121/1.4950751

    Article  Google Scholar 

  19. Ma, F., Huang, M., Wu, J.H.: Ultrathin lightweight plate-type acoustic metamaterials with positive lumped coupling resonant. J. Appl. Phys. 121(1), 015102 (2017). https://doi.org/10.1063/1.4972839

    Article  Google Scholar 

  20. Bilal, O.R., Hussein, M.I.: Trampoline metamaterial: local resonance enhancement by springboards. Appl. Phys. Lett. 103(11), 111901 (2013). https://doi.org/10.1063/1.4820796

    Article  Google Scholar 

  21. Ma, J., Hou, Z., Assouar, B.M.: Opening a large full phononic band gap in thin elastic plate with resonant units. J. Appl. Phys. 115(9), 093508 (2014). https://doi.org/10.1063/1.4867617

    Article  Google Scholar 

  22. Li, Y., Chen, T., Wang, X., Xi, Y., Liang, Q.: Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial. Phys. Lett. A 379(5), 412–416 (2015). https://doi.org/10.1016/j.physleta.2014.11.028

    Article  Google Scholar 

  23. Li, Y., Zhu, L., Chen, T.: Plate-type elastic metamaterials for low-frequency broadband elastic wave attenuation. Ultrasonics 73, 34–42 (2017). https://doi.org/10.1016/j.ultras.2016.08.019

    Article  Google Scholar 

  24. Wang, Y.F., Wang, Y.S., Su, X.X.: Large bandgaps of two-dimensional phononic crystals with cross-like holes. J. Appl. Phys. 110, 113520 (2011). https://doi.org/10.1063/1.3665205

    Article  Google Scholar 

  25. Wang, Y.F., Wang, Y.S.: Complete bandgaps in two-dimensional phononic crystal slabs with resonators. J. Appl. Phys. 114, 043509 (2013). https://doi.org/10.1063/1.4816273

    Article  Google Scholar 

  26. Wang, Y.F., Wang, Y.S.: Multiple wide complete bandgaps of two-dimensional phononic crystal slabs with cross-like holes. J. Sound Vib. 332(8), 2019–2037 (2013). https://doi.org/10.1016/j.jsv.2012.11.031

    Article  Google Scholar 

  27. Wang, Y.F., Wang, Y.S., Zhang, C.: Two-dimensional locally resonant elastic metamaterials with chiral comb-like interlayers: bandgap and simultaneously double negative properties. J. Acoust. Soc. Am. 139(6), 3311–3319 (2016). https://doi.org/10.1121/1.4950766

    Article  Google Scholar 

  28. Beli, D., Arruda, J.R.F., Ruzzene, M.: Wave propagation in elastic metamaterial beams and plates with interconnected resonators. Int. J. Solids Struct. (2018). https://doi.org/10.1016/j.ijsolstr.2018.01.027

  29. Baravelli, E., Ruzzene, M.: Internally resonating lattices for bandgap generation and low-frequency vibration control. J. Sound Vib. 332, 6562–6579 (2013). https://doi.org/10.1016/j.jsv.2013.08.014

    Article  Google Scholar 

  30. Peng, H., Pai, P.F.: Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression. Int. J. Mech. Sci. 89, 350–361 (2014). https://doi.org/10.1016/j.ijmecsci.2014.09.018

    Article  Google Scholar 

  31. Gao, N., Wu, J.H., Yu, L.: Research on bandgaps in two-dimensional phononic crystal with two resonators. Ultrasonics 56, 287–293 (2015). https://doi.org/10.1016/j.ultras.2014.08.006

    Article  Google Scholar 

  32. Li, Y., Chen, T., Wang, X., Yu, K., Song, R.: Band structures in two-dimensional phononic crystals with periodic Jerusalem cross slot. Phys. B 456, 261–266 (2015). https://doi.org/10.1016/j.physb.2014.08.035

    Article  Google Scholar 

  33. Frandsen, N.M.M., Bilal, O.R., Jensen, J.S., Hussein, M.I.: Inertial amplification of continuous structures: large band gaps from small masses. J. Appl. Phys. 119, 124902 (2016). https://doi.org/10.1063/1.4944429

    Article  Google Scholar 

  34. Li, B., Tan, K.T.: Asymmetric wave transmission in a diatomic acoustic/elastic metamaterial. J. Appl. Phys. 120, 075103 (2016). https://doi.org/10.1063/1.4961209

    Article  Google Scholar 

  35. Qureshi, A., Li, B., Tan, K.T.: Numerical investigation of band gaps in 3D printed cantilever-in-mass metamaterials. Sci. Rep. 6, 28314 (2016). https://doi.org/10.1038/srep28314

    Article  Google Scholar 

  36. Colquitt, D.J., Colombi, A., Craster, R.V., Roux, P., Guenneau, S.R.L.: Seismic metasurfaces: sub-wavelength resonators and Rayleigh wave interaction. J. Mech. Phys. Solids 99, 379–393 (2017). https://doi.org/10.1016/j.jmps.2016.12.004

    Article  MathSciNet  Google Scholar 

  37. Love, A.E.H.: The small free vibrations and deformations of elastic shells. Philos. Trans. A Math. Phys. Eng. Sci. 179, 491–549 (1888). https://doi.org/10.1098/rsta.1888.0016

    Article  MATH  Google Scholar 

  38. Osterberg, H., Cookson, J.W.: A theory of two-dimensional longitudinal and flexural vibrations in rectangular isotropic plates. J. Appl. Phys. 6, 234–246 (1935). https://doi.org/10.1063/1.1745325

    MATH  Google Scholar 

Download references

Acknowledgements

HH Huang acknowledges the support (Grant No. 106-2221-E-002-018-MY3) provided by the Ministry of Science and Technology (MOST), Taiwan.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hsin-Haou Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, JH., Huang, HH. Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators. Arch Appl Mech 88, 1263–1274 (2018). https://doi.org/10.1007/s00419-018-1371-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1371-0

Keywords

Navigation