Skip to main content
Log in

Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A new model for determining band gaps for wave propagation in two-dimensional (2-D) periodic composite structures is developed using a modified couple stress theory. The general equation of motion and boundary conditions in the elasto-dynamics of the modified couple stress theory are first derived by a variational formulation based on Hamilton’s principle. The in-plane and anti-plane wave equations incorporating microstructure effects are then obtained explicitly from the general equation of motion. The plane wave expansion method and the Bloch theorem for periodic media are used to solve the in-plane and anti-plane wave equations, which are reduced to an eigenvalue problem in each case. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new model recovers the classical elasticity-based model when microstructure effects are not considered. To quantitatively illustrate the newly developed model, a parametric study is conducted for 2-D periodic composite structures containing circular and square inclusions. The numerical results reveal that the microstructure effects on the band gaps are significant only when the unit cell size is small for both the composite structures. In addition, it is found that the volume fraction has a significant effect on the band gap size, and the inclusion shape has a large influence on the band gaps.

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

Similar content being viewed by others

References

  1. Billon, K., Zampetakis, I., Scarpa, F., Ouisse, M., Sadoulet-Reboul, E., Collet, M., Perriman, A., Hetherington, A.: Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials. Compos. Struct. 160, 1042–1050 (2017)

    Article  Google Scholar 

  2. Cai, B., Wei, P.: Surface/interface effects on dispersion relations of 2D phononic crystals with parallel nanoholes or nanofibers. Acta Mech. 224, 2749–2758 (2013)

    Article  MathSciNet  Google Scholar 

  3. Cao, Y., Hou, Z., Liu, Y.: Convergence problem of plane-wave expansion method for phononic crystals. Phys. Lett. A 327(2), 247–253 (2004)

    Article  Google Scholar 

  4. Chen, A.L., Wang, Y.S.: Size-effect on band structures of nanoscale phononic crystals. Phys. E 44(1), 317–321 (2011)

    Article  Google Scholar 

  5. Chen, Y., Wang, L.: Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps. Appl. Phys. Lett. 105, 191907-1–191907-5 (2014)

    Google Scholar 

  6. Christensen, J., Kadic, M., Kraft, O., Wegener, M.: Vibrant times for mechanical metamaterials. MRS Commun. 5(03), 453–462 (2015)

    Article  Google Scholar 

  7. Ellis, R.W., Smith, C.W.: A thin-plate analysis and experimental evaluation of couple-stress effects. Exp. Mech. 7, 372–380 (1967)

    Article  Google Scholar 

  8. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  9. Gao, X.-L.: A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. 226, 457–474 (2015)

    Article  MathSciNet  Google Scholar 

  10. Gao, X.-L., Huang, J.X., Reddy, J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224, 2699–2718 (2013)

    Article  MathSciNet  Google Scholar 

  11. Gao, X.-L., Ma, H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)

    Article  MathSciNet  Google Scholar 

  12. Gao, X.-L., Mall, S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38, 855–874 (2001)

    Article  Google Scholar 

  13. Gao, X.-L., Park, S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)

    Article  Google Scholar 

  14. Gao, X.-L., Zhang, G.Y.: A microstructure- and surface energy-dependent third-order shear deformation beam model. Zeitschrift für angewandte Mathematik und Physik ZAMP 66, 1871–1894 (2015)

    Article  MathSciNet  Google Scholar 

  15. Gao, X.-L., Zhang, G.Y.: A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects. Proc. R. Soc. A 472, 20160275-1–20160275-25 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Gourgiotis, P.A., Georgiadis, H.G.: Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J. Mech. Phys. Solids 57, 1898–1920 (2009)

    Article  MathSciNet  Google Scholar 

  17. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  Google Scholar 

  18. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  Google Scholar 

  19. Hsu, J.-C., Wu, T.-T.: Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates. Phys. Rev. B 74, 144303-1–144303-8 (2006)

    Google Scholar 

  20. Ke, L.L., Wang, Y.S.: Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos. Struct. 93, 342–350 (2011)

    Article  Google Scholar 

  21. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50, 256–267 (2012)

    Article  MathSciNet  Google Scholar 

  22. Kittel, C.: Introduction to Solid State Physics, 8th edn. Wiley, New York (2004)

    MATH  Google Scholar 

  23. Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71(13), 2022–2025 (1993)

    Article  Google Scholar 

  24. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  Google Scholar 

  25. Li, L.: Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A 13(9), 1870–1876 (1996)

    Article  Google Scholar 

  26. Li, Y., Wei, P., Zhou, Y.: Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity. Acta Mech. 227, 1005–1023 (2016)

    Article  MathSciNet  Google Scholar 

  27. Liu, W., Chen, J., Liu, Y., Su, X.: Effect of interface/surface stress on the elastic wave band structure of two-dimensional phononic crystals. Phys. Lett. A 376, 605–609 (2012)

    Article  Google Scholar 

  28. Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MathSciNet  Google Scholar 

  29. Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)

    Article  Google Scholar 

  30. Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Wave propagation in relaxed micromorphic continua: modelling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27, 551–570 (2015)

    Article  MathSciNet  Google Scholar 

  31. Matlack, K.H., Bauhofer, A., Krödel, S., Palermo, A., Daraio, C.: Composite 3D-printed meta-structures for low frequency and broadband vibration absorption. Proc. Natl. Acad. Sci. 113(30), 8386–8390 (2016)

    Article  Google Scholar 

  32. Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)

    Article  Google Scholar 

  33. Nikolov, S., Han, C.-S., Raabe, D.: On the origin of size effects in small-strain elasticity of solid polymers. Int. J. Solids Struct. 44, 1582–1592 (2007)

    Article  Google Scholar 

  34. Park, S.K., Gao, X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)

    Article  Google Scholar 

  35. Park, S.K., Gao, X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift für angewandte Mathematik und Physik ZAMP 59, 904–917 (2008)

    Article  MathSciNet  Google Scholar 

  36. Phani, A.S., Woodhouse, J., Fleck, N.A.: Wave propagation in two-dimensional periodic lattices. J. Acoust. Soc. Am. 119(4), 1995–2005 (2006)

    Article  Google Scholar 

  37. Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, Hoboken, New Jersey (2002)

    Google Scholar 

  38. Sigalas, M.M.: Elastic wave band gaps and defect states in two-dimensional composites. J. Acoust. Soc. Am. 101, 1256–1261 (1997)

    Article  Google Scholar 

  39. Sigalas, M.M., Economou, E.N.: Elastic waves in plates with periodically placed inclusions. J. Appl. Phys. 75(6), 2845–2850 (1994)

    Article  Google Scholar 

  40. Susa, N.: Large absolute and polarization-independent photonic band gaps for various lattice structures and rod shapes. J. Appl. Phys. 91, 3501–3510 (2002)

    Article  Google Scholar 

  41. Suzuki, T., Yu, P.K.L.: Complex elastic wave band structures in three-dimensional periodic elastic media. J. Mech. Phys. Solids 46, 115–138 (1998)

    Article  Google Scholar 

  42. Tanaka, Y., Tomoyasu, Y., Tamura, S.-I.: Band structure of acoustic waves in phononic lattices: two-dimensional composites with large acoustic mismatch. Phys. Rev. B 62, 7387–7392 (2000)

    Article  Google Scholar 

  43. Wang, L.: Size-dependent vibration characteristics of fluid-conveying microtubes. J. Fluids Struct. 26, 675–684 (2010)

    Article  Google Scholar 

  44. Wang, Y.-S.: Nonlocal elastic analogy for wave propagation in periodic layered composites. Mech. Res. Commun. 26(6), 719–723 (1999)

    Article  Google Scholar 

  45. Wang, Y.-Z., Li, F.-M., Huang, W.-H., Wang, Y.-S.: Effects of inclusion shapes on the band gaps in two-dimensional piezoelectric phononic crystals. J. Phys. Condens. Matter 19, 496204-1–496204-9 (2007)

    Google Scholar 

  46. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  Google Scholar 

  47. Zhang, G.Y., Gao, X.-L., Bishop, J.E., Fang, H.E.: Band gaps for elastic wave propagation in a periodic composite beam structure incorporating microstructure and surface energy effects. Compos. Struct. 189, 263–272 (2018)

    Article  Google Scholar 

  48. Zhang, G.Y., Gao, X.-L., Guo, Z.Y.: A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium. Acta Mech. 228, 3811–3825 (2017)

    Article  MathSciNet  Google Scholar 

  49. Zhang, G.Y., Gao, X.-L., Wang, J.Z.: A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects. Acta Mech. 226, 4073–4085 (2015)

    Article  MathSciNet  Google Scholar 

  50. Zhen, N., Wang, Y.S., Zhang, C.: Surface/interface effect on band structures of nanosized phononic crystals. Mech. Res. Commun. 46, 81–89 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof.George J.Weng and two anonymous reviewers for their encouragement and helpful comments on an earlier version of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to X.-L. Gao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, G.Y., Gao, XL. & Ding, S.R. Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech 229, 4199–4214 (2018). https://doi.org/10.1007/s00707-018-2207-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2207-2

Navigation