Advertisement

Acta Mechanica

, Volume 230, Issue 6, pp 2187–2200 | Cite as

Dispersion feature of elastic waves in a 1-D phononic crystal with consideration of couple stress effects

  • Yueqiu LiEmail author
  • Peijun Wei
  • Changda Wang
Original Paper
  • 37 Downloads

Abstract

The couple stress effects upon the propagation behavior and the dispersive feature of Bloch waves in the periodically laminated structure are studied in this paper. First, the various modes of wave motion in the elastic solid with consideration of the couple stress are formulated. Apart from the dispersive P wave, SV wave, and SH wave, there are two evanescent waves, namely SS wave and SSH wave. These modes of wave motion are considered to formulate the state vector and further the transfer matrix in each layer. Then, the non-classic interface continuity conditions with consideration of couple stress effects are used to derive the transfer matrix of the state vector in a typical single cell. At last, the Bloch theorem is used to give the dispersive equations of Bloch waves in the periodical structure. The in-plane Bloch waves and the anti-plane Bloch waves which propagate either obliquely or vertically are both considered in the present work. The numerical results are obtained by solving the dispersive equations and shown graphically. The influences of the couple stress effects on the propagation and the dispersive feature of Bloch waves are discussed based on the numerical results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The work is supported by Fundamental Research Funds for the Central Universities (FRF-BR-15-026A), National Natural Science Foundation of China (No. 11872105), National Natural Science Foundation of China (No. 51701099), State science and technology support program (No. 2013BAK12B08) and HeiLongJiang Natural Science Fund (No. B2015019), and The Fundamental Research Funds in HeiLongJiang Provincial Universities (No. 135109232).

References

  1. 1.
    Chen, A.L., Wang, Y.S.: Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals. Physica B: Condens. Matter 392, 369–378 (2007)CrossRefGoogle Scholar
  2. 2.
    Chen, A.L., Wang, Y.S., Zhang, C.Z.: Wave propagation in one-dimensional solid-fluid quasi-periodic and aperiodic phononic crystals. Physica B: Condens. Matter 407, 324–329 (2012)CrossRefGoogle Scholar
  3. 3.
    Eringen, A.C.: Mechanics of micromorphic materials. In: Proceedings of the Eleventh International Congress of Applied Mechanics Munich (Germany), Springer, Berlin (1966)Google Scholar
  4. 4.
    Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–924 (1966)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2001)zbMATHGoogle Scholar
  7. 7.
    Fomenko, S.I., Golub, M.V., Zhang, C.Z., Bui, T.Q., Wang, Y.S.: In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals. Int. J. Solids Struct. 51, 2491–2503 (2014)CrossRefGoogle Scholar
  8. 8.
    Gourgiotis, P.A., Georgiadis, H.G., Neocleous, I.: On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity. Wave Motion 50, 437–455 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gourgiotis, P.A., Georgiadis, H.G.: Torsional and SH surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin–Mindlin gradient theory. Int. J. Solids Struct. 62, 217–228 (2015)CrossRefGoogle Scholar
  10. 10.
    Graff, K.F., Pao, Y.H.: The effects of couple-stresses on the propagation and reflection of plane waves in an elastic half-space. J. Sound Vib. 6, 217–229 (1967)CrossRefGoogle Scholar
  11. 11.
    Guo, X., Wei, P.J., Li, L.: Dispersion relations of elastic waves in one-dimensional piezoelectric phononic crystal with mechanically and dielectrically imperfect interfaces. Mech. Mater. 93, 168–183 (2016)CrossRefGoogle Scholar
  12. 12.
    Hussein, M.I., Khajehtourian, R.: Nonlinear Bloch waves and balance between hardening and softening dispersion. Proc. R. Soc. A 474, 20180173 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Khajehtourian, R., Hussein, M.I.: Dispersion characteristics of a nonlinear elastic metamaterial. AIP Adv. 4, 124308 (2014)CrossRefGoogle Scholar
  14. 14.
    Kim, E., Kim, Y.H.N., Yang, J.: Nonlinear stress wave propagation in 3D woodpile elastic metamaterials. Int. J. Solids Struct. 58, 128–135 (2015)CrossRefGoogle Scholar
  15. 15.
    Kim, E., Li, F., Chong, C., Theocharis, G., Yang, J., Kevrekidis, P.G.: Highly nonlinear wave propagation in elastic woodpile periodic structures. Phys. Rev. Lett. 114, 118002 (2015)CrossRefGoogle Scholar
  16. 16.
    Koiter, W.T.: Couple-stresses in the theory of elasticity: I and II. Ned. Akad. Wet. Proc. Ser. B 67, 17–44 (1964)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kumar, R., Kumar, K.: Reflection and transmission at the boundary surface of modified couple stress thermoelastic media. Int. J. Appl. Mech. Eng. 21, 61–81 (2016)CrossRefGoogle Scholar
  18. 18.
    Lan, M., Wei, P.J.: Laminated piezoelectric phononic crystal with imperfect interfaces. J. Appl. Phys. 111, 013505 (2012)CrossRefGoogle Scholar
  19. 19.
    Lan, M., Wei, P.J.: Band gap of piezoelectric/piezomagnetic phononic crystal with graded interlayer. Acta Mech. 225, 1779–1794 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lebon, F., Rizzoni, R.: Higher order interfacial effects for elastic waves in one dimensional phononic crystals via the Lagrange-Hamilton’s principle. Eur. J. Mech. A/Solids 67, 58–70 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Lu, M.H., Feng, L., Chen, Y.F.: Phononic crystals and acoustic metamaterials. Mater. Today 12, 34–42 (2009)CrossRefGoogle Scholar
  23. 23.
    Lydon, J., Jayaprakash, K.R., Ngo, D., Starosvetsky, Y., Vakakis, A.F., Daraio, C.: Frequency bands of strongly nonlinear homogeneous granular systems. Phys. Rev. E 88, 012206 (2013)CrossRefGoogle Scholar
  24. 24.
    Lydon, J., Theocharis, G., Daraio, C.: Nonlinear resonances and energy transfer in finite granular chains. Phys. Rev. E 91, 023208 (2015)CrossRefGoogle Scholar
  25. 25.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple stress in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)CrossRefGoogle Scholar
  26. 26.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)CrossRefzbMATHGoogle Scholar
  28. 28.
    Pang, Y., Jiao, F.Y., Liu, J.X.: Propagation behavior of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with line defects. Acta Mech. Sin. 30, 703–713 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Parfitt, V.R., Eringen, A.C.: Reflection of plane waves from the flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am. 45, 1258–1272 (1969)CrossRefGoogle Scholar
  30. 30.
    Qiu, C., Liu, Z., Shi, J., Chan, C.T.: Directional acoustic source based on the resonant cavity of two-dimensional phononic crystals. Appl. Phys. Lett. 86, 224105 (2005)CrossRefGoogle Scholar
  31. 31.
    Sun, J.Z., Wei, P.J.: Band gaps of 2D phononic crystal with imperfect interface. Mech. Adv. Mater. Struct. 21, 107–116 (2014)CrossRefGoogle Scholar
  32. 32.
    Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, C.D., Wei, P.J., Zhang, P., Li, Y.Q.: Influences of a visco-elastically supported boundary on reflected waves in a couple-stress elastic half-space. Arch. Mech. 69, 131–156 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Wang, C.D., Chen, X.J., Wei, P.J., Li, Y.Q.: Reflection of elastic waves at the elastically supported boundary of a couple stress elastic half-space. Acta Mech. Solida Sin. 30, 154–164 (2017)CrossRefGoogle Scholar
  35. 35.
    Wang, C.D., Chen, X.J., Wei, P.J., Li, Y.Q.: Reflection and transmission of elastic waves through a couple-stress elastic slab sandwiched between two half-spaces. Acta Mech. Sin. 33, 1022–1039 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, Y.Z., Li, F.M., Kishimoto, K., Wang, Y.S., Huang, W.H.: Elastic wave band gaps in magnetoelectroelastic phononic crystals. Wave Motion 46, 47–56 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, Y.Z., Li, F.M., Kishimoto, K., Wang, Y.S., Huang, W.H.: Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress. Eur. J. Mech. A-Solids 29, 182–189 (2010)CrossRefGoogle Scholar
  38. 38.
    Wang, Y.Z., Li, F.M., Wang, Y.S.: Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain. Int. J. Mech. Sci. 106, 357–362 (2016)CrossRefGoogle Scholar
  39. 39.
    Wang, Y.Z., Wang, Y.S.: Active control of elastic wave propagation in nonlinear phononic crystals consisting of diatomic lattice chain. Wave Motion 78, 1–8 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wei, P.J., Zhao, Y.P.: The influence of viscosity on band gaps of 2D phononic crystal. Mech. Adv. Mater. Struct. 17, 383–392 (2010)CrossRefGoogle Scholar
  41. 41.
    Yang, S., Page, J., Liu, Z., Cowan, M., Chan, C., Sheng, P.: Focusing of sound in a 3D phononic crystal. Phys. Rev. Lett. 93, 024301 (2004)CrossRefGoogle Scholar
  42. 42.
    Zhan, Z.Q., Wei, P.J.: Influences of anisotropy on band gaps of 2D phononic crystal. Acta. Mech. Solida Sin. 23, 182–188 (2010)CrossRefGoogle Scholar
  43. 43.
    Zhan, Z.Q., Wei, P.J.: Band gaps of three-dimensional phononic crystal with anisotropic spheres. Mech. Adv. Mater. Struct. 21, 245–254 (2014)CrossRefGoogle Scholar
  44. 44.
    Zhang, P., Wei, P.J., Li, Y.Q.: The elastic wave propagation through the finite and infinite periodic laminated structure of micropolar elasticity. Compos. Struct. 200, 358–370 (2018)CrossRefGoogle Scholar
  45. 45.
    Zhao, Y.P., Wei, P.J.: The band gap of 1D viscoelastic phononic crystal. Comput. Mater. Sci. 46, 603–606 (2009)CrossRefGoogle Scholar
  46. 46.
    Zheng, M., Wei, P.J.: Band gaps of elastic waves 1-D phononic crystals with imperfect interfaces. Int. J. Miner. Metall. Mater. 16, 608–614 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsQiqihar UniversityQiqiharChina
  2. 2.Department of Applied MechanicsUniversity of Science and Technology BeijingBeijingChina
  3. 3.State Key Laboratory of High-Efficient Mining and Safety of Metal MinesUniversity of Science and Technology BeijingBeijingChina

Personalised recommendations