Band-Gap Properties of Prestressed Structures

Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 166)

Abstract

The design of periodic and quasiperiodic structures possessing innovative filtering properties for elastic waves opens the way to the realization of elastic metamaterials. In these structures prestress has a strong influence, ‘shifting’ in frequency, but also ‘annihilating’ or ‘nucleating’ band gaps. The effects of prestress are demonstrated with examples involving flexural waves in periodic and quasiperiodic beams and periodic plates. Results highlight that prestress can be employed as a ‘tuning parameter’ for continuously changing vibrational properties of elastic metamaterials.

Keywords

Elastic Foundation Elementary Cell Fibonacci Sequence Flexural Wave Dispersion Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

M.G. and M.B. gratefully acknowledge the support of Italian Ministry of Education, University and Research (PRIN grant No. 2009XWLFKW); D.B. and A.B.M. gratefully acknowledge the support from the European Union FP7 under contract No. PIAP-GA-2011-286110-INTERCER2.

References

  1. 1.
    Aynaou, H., El Boudouti, E.H., Djafari-Rouhani, B., Akjouj, A., Velasco, V.R.: Propagation and localization of acoustic waves in Fibonacci phononic circuits. J. Phys. Condens. Matter 17, 4245–4262 (2005) CrossRefGoogle Scholar
  2. 2.
    Bacon, M.D., Dean, P., Martin, J.L.: Proc. Phys. Soc. 80, 174 (1962) CrossRefGoogle Scholar
  3. 3.
    Bigoni, D., Capuani, D., Bonetti, P., Colli, S.: A novel boundary element approach to time-harmonic dynamics of incremental non-linear elasticity: The role of pre-stress on structural vibrations and dynamic shear banding. Comput. Methods Appl. Mech. Eng. 196, 4222 (2007) CrossRefGoogle Scholar
  4. 4.
    Bigoni, D., Capuani, D.: Time-harmonic Green’s function and boundary integral formulation for incremental nonlinear elasticity: Dynamics of wave patterns and shear bands. J. Mech. Phys. Solids 53, 1163 (2005) CrossRefGoogle Scholar
  5. 5.
    Bigoni, D., Gei, M., Movchan, A.B.: Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics. J. Mech. Phys. Solids 56, 2494–2520 (2008) CrossRefGoogle Scholar
  6. 6.
    Chen, A.L., Wang, Y.S.: Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals. Physica B 392, 369–378 (2007) CrossRefGoogle Scholar
  7. 7.
    Cremer, L., Leilich, H.O.: Zur theorie der biegekettenleiter. Arch. Elektr. Übertrag. 7, 261 (1953) Google Scholar
  8. 8.
    Feynman, R.: The Feynman Lectures on Physics, vol. 2. Addison-Wesley, Reading (1965) Google Scholar
  9. 9.
    Gei, M.: Elastic waves guided by a material interface. Eur. J. Mech. A, Solids 27, 328–345 (2008) CrossRefGoogle Scholar
  10. 10.
    Gei, M.: Wave propagation in quasiperiodic structures: Stop/pass band distribution and prestress effects. Int. J. Solids Struct. 47, 3067–3075 (2010) CrossRefGoogle Scholar
  11. 11.
    Gei, M., Bigoni, D., Franceschini, G.: Thermoelastic small-amplitude wave propagation in nonlinear elastic multilayer. Math. Mech. Solids 9, 555–568 (2004) CrossRefGoogle Scholar
  12. 12.
    Gei, M., Movchan, A.B., Bigoni, D.: Band-gap shift and defect induced annihilation in prestressed elastic structures. J. Appl. Phys. 105, 063507 (2009) CrossRefGoogle Scholar
  13. 13.
    Gei, M., Ogden, R.W.: Vibration of a surface-coated elastic block subject to bending. Math. Mech. Solids 7, 607–629 (2002) CrossRefGoogle Scholar
  14. 14.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975) CrossRefGoogle Scholar
  15. 15.
    Hladky-Hennion, A.-C., Vasseur, J., Dubus, B., Djafari-Rouhani, B., Ekeom, D., Morvan, B.: J. Appl. Phys. 104, 094206 (2008) CrossRefGoogle Scholar
  16. 16.
    Hou, Z., Wu, F., Liu, Y.: Acoustic wave propagating in one-dimensional Fibonacci binary composite systems. Physica B 344, 391–397 (2004) CrossRefGoogle Scholar
  17. 17.
    John, S.: Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486–2489 (1987) CrossRefGoogle Scholar
  18. 18.
    King, P.D.C., Cox, T.J.: Acoustic band gaps in periodically and quasiperiodically modulated waveguides. J. Appl. Phys. 102, 014908 (2007) CrossRefGoogle Scholar
  19. 19.
    Kohmoto, M., Kadanoff, L.P., Tang, C.: Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983) CrossRefGoogle Scholar
  20. 20.
    Kohmoto, M., Oono, Y.: Cantor spectrum for an almost periodic Schroedinger equation and a dynamical map. Phys. Lett. A 102, 145–148 (1984) CrossRefGoogle Scholar
  21. 21.
    Kohmoto, M., Sutherland, B., Iguchi, K.: Localization in optics: Quasiperiodic media. Phys. Rev. Lett. 58, 2436–2438 (1987) CrossRefGoogle Scholar
  22. 22.
    Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022–2025 (1993) CrossRefGoogle Scholar
  23. 23.
    Lin, Y.K.: Free vibrations of a continuous beam on elastic supports. Int. J. Mech. Sci. 4, 409–423 (1962) CrossRefGoogle Scholar
  24. 24.
    Liu, Z., Chan, C. T, Sheng, P.: Analytic model of phononic crystals with local resonances. Phys. Rev. B 71, 014103 (2005) CrossRefGoogle Scholar
  25. 25.
    Liu, Z., Zhang, W.: Bifurcation in band-gap structures and extended states of piezoelectric Thue-Morse superlattices. Phys. Rev. B 75, 064207 (2007) CrossRefGoogle Scholar
  26. 26.
    Mead, D.J.: Wave propagation in continuous periodic structures: Research contributions from Southampton. J. Sound Vib. 190, 495 (1996) CrossRefGoogle Scholar
  27. 27.
    Mead, D.J.: Wave propagation and natural modes in periodic systems. II. Multi-coupled systems, with and without damping. J. Sound Vib. 40, 19 (1975) CrossRefGoogle Scholar
  28. 28.
    Miles, J.W.: Vibrations of beams on many supports. J. Eng. Mech. 82, 1–9 (1956) Google Scholar
  29. 29.
    Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. Lond. A 463, 855 (2007) CrossRefGoogle Scholar
  30. 30.
    Movchan, A.B., Slepyan, L.I.: Band gap Green’s functions and localized oscillations. Proc. R. Soc. Lond. A 463, 2709 (2007) CrossRefGoogle Scholar
  31. 31.
    Ogden, R.W., Steigmann, D.J.: Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation. J. Mech. Phys. Solids 50, 1869–1896 (2002) CrossRefGoogle Scholar
  32. 32.
    Page, J.H., Sukhovich, A., Yang, S., Cowan, M.L., Van Der Biest, F., Tourin, A., Fink, M., Liu, Z., Chan, C.T., Sheng, P.: Phys. Status Solidi B 241, 3454 (2004) CrossRefGoogle Scholar
  33. 33.
    Parnell, W.J.: Effective wave propagation in a prestressed nonlinear elastic composite bar. IMA J. Appl. Math. 72, 223–244 (2007) CrossRefGoogle Scholar
  34. 34.
    Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966 (2000) CrossRefGoogle Scholar
  35. 35.
    Sheng, P., Zhang, X.X., Liu, Z., Chan, C.T.: Physica B 338, 201 (2003) CrossRefGoogle Scholar
  36. 36.
    Sigalas, M.M., Economou, E.N.: Elastic and acoustic-wave band-structure. J. Sound Vib. 158, 377–382 (1992) CrossRefGoogle Scholar
  37. 37.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A 453, 853–877 (1997) CrossRefGoogle Scholar
  38. 38.
    Timoshenko, S.P., Weaver, W., Young, D.H.: Vibration Problems in Engineering. Wiley, New York (1974) Google Scholar
  39. 39.
    Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987) CrossRefGoogle Scholar
  40. 40.
    Yang, S., Page, J.H., Liu, Z., Cowan, M.L., Chan, C.T., Sheng, P.: Phys. Rev. Lett. 93, 024301 (2004) CrossRefGoogle Scholar
  41. 41.
    Zhang, X., Liu, Z.: Appl. Phys. Lett. 85, 341 (2004) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Civil, Environmental and Mechanical EngineeringUniversity of TrentoTrentoItaly
  2. 2.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

Personalised recommendations