Skip to main content
Log in

Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. For example, \((A\cdot v)_{i}=A_{ij}v_{j}\), \((A\cdot B)_{ik}=A_{ij}B_{jk}\), \(A:B=A_{ij}B_{ji}\), \((C\cdot B)_{ijk}=C_{ijp}B_{pk}\), \((C:B)_{i}=C _{ijp}B_{pj}\), \(\left . \langle v,w\right . \rangle=v\cdot w=v_{i}w _{i}\), \(\left . \langle A,B\right . \rangle=A_{ij}B_{ij}\) etc.

  2. This same energy could be used to describe, from a macroscopic point of view, the behavior of band-gap metamaterials obtained using piezoelectric patches, as those presented, e.g., in [54].

  3. In what follows, we will not differentiate anymore the Lagrangian space variable \(X\) and the Eulerian one \(x\). In general, such undifferentiated space variable will be denoted as \(x=(x_{1},x_{2},x _{3})^{T}\).

  4. We retain the first set of values of the micro-inertiae that allows us to obtain a good fitting of the dispersion curves. Given the non-linearity of the relationships linking the parameters of the relaxed model, it cannot be excluded a priori that other values of the micro-inertiae may exist which give an equivalently good fitting. Nevertheless such values would have to be excluded since they would correspond to artificially high values of the micro-inertiae.

  5. It can be checked that the expressions (21) for \(\mu_{\mathrm{micro}}\) and \(\lambda{}_{ \mathrm{micro}}\) together with a choice of \(\eta_{1}\) and \(\eta_{3}\) complying with the conditions (22) imply that \(\mu_{\mathrm{micro}}>0\) and \(3\lambda_{\mathrm{micro}}+2 \mu_{\mathrm{micro}}>0\). Moreover, such conditions on \(\mu_{\mathrm{micro}}\) and \(\lambda{}_{\mathrm{micro}}\) also imply that, given Eqs. (20), \(\mu_{e}>0\) and \(3\lambda_{e}+2\mu_{e}>0\). This means that, in the end, the only fact of using the restrictions (22) and of additionally imposing \(\eta_{2}\geq0\), imply positive definiteness of the strain energy density.

References

  1. Armenise, M.N., Campanella, C.E., Ciminelli, C., Dell’Olio, F., Passaro, V.M.N.: Phononic and photonic band gap structures: Modelling and applications. Phys. Proc. 3(1), 357–364 (2010)

    Article  ADS  Google Scholar 

  2. Auriault, J.L., Boutin, C.: Long wavelength inner-resonance cut-off frequencies in elastic composite materials. Int. J. Solids Struct. 49(23–24), 3269–3281 (2012)

    Article  Google Scholar 

  3. Barbagallo, G., d’Agostino, M.V., Abreu, R., Ghiba, I.-D., Madeo, A., Neff, P.: Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics. Preprint arXiv:1601.03667 (2016)

  4. Bigoni, D., Guenneau, S., Movchan, A.B., Brun, M.: Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization. Phys. Rev. B 87(17), 174303 (2013)

    Article  ADS  Google Scholar 

  5. Blanco, A., Chomski, E., Grabtchak, S., Ibisate, M., John, S., Leonard, S.W., Lopez, C., Meseguer, F., Miguez, H., Mondia, J.P., Ozin, G.A., Toader, O., van Driel, H.M.: Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres. Nature 405(6785), 437–440 (2000)

    Article  ADS  Google Scholar 

  6. Boutin, C., Hans, S.: Homogenisation of periodic discrete medium: Application to dynamics of framed structures. Comput. Geotech. 30(4), 303–320 (2003)

    Article  Google Scholar 

  7. Boutin, C., Hans, S., Chesnais, C.: Generalized beams and continua. Dynamics of reticulated structures. In: Mechanics of Generalized Continua. Advances in Mechanics and Mathematics, vol. 21, pp. 131–141. Springer, New York (2010)

    Chapter  Google Scholar 

  8. Boutin, C., Soubestre, J.: Generalized inner bending continua for linear fiber reinforced materials. Int. J. Solids Struct. 48(3–4), 517–534 (2011)

    Article  MATH  Google Scholar 

  9. Brun, M., Guenneau, S., Movchan, A.B., Bigoni, D.: Dynamics of structural interfaces: Filtering and focussing effects for elastic waves. J. Mech. Phys. Solids 58(9), 1212–1224 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Physica A 322, 359–376 (2003)

    Article  ADS  MATH  Google Scholar 

  11. Chen, Y., Lee, J.D., Eskandarian, A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41(8), 2085–2097 (2004)

    Article  MATH  Google Scholar 

  12. Collet, M., Ouisse, M., Ruzzene, M., Ichchou, M.: Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems. Int. J. Solids Struct. 48(20), 2837–2848 (2011)

    Article  Google Scholar 

  13. Colombi, A., Colquitt, D.J., Roux, P., Guenneau, S., Craster, R.V.: A seismic metamaterial: The resonant metawedge. Sci. Rep. 6(7249), 27717 (2016)

    Article  ADS  Google Scholar 

  14. Colquitt, D.J., Brun, M., Gei, M., Movchan, A.B., Movchan, N.V., Jones, I.S.: Transformation elastodynamics and cloaking for flexural waves. J. Mech. Phys. Solids 72, 131–143 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Craster, R.V., Guenneau, S.: Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. Springer Series in Materials Science. Springer, Berlin (2013), 332 pp

    Book  Google Scholar 

  16. d’Agostino, M.V., Barbagallo, G., Ghiba, I.-D., Madeo, A., Neff, P.: A panorama of dispersion curves for the isotropic weighted relaxed micromorphic model. Preprint arXiv:1610.03296 (2016)

  17. Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)

    Book  MATH  Google Scholar 

  18. Fan, Y., Collet, M., Ichchou, M., Li, L., Bareille, O., Dimitrijevic, Z.: A wave-based design of semi-active piezoelectric composites for broadband vibration control. Smart Mater. Struct. 25(5), 055032 (2016)

    Article  ADS  Google Scholar 

  19. Fan, Y., Collet, M., Ichchou, M., Li, L., Bareille, O., Dimitrijevic, Z.: Energy flow prediction in built-up structures through a hybrid finite element/wave and finite element approach. Mech. Syst. Signal Process. 66–67, 137–158 (2016)

    Article  Google Scholar 

  20. Florescu, M., Torquato, S., Steinhardt, P.J.: Complete band gaps in two-dimensional photonic quasicrystals. Phys. Rev. B, Condens. Matter Mater. Phys. 80(15), 1–7 (2009)

    Article  Google Scholar 

  21. Florescu, M., Torquato, S., Steinhardt, P.J.: Designer disordered materials with large, complete photonic band gaps. Proc. Natl. Acad. Sci. USA 106(49), 20658–20663 (2009)

    Article  ADS  Google Scholar 

  22. Ghiba, I.-D., Neff, P., Madeo, A., Placidi, L., Rosi, G.: The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math. Mech. Solids 20(10), 1171–1197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gonella, S., Greene, M.S., Liu, W.K.: Characterization of heterogeneous solids via wave methods in computational microelasticity. J. Mech. Phys. Solids 59(5), 959–974 (2011)

    Article  ADS  MATH  Google Scholar 

  24. Haberko, J., Scheffold, F.: Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials. Opt. Express 21(1), 1057–1065 (2013)

    Article  ADS  Google Scholar 

  25. Huang, J., Shi, Z.: Attenuation zones of periodic pile barriers and its application in vibration reduction for plane waves. J. Sound Vib. 332(19), 4423–4439 (2013)

    Article  ADS  Google Scholar 

  26. Jiménez, N., Huang, W., Romero-García, V., Pagneux, V., Groby, J.-P.: Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption. Appl. Phys. Lett. 109(12), 121902 (2016)

    Article  ADS  Google Scholar 

  27. Lin, S.-Y., Fleming, J.G.: A three-dimensional optical photonic crystal. J. Lightwave Technol. 17(11), 1944–1947 (1999)

    Article  ADS  Google Scholar 

  28. 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)

    Article  ADS  Google Scholar 

  29. Madeo, A., Barbagallo, G., d’Agostino, M.V., Placidi, L., Neff, P.: First evidence of non-locality in real band-gap metamaterials: Determining parameters in the relaxed micromorphic model. Proc. R. Soc. A, Math. Phys. Eng. Sci. 472(2190), 20160169 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Madeo, A., Neff, P., Aifantis, E.C., Barbagallo, G., d’Agostino, M.V.: On the role of micro-inertia in enriched continuum mechanics. Preprint arXiv:1607.07385 (2016)

  31. Madeo, A., Neff, P., d’Agostino, M.V., Barbagallo, G.: Complete band gaps including non-local effects occur only in the relaxed micromorphic model. C. R., Méc. 344(11–12), 784–796 (2016)

    Article  Google Scholar 

  32. Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Band gaps in the relaxed linear micromorphic continuum. Z. Angew. Math. Mech. 95(9), 880–887 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Wave propagation in relaxed micromorphic continua: Modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27(4–5), 551–570 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Madeo, A., Neff, P., Ghiba, I.-D., Rosi, G.: Reflection and transmission of elastic waves in non-local band-gap metamaterials: A comprehensive study via the relaxed micromorphic model. J. Mech. Phys. Solids 95, 441–479 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  35. Maldovan, M.: Sound and heat revolutions in phononics. Nature 503(7475), 209–217 (2013)

    Article  ADS  Google Scholar 

  36. Man, W., Florescu, M., Matsuyama, K., Yadak, P., Nahal, G., Hashemizad, S., Williamson, E., Steinhardt, P., Torquato, S., Chaikin, P.: Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast. Opt. Express 21(17), 19972–19981 (2013)

    Article  ADS  Google Scholar 

  37. Martínez-Sala, R., Sancho, J., Sánchez, J.V., Gómez, V., Llinares, J., Meseguer, F.: Sound attenuation by sculpture. Nature 378(6554), 241 (1995)

    Article  ADS  Google Scholar 

  38. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  39. Miniaci, M., Krushynska, A., Bosia, F., Pugno, N.M.: Large scale mechanical metamaterials as seismic shields. New J. Phys. 18(8), 83041 (2016)

    Article  Google Scholar 

  40. Misseroni, D., Colquitt, D.J., Movchan, A.B., Movchan, N.V., Jones, I.S.: Cymatics for the cloaking of flexural vibrations in a structured plate. Sci. Rep. 6, 23929 (2016)

    Article  ADS  Google Scholar 

  41. Morandi, F., Miniaci, M., Marzani, A., Barbaresi, L., Garai, M.: Standardised acoustic characterisation of sonic crystals noise barriers: Sound insulation and reflection properties. Appl. Acoust. 114, 294–306 (2016)

    Article  Google Scholar 

  42. Morvan, B., Tinel, A., Hladky Hennion, A.C., Vasseur, J., Dubus, B.: Experimental demonstration of the negative refraction of a transverse elastic wave in a two dimensional solid phononic crystal. Appl. Phys. Lett. 96(10), 2008–2011 (2010)

    Article  Google Scholar 

  43. Neff, P.: The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Mech. 86(11), 892–912 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Neff, P., Ghiba, I.-D., Lazar, M., Madeo, A.: The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q. J. Mech. Appl. Math. 68(1), 53–84 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: The relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26(5), 639–681 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85(18), 3966–3969 (2000)

    Article  ADS  Google Scholar 

  47. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Pham, K., Kouznetsova, V.G., Geers, M.G.D.: Transient computational homogenization for heterogeneous materials under dynamic excitation. J. Mech. Phys. Solids 61(11), 2125–2146 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Spadoni, A., Ruzzene, M., Gonella, S., Scarpa, F.: Phononic properties of hexagonal chiral lattices. Wave Motion 46(7), 435–450 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sridhar, A., Kouznetsova, V.G., Geers, M.G.D.: Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Comput. Mech. 57(3), 423–435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. 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)

    Article  ADS  Google Scholar 

  52. Eli, Y.: Photonic band-gap structures. J. Opt. Soc. Am. B 10(2), 283–295 (1993)

    Google Scholar 

  53. Yamamoto, N., Noda, S.: Fabrication and optical properties of one period of a three-dimensional photonic crystal operating in the 5–10 micron wavelength region. Jpn. J. Appl. Phys. 38(2), 1282–1285 (1999)

    Article  ADS  Google Scholar 

  54. Yi, K., Collet, M., Ichchou, M., Li, L.: Flexural waves focusing through shunted piezoelectric patches. Smart Mater. Struct. 25(7), 075007 (2016)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

Angela Madeo thanks the Institut Universitaire de France for its recognition and financial support, INSA-Lyon for the funding of the BQR 2016 “Caractérisation mécanique inverse des métamatériaux: modélisation, identification expérimentale des paramétres et évolutions possibles”, as well as the CNRS-INSIS for the funding of the PEPS project. Kévin Billon’s contribution has been financed by The French National Research Agency under grant number ANR-12-JS09-008-COVIA, in cooperation with the Labex ACTION program (ANR-11-LABX-0001-01).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Angela Madeo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Madeo, A., Collet, M., Miniaci, M. et al. Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia. J Elast 130, 59–83 (2018). https://doi.org/10.1007/s10659-017-9633-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-017-9633-6

Keywords

Mathematics Subject Classification (2000)

Navigation