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.
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Notes
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.
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].
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 \(x=(x_{1},x_{2},x _{3})^{T}\).
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.
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
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)
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)
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)
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)
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)
Boutin, C., Hans, S.: Homogenisation of periodic discrete medium: Application to dynamics of framed structures. Comput. Geotech. 30(4), 303–320 (2003)
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)
Boutin, C., Soubestre, J.: Generalized inner bending continua for linear fiber reinforced materials. Int. J. Solids Struct. 48(3–4), 517–534 (2011)
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)
Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Physica A 322, 359–376 (2003)
Chen, Y., Lee, J.D., Eskandarian, A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41(8), 2085–2097 (2004)
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)
Colombi, A., Colquitt, D.J., Roux, P., Guenneau, S., Craster, R.V.: A seismic metamaterial: The resonant metawedge. Sci. Rep. 6(7249), 27717 (2016)
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)
Craster, R.V., Guenneau, S.: Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. Springer Series in Materials Science. Springer, Berlin (2013), 332 pp
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)
Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)
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)
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)
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)
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)
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)
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)
Haberko, J., Scheffold, F.: Fabrication of mesoscale polymeric templates for three-dimensional disordered photonic materials. Opt. Express 21(1), 1057–1065 (2013)
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)
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)
Lin, S.-Y., Fleming, J.G.: A three-dimensional optical photonic crystal. J. Lightwave Technol. 17(11), 1944–1947 (1999)
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)
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)
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)
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)
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)
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)
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)
Maldovan, M.: Sound and heat revolutions in phononics. Nature 503(7475), 209–217 (2013)
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)
Martínez-Sala, R., Sancho, J., Sánchez, J.V., Gómez, V., Llinares, J., Meseguer, F.: Sound attenuation by sculpture. Nature 378(6554), 241 (1995)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)
Miniaci, M., Krushynska, A., Bosia, F., Pugno, N.M.: Large scale mechanical metamaterials as seismic shields. New J. Phys. 18(8), 83041 (2016)
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)
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)
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)
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)
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)
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)
Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85(18), 3966–3969 (2000)
Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006)
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)
Spadoni, A., Ruzzene, M., Gonella, S., Scarpa, F.: Phononic properties of hexagonal chiral lattices. Wave Motion 46(7), 435–450 (2009)
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)
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)
Eli, Y.: Photonic band-gap structures. J. Opt. Soc. Am. B 10(2), 283–295 (1993)
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)
Yi, K., Collet, M., Ichchou, M., Li, L.: Flexural waves focusing through shunted piezoelectric patches. Smart Mater. Struct. 25(7), 075007 (2016)
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).
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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
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DOI: https://doi.org/10.1007/s10659-017-9633-6
Keywords
- Microstructure
- Metamaterials
- Phononic crystals
- Relaxed micromorphic model
- Gradient micro-inertia
- Free micro-inertia
- Complete band-gaps
- Fitting of the elastic coefficients
- Inverse approach