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Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps

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Abstract

In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013), Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus μ c , which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

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Author notes

  1. G. Rosi

    Present address: Laboratoire Modélisation Multi Echelle, MSME UMR 8208 CNRS, Université Paris-Est, 61 Avenue du Géneral de Gaulle, 94010, Créteil Cedex, France

Authors and Affiliations

  1. Université de Lyon-INSA, 20 Av. Albert Einstein, 69100, Villeurbanne Cedex, France

    A. Madeo

  2. International Center M&MOCS “Mathematics and Mechanics of Complex Systems”, Palazzo Caetani, Cisterna di Latina, Italy

    A. Madeo, P. Neff, I. D. Ghiba, L. Placidi & G. Rosi

  3. Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    P. Neff

  4. Department of Mathematics, University “A.I. Cuza”, Blvd. Carol I, no. 11, 700506, Iasi, Romania

    I. D. Ghiba

  5. Octav Mayer Institute of Mathematics, Romanian Academy, 700505, Iasi, Romania

    I. D. Ghiba

  6. Institute of Solid Mechanics, Romanian Academy, 010141, Bucharest, Romania

    I. D. Ghiba

  7. Università Telematica Internazionale Uninettuno, Corso V. Emanuele II 39, 00186, Rome, Italy

    L. Placidi

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Communicated by Andreas Öchsner.

“Sans la curiosité de l’ésprit, que serions-nous?” Marie Curie.

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Madeo, A., Neff, P., Ghiba, I.D. et al. Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Continuum Mech. Thermodyn. 27, 551–570 (2015). https://doi.org/10.1007/s00161-013-0329-2

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