Complete band gaps including non-local effects occur only in the relaxed micromorphic model

Angela Madeo; Patrizio Neff; Marco Valerio d'Agostino; Gabriele Barbagallo

doi:10.1016/j.crme.2016.07.002

Complete band gaps including non-local effects occur only in the relaxed micromorphic model

Angela Madeo ^1,² ; Patrizio Neff ³ ; Marco Valerio d'Agostino ¹ ; Gabriele Barbagallo ⁴

¹ LGCIE, INSA-Lyon, Université de Lyon, 20, avenue Albert-Einstein, 69621 Villeurbanne cedex, France
² IUF, Institut universitaire de France, 1, rue Descartes, 75231 Paris Cedex 05, France
³ Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Mathematik-Carrée, Thea-Leymann-Straße 9, 45127 Essen, Germany
⁴ LaMCoS-CNRS & LGCIE, INSA-Lyon, Université de Lyon, 20, avenue Albert-Einstein, 69621 Villeurbanne cedex, France

Comptes Rendus. Mécanique, Volume 344 (2016) no. 11-12, pp. 784-796.

Résumé

In this paper, we substantiate the claim implicitly made in previous works that the relaxed micromorphic model is the only linear, isotropic, reversibly elastic, nonlocal generalized continuum model able to describe complete band-gaps on a phenomenological level. To this end, we recapitulate the response of the standard Mindlin–Eringen micromorphic model with the full micro-distortion gradient ∇P, the relaxed micromorphic model depending only on the Curl P of the micro-distortion P, and a variant of the standard micromorphic model, in which the curvature depends only on the divergence Div P of the micro distortion. The Div-model has size-effects, but the dispersion analysis for plane waves shows the incapability of that model to even produce a partial band gap. Combining the curvature to depend quadratically on Div P and Curl P shows that such a model is similar to the standard Mindlin–Eringen model, which can eventually show only a partial band gap.

Reçu le : 2016-02-13
Accepté le : 2016-07-30
Publié le : 2016-09-05

DOI : 10.1016/j.crme.2016.07.002

Mots clés : Relaxed micromorphic model, Band gaps, Generalized continuum models, Long wavelength limit, Macroscopic consistency, Cauchy continuum, Homogenization, Multi-scale modeling, Parameter identification, Non-redundant model

Affiliations des auteurs :

Angela Madeo ^{1, 2} ; Patrizio Neff ³ ; Marco Valerio d'Agostino ¹ ; Gabriele Barbagallo ⁴

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     author = {Angela Madeo and Patrizio Neff and Marco Valerio d'Agostino and Gabriele Barbagallo},
     title = {Complete band gaps including non-local effects occur only in the relaxed micromorphic model},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {784--796},
     publisher = {Elsevier},
     volume = {344},
     number = {11-12},
     year = {2016},
     doi = {10.1016/j.crme.2016.07.002},
     language = {en},
}

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Angela Madeo; Patrizio Neff; Marco Valerio d'Agostino; Gabriele Barbagallo. Complete band gaps including non-local effects occur only in the relaxed micromorphic model. Comptes Rendus. Mécanique, Volume 344 (2016) no. 11-12, pp. 784-796. doi : 10.1016/j.crme.2016.07.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.07.002/

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