[Théorème de Bloch généralisé pour les métamatériaux visqueux : dispersion et propriétés effectives fondées sur les fréquences et nombres d'onde simultanément complexes]
Les courbes de dispersion des matériaux périodiques amortis sont habituellement basées soit sur des fréquences réelles en fonction de nombres d'onde complexes, soit sur des nombres d'onde réels en fonction de fréquences complexes. Le premier cas correspond à la propagation d'ondes harmoniques, dont la fréquence d'excitation est imposée, et dont l'atténuation due à la dissipation survient uniquement dans l'espace, en même temps que l'atténuation spatiale due à la diffraction de Bragg. Le second cas concerne la propagation d'ondes libres dont l'atténuation est due à une perte d'énergie dans le temps, en plus de l'atténuation spatiale causée par la diffraction de Bragg. Dans cet article, nous développons un algorithme pour des systèmes unidimensionnels afin d'obtenir—pour le mouvement d'ondes libres amorties—les courbes de dispersion fondées sur des fréquences et des nombres d'onde qui sont autorisés à être simultanément complexes. Cette application généralisée du théorème de Bloch fournit une structure de bandes qui décrit pleinement tous les mécanismes d'atténuation, dans l'espace comme dans le temps. L'algorithme est appliqué à un metamatériau à résonance locale (masse incluse dans une masse) amorti de façon visqueuse. Une masse effective dépendant de la fréquence est également obtenue pour cette chaine infinie amortie.
It is common for dispersion curves of damped periodic materials to be based on real frequencies as a function of complex wavenumbers or, conversely, real wavenumbers as a function of complex frequencies. The former condition corresponds to harmonic wave motion where a driving frequency is prescribed and where attenuation due to dissipation takes place only in space alongside spatial attenuation due to Bragg scattering. The latter condition, on the other hand, relates to free wave motion admitting attenuation due to energy loss only in time while spatial attenuation due to Bragg scattering also takes place. Here, we develop an algorithm for 1D systems that provides dispersion curves for damped free wave motion based on frequencies and wavenumbers that are permitted to be simultaneously complex. This represents a generalized application of Bloch's theorem and produces a dispersion band structure that fully describes all attenuation mechanisms, in space and in time. The algorithm is applied to a viscously damped mass-in-mass metamaterial exhibiting local resonance. A frequency-dependent effective mass for this damped infinite chain is also obtained.
Mot clés : Ondes amorties, Dispersion complexe, Structure de bandes complexe, Cristaux phononiques, Métamatériaux acoustiques, Matériaux périodiques
Michael J. Frazier 1 ; Mahmoud I. Hussein 1
@article{CRPHYS_2016__17_5_565_0,
author = {Michael J. Frazier and Mahmoud I. Hussein},
title = {Generalized {Bloch's} theorem for viscous metamaterials: {Dispersion} and effective properties based on frequencies and wavenumbers that are simultaneously complex},
journal = {Comptes Rendus. Physique},
pages = {565--577},
publisher = {Elsevier},
volume = {17},
number = {5},
year = {2016},
doi = {10.1016/j.crhy.2016.02.009},
language = {en},
}
TY - JOUR AU - Michael J. Frazier AU - Mahmoud I. Hussein TI - Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex JO - Comptes Rendus. Physique PY - 2016 SP - 565 EP - 577 VL - 17 IS - 5 PB - Elsevier DO - 10.1016/j.crhy.2016.02.009 LA - en ID - CRPHYS_2016__17_5_565_0 ER -
%0 Journal Article %A Michael J. Frazier %A Mahmoud I. Hussein %T Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex %J Comptes Rendus. Physique %D 2016 %P 565-577 %V 17 %N 5 %I Elsevier %R 10.1016/j.crhy.2016.02.009 %G en %F CRPHYS_2016__17_5_565_0
Michael J. Frazier; Mahmoud I. Hussein. Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex. Comptes Rendus. Physique, Volume 17 (2016) no. 5, pp. 565-577. doi : 10.1016/j.crhy.2016.02.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2016.02.009/
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