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Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex

Frazier, Michael J. and Hussein, Mahmoud I. (2016) Generalized Bloch's theorem for viscous metamaterials: Dispersion and effective properties based on frequencies and wavenumbers that are simultaneously complex. Comptes Rendus Physique, 17 (5). pp. 565-577. ISSN 1631-0705. http://resolver.caltech.edu/CaltechAUTHORS:20160429-083703153

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Abstract

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.crhy.2016.02.009 DOIArticle
http://www.sciencedirect.com/science/article/pii/S1631070516300020PublisherArticle
http://arxiv.org/abs/1601.00683arXivDiscussion Paper
Alternate Title:Théorème de Bloch généralisé pour les métamatériaux visqueux : dispersion et propriétés effectives fondées sur les fréquences et nombres d'onde simultanément complexes
Additional Information:© 2016 Académie des sciences. Published by Elsevier Masson SAS. Available online 17 March 2016. This research has been supported by the National Science Foundation Graduate Research Fellowship Grant No. DGE 1144083 and CAREER Grant No. 1254931. Support was also provided by the Department of Education GAANN program.
Funders:
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1144083
NSF1254931
Department of EducationUNSPECIFIED
Subject Keywords:Damped waves; Complex dispersion; Complex band structure; Phononic crystals; Acoustic metamaterials; Periodic materials
Record Number:CaltechAUTHORS:20160429-083703153
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20160429-083703153
Official Citation: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, Issue 5, May 2016, Pages 565-577, ISSN 1631-0705, http://dx.doi.org/10.1016/j.crhy.2016.02.009. (http://www.sciencedirect.com/science/article/pii/S1631070516300020)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66546
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:29 Apr 2016 19:46
Last Modified:29 Apr 2016 19:46

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