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. doi:10.1016/j.crhy.2016.02.009. https://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 | ||||||||||||
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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. | ||||||||||||
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Subject Keywords: | Damped waves; Complex dispersion; Complex band structure; Phononic crystals; Acoustic metamaterials; Periodic materials | ||||||||||||
Issue or Number: | 5 | ||||||||||||
DOI: | 10.1016/j.crhy.2016.02.009 | ||||||||||||
Record Number: | CaltechAUTHORS:20160429-083703153 | ||||||||||||
Persistent URL: | https://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: | 10 Nov 2021 23:59 |
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