Analytical approximations for low frequency band gaps
in periodic arrays of elastic shells

Krynkin, A; Umnova, O; Taherzadeh, S; Attenborough, K

Abstract

This paper presents and compares three analytical methods for calculating low frequency band
gap boundaries in doubly periodic arrays of resonating thin elastic shells. It is shown that both
Foldy-type equations (derived with lattice sum expansions in the vicinity of its poles) and a
self-consistent scheme could be used to predict boundaries of low-frequency (below the first Bragg
band gap) band gaps due to axisymmetric (n ¼ 0) and dipolar (n ¼ 1) shell resonances. The accuracy
of the former method is limited to low filling fraction arrays, however, as the filling fraction increases
the application of the matched asymptotic expansions could significantly improve approximations of
the upper boundary of band gap related to axisymmetric resonance. The self-consistent scheme is
shown to be very robust and gives reliable results even for dense arrays with filling fractions around
70%. The estimates of band gap boundaries can be used in analyzing the performance of periodic
arrays (in terms of the band gap width) without using full semi-analytical and numerical models. The
results are used to predict the dependence of the position and width of the low frequency band gap on
the properties of shells and their periodic arrays.

Citation

in periodic arrays of elastic shells. ˜The œJournal of the Acoustical Society of America (Online), 133(2), https://doi.org/10.1121/1.4773257

• Journal Article Type: Article
• Publication Date: Feb 1, 2013
• Deposit Date: May 21, 2015
• Publicly Available Date: Apr 5, 2016
• Journal: The Journal of the Acoustical Society of America (JASA)
• Print ISSN: 0001-4966
• Peer Reviewed: Peer Reviewed
• Volume: 133
• Issue: 2
• DOI: https://doi.org/10.1121/1.4773257
• Publisher URL: http://dx.doi.org/10.1121/1.4773257
• Related Public URLs: http://scitation.aip.org/content/asa/journal/jasa
• Additional Information: Access Information : Copyright (2015) Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.; Funders : Engineering and Physical Sciences Research Council (EPSRC)