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

TL;DR

This paper develops and compares three analytical methods to predict low frequency band gaps in periodic arrays of elastic shells, providing accurate tools for designing such materials with specific wave filtering properties.

Contribution

It introduces and evaluates three analytical approaches—lattice sum expansions, matched asymptotic expansions, and a self-consistent scheme—for calculating band gap boundaries in elastic shell arrays.

Findings

Lattice sum expansions predict band gaps accurately for low filling fractions.

Matched asymptotic expansions improve upper band gap boundary estimates.

Self-consistent scheme reliably predicts band gaps even in dense arrays.

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 lattice sum expansions in the vicinity of its poles and self consistent scheme could be used to predict boundaries of low-frequency band gaps due to axysimmetric (n=0) and dipolar (n=1) shell resonances. The accuracy of the former method is limited to low filling fraction arrays, however the application of the matched asymptotic expansions could significantly improve approximations of the upper band gap boundary due to axysimmetric resonance. The self-consistent scheme is shown to be very robust and gives reliable results even for dense arrays with filling fraction higher than 40%. 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.