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

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=0) and dipolar (n=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.

Comparisons of two effective medium approaches for predicting sound scattering by periodic arrays of elastic shells

“…The detailed description of the model can be found in a previous work, 8 so here it is only briefly outlined. Consider an effective medium as a replacement of a doubly periodic array of elastic shells in air.…”

“…Analytical approximations for low frequency bandgap boundaries in the arrays of elastic shells are presented in Ref. 8. It is shown that as a result of the variation in bandgap width with filling fraction of scatterers in the array, the low transmission bands are wider in denser arrays.…”

Bloch-wave expansion technique for predicting wave reflection and transmission in two-dimensional phononic crystals

“…4, use Fourier expansions for the wave field and periodic material properties to compute the dispersion behavior, yet can experience convergence issues for highly contrasting material properties. 9 Chen and Ye 3 and Krynkin et al 10 considered spherical harmonics via separation of variables to compute the dispersion relationship for given fluid, and respectively, fluid-solid crystals. Alternatively, transfer matrices 7 are well-suited for one-dimensional systems whereas finite difference techniques can also be used to numerically compute the dispersion of complex or finite periodic structures.…”

Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays

“…Other Helmholtz equation studies of this type include work on two-dimensional arrays of thick-walled split-ring resonators [16] and two-dimensional arrays of closely packed solid cylinders [17]. Estimates for the upper and lower bounds of the first band gap in elastic resonator array problems have also been considered [18]. Research on resonator arrays has also been conducted extensively for Maxwell's equations, including a numerical study on determining effective optical constants of a two-dimensional array of infinitesimally thin split-ring resonators [19].…”