Theoretical study of platonic crystals with periodically structured N -beam resonators

Abstract

A multiple scattering theory is applied to study the properties of flexural waves propagating in a plate with periodically structured N-beam resonators. Each resonator consists of a circular hole containing an inner disk connected to background plate with N rectangular beams. The Bloch theorem is employed to obtain the band structure of a two-dimensional lattice containing a single resonator per unit cell. Also, a numerical algorithm has been developed to get the transmittance through resonator slabs infinitely long in the direction perpendicular to the incident wave. For the numerical validation, a square lattice of 2-beam resonators has been comprehensively analyzed. Its band structure exhibits several flat bands, indicating the existence of local resonances embedded in the structure. Particularly, the one featured as the fundamental mode of the inner disk opens a bandgap at low frequencies. This mode has been fully described in terms of a simple spring-mass model. As a practical application of the results obtained, a homogenization approach has been employed to design a focusing lens for flexural waves, where the index gradient is obtained by adjusting the orientation of the resonators beams. Numerical experiments performed within the framework of a three-dimensional finite element method have been employed to discuss the accuracy of the models described here.