A study of frequency band structure in two-dimensional homogeneous anisotropic phononic K3-metamaterials

Abstract

Phononic metamaterials are synthesised materials in which locally resonant units are arranged in a particular geometry of a substratum lattice and connected in a predefined topology. This study investigates dispersion surfaces in two-dimensional anisotropic acoustic metamaterials involving mass-in-mass units connected by massless springs in K3 topology. The reasons behind the particular choice of this topology are explained. Two sets of solutions for the eigenvalue problem $| {\boldsymbol{D}}({\omega }^{2},{\boldsymbol{k}})| =0$ are obtained and the existence of absolutely different mechanisms of gap formation between acoustic and optical surface frequencies is shown as a bright display of quantum effects like strong coupling, energy splitting, and level crossings in classical mechanical systems. It has been concluded that a single dimensionless parameter i.e. relative mass controls the order of formation of gaps between different frequency surfaces. If the internal mass of the locally resonant mass-in-mass unit, $m,$ increases relative to its external mass, $M,$ then the coupling between the internal and external vibrations in the whole system rises sharply, and a threshold ${\mu }^{* }$ is reached so that for $m/M\gt {\mu }^{* }$ the optical vibrations break the continuous spectrum of 'acoustic phonons' creating the gap between them for any value of other system parameters. The methods to control gap parameters and polarisation properties of the optical vibrations created over these gaps were investigated. Dependencies of morphology and width of gaps for several anisotropic cases have been expounded and the physical meaning of singularity at the point of tangential contact between two adjacent frequency surfaces has been provided. Repulsion between different frequency band curves, as planar projections of surfaces, has been explained. The limiting case of isotropy has been discussed and it has been shown that, in the isotropic case, the lower gap always forms, irrespective of the value of relative mass.