Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

We investigate the band-gap structure of the frequency spectrum for elastic waves in a high-contrast, two-component periodic elastic medium. We consider two-dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. In this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large. The main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator-valued functions in the complex plane such as Generalized Rouch\'{e}'s theorem. We establish a connection between the band structures and the Dirichlet eigenvalue problem on the elementary hole. We also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is, wider the band-gap is, provided that the criterion is fulfilled. This phenomenon was reported by Economou and Sigalas who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps. Our analysis in this paper agrees with this experimental finding.