高对比度声子晶体的渐近分析及带隙开放判据
Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening
摘要：
本文研究了高对比度双组分周期弹性介质中弹性波频谱的带隙结构。我们考虑二维声子晶体的背景介质是由一个阵列的孔周期性沿两个正交坐标轴穿孔。本文建立了当剪切模量和密度模量的差值变大时声子带结构色散关系的完全渐近公式。主要内容是谐振动线弹性方程解的积分方程公式和关于亚纯算子值函数在复平面上的特征值的几个定理，如广义Rouché定理。我们建立了带结构和基本孔上的Dirichlet本征值问题之间的联系。我们还提供了一个标准，表现出间隙的能带结构中的矩阵的密度越小，带隙越宽，只要满足该标准。Economou和Sigalas报道了这种现象，他们观察到，与夹杂物相比，其基体具有较低密度和较高剪切模量的周期性弹性复合材料产生更好的开放间隙。我们在本文中的分析同意这一实验发现。
英文摘要：
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 Rouché'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.
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