Research On Material Parameters Determining The Band Gaps Of Phononic Crystals

Phononic crystals (PCs) are a kind of acoustic functional materials which are composed of artificially periodic structures and exhibit elastic/acoustic wave band gaps (BGs). They have plenty of unique acoustic properties and designability, and thus have prospective applications in many fields. Design of phononic crystals through engineering of band gaps has been a key research field.Phononic band gaps are determined by many factors, including material and structural parameters. This thesis focuses on the influence of the material parameters on the band gaps. We will begin our analysis with basic wave equations and derive the material parameters directly determining band gaps for solid/solid, fluid/fluid and solid/fluid phononic crystals. Then band gaps are calculated by using the plane wave expansion (PWE) method for 2D solid/solid PCs and for 2D and 3D fluid/fluid PCs, and by using the Dirichlet-to-Neumann (DtN) map method for 2D solid/fluid PCs. For all of the above systems, the influences of the material parameters on the first band gap which is of the most practical importance are discussed by examining variation of the normalized band-gap width (i.e. gap width to mid-gap frequency ratio), mid-gap frequency and upper and lower band-gap edges. The results show:1. The material parameters determining the band gaps of 1D solid/solid, fluid/fluid and solid/fluid PCs are the transverse or longitudinal impedance ratio and velocity ratio. The impedance ratio predominantly determines the appearance of and width of the band gaps. There is no band gap when the impedance ratio is 1. Band gaps appear with the normalized width increasing when the value of the impedance ratio becomes more deviating from 1.2. The material parameters determining the band gaps of 2D solid/solid PCs are the mass density ratio and shear modulus ratio (or equivalently, the impedance ratio and velocity ratio) of the scatterers and the host for the anti-plane shear wave modes, and besides also include Poisson's ratios of the two components for the in-plane mixed wave modes or in 3D PCs. The results calculated by PWE method for 2D PCs show that the mass density ratio plays a more important role on band gaps for the anti-plane modes, and that both mass density ratio and shear modulus ratio act the same role for the in-plane modes. Band gaps for all wave modes easily appear at large density ratios and modulus ratios (or equivalently large impedance ratios with the velocity ratios close to 1), and their normalized gap widths become wider with both of these two parameters increasing. Band gaps may also appear at large density ratios and small modulus ratios, and disappear when these two ratios approach to 1 for both the square lattice and triangle lattice. However, no band gap appears in a square lattice with small density ratios and large modulus ratios, or in a triangle lattice with these two ratios being small. The mid-gap frequencies of the band gaps all become lower with velocity ratios decreasing. In addition, Poisson's ratios play a slight influence. The Poisson's ratio of the host material affects the band gaps a little more than that of the scatterers. The band gaps are a little more sensitive to Poisson's ratios for a triangle lattice than for a square lattice.3. The material parameters determining the band gaps of 2D and 3D fluid/fluid PCs are the mass density ratio and bulk modulus ratio (or equivalently, the impedance ratio and velocity ratio) of the scatterers and the host. The results calculated by PWE method show that the bulk modulus ratio has more influences on the band gaps than the density ratio. Band gaps easily appear at small density ratios and modulus ratios (or equivalently small impedance ratios with the values of velocity ratios close to 1), and their normalized gap widths become wider with both of these ratios decreasing. In 2D fluid/fluid PCs, band gaps may also appear at large density ratios and small modulus ratios, and disappear when these two ratios approach to 1, for both the square lattice and triangle lattice. But no band gap appears in a square lattice with small density ratios and large modulus ratios, or in a triangle lattice with these two ratios being large. Band gaps may also appear in a 3D fluid/fluid PC with large density ratios and small bulk modulus ratios in all lattices of simple cubic, body-centered cubic and face-centered cubic. In the case of the same filling friction, the normalized band gap width of a body-centered cubic system is bigger than that of a simple cubic system, but smaller than that of a face-centered cubic system. As a solid/solid system, the mid-gap frequencies of the band gaps become lower with the velocity ratios decreasing.4. The material parameters determining the band gaps of 2D solid/fluid PCs are the longitudinal wave impedance ratio and velocity ratio (or equivalently, the mass density ratio and modulus ratio) of the scatterers and the host, and Poisson's ratio of solid scatterers. the calculated results for a square lattice by using Dirichlet-to-Neumann map method (DtN method) show that when the velocity ratio is more than 1 and modulus ratio less than 1, the normalized gap widths increase and the mid-gap frequencies decrease as the modulus ratio becomes smaller. When the velocity ratio and density ratio are both bigger than 1, the band gaps appear with the impedance ratio increases. Their normalized gap widths are increasing rapidly to a nearly constant value with the material parameters varying in a small range. The Poisson's ratio of the solid scatterers affects the band gaps slightly. A little bigger normalized gap width can be obtained at small values of the Poisson's ratio.Illustrations of the normalized gap widths, mid-gap frequencies and upper and lower band gap edges varying with the material parameters demonstrate the influences of the material parameters on the band gaps of PCs and are of great help in engineering of band gaps.