Study Of The Band Structure Of Phononic Crystals

Phononic crystals is an elastic composite material composed by different elastic units arranged in a periodical sequence. Prohibited band appears as elastic waves (acoustic waves) propagate in it. Investigation of the energy band structure of the phononic crystal is, therefore, of great importance for both theoretical studies and applications. In this paper, we calculate the energy band structure of the phononic crystal by making use of the method of plane wave expansion and investigate the influence of crystal structure, the properties of elastic units and the filling and shape of scatterer on the band gaps in an attempt to provide theoretical guidance to the preparation of phononic crystals. This paper consists of the following five chapters: In chapter 1, we introduce the basic concepts on phononic crystals,research background related to phononic crystal and recent research activities at home and abroad. In chapter 2, based on the elastic wave equation, we introduce the traditional method for the calculation of band structure, namely the method of plane wave expansion. In addition, theoretic models to study 2D and 3D phononic crystals are presented. In chapter 3, we have calculated the band structure of binary 2D phononic crystal systems arranged in square lattice and consisting of solid scatterer with different shape embedded in solid matrix. The outcomes show that, (1) the maximums of the band gaps with the lowest frequency do not always appear at a certain shape of the scatterers'cross-section when the filling fraction changes; (2) The largest band gap appears when the proportion of length and width of scatterers'rectangle cross-section is 1.2 for XY mode and is 1 for Z mode; (3) When the cross-section is square with revolving angle θ, it is more favorable at θ=45o for the appearance of the band gaps. In chapter 4, the energy band structures of binary 3D phononic crystals of three kinds of structures (simple-cubic (SC), face-centered-cubic (FCC) and body-centered-cubic (BCC)) were calculated. Three different shapes of scatterers (cube, cuboid and sphere) are taken into account. It is shown that, (1) the computational results are almost the same for SC,FCC and BCC structures when the scatterers are cuboid. The band gap width will increase at first and then decrease as the ratio of height and length increases. The largest band gap corresponds to the case of the ratio 1; (2) When the scatterers are cube with revolving angle θ(0o≤θ≤45o), the (lowest) band gap width will decrease as the angle θincreases for SC and FCC structures. Totally differently, the largest gap for BCC structure appears at θ= 45o; (3) The (lowest) band gap is the largest when scatterers are arranged in FCC structure and the band gap is the smallest when scatterers are arranged in SC structure. Finally, a summary of our works and outlook for future works on the investigation of phononic crystals are presented in chapter 5.