Analysis Of The Band-Gap Structures Of Two-Dimensional Phononic Crystals Based On The Wavelet Theory

Phononic crystal (PC) is a kind of functional materials with periodic structures which exhibit acoustic or elastic wave band gaps (BGs). Recently, it has become one of the active fast-developing research fields in condensed physics, acoustics, mechanics and other related area due to its unique properties and many potential applications.As the physical process of elastic waves in a phonomc crystal is very complex, it is relatively hard to analyze a phononic crystal in an explicit way. People usually analyze a phononic crystal through numerical simulations. Therefore, it is essential to study numerical methods in the theoretical research of phononic crystals. Among several theoretical methods proposed for phononic crystals, the plane wave expansion method (PWE) is the one that is proposed first and is used most commonly. However, it has the disadvantage of slow convergence in calculating the band structures, field distribution, etc, especially when dealing with the large elastic mismatch systems. It even fails to calculate the mixed solid-liquid systems. In this thesis, we introduce the wavelet theory to overcome these difficulties effectively. In this method, the elastic constants and the wave fields are expanded in the wavelet bases. Then by using the variational theory, the elastic wave equations are reduced to an eigenvalue problem. Numerical solution can be obtained by adopting the wavelet integral technique. The new method has a better convergence than that of the PWE. And it can be used to calculate the band structures of the mixed solid-liquid systems accurately for which the PWE fails. With the new method, the following problems are investigated:(1) The band structures of bulk waves in one and two-dimensional solid-solid, solid-liquid and liquid-liquid systems are calculated in detail. The validity and advantages of the wavelet method are discussed by comparing with the existing results. In addition, the band structures of three typical honeycomb materials are also analyzed.(2) The properties of surface waves (SW) or pseudo surface waves (PSW) for two types of phononic crystals, solid-solid and solid-liquid systems, are calculated. The influence of various factors on the surface waves is systemically discussed. Especially we calculate successfully the band structures of the surface waves for the large acoustic mismatch systems and the mixed solid-liquid systems where the PWE fails.(3) Combined with the supercell technique, the developed wavelet method is extended from the ideal phononic crystals to those with point- or line-defects. The defect modes for both bulk waves and surface waves are calculated.The results show:(1) The wavelet method can be effectively used to calculate the band structures of the bulk waves for one- and two-dimensional phononic crystals. Compared with the PWE, the wavelet method converges faster. Furthermore, it can overcome the disadvantages of the PWE, e.g. it can be used to calculate the band structures of the mixed solid-liquid systems, and remove the unphysical flat bands efficiently.(2) The wavelet method can also be used to calculate the band structures of surface waves in various phononic crystals, including the large acoustic mismatch systems and the mixed solid-liquid systems where the PWE fails. The detailed calculations show that the acoustic impedance ratio (AIR) has significant effect on the surface waves: For the systems with AIR≈1, neither complete band gaps nor directional band gaps do exist for the bulk waves, but the SW-SW or PSW-PSW band gaps or the SW-PSW band gaps exist; for the systems with a medium or high AIR, there may exists a critical filling fraction value at which the SW and PSW will interchange; the PSW is dominant over SW in the systems with a higher AIR. The lattice structure also has significant effect on the surface waves, but the shape of scatters does not. The band gap width for both bulk and surface waves increases as the filling fraction increases.(3) Combined with the supercell technique, the wavelet method can be used to calculate the band structues of bulk and surface waves in phononic crystals with the point- and line-defects. The method is proved very effective although more wavelet basis functions are required in calculation. The properties of the defect modes obtained by the PWE may be verified and reproduced by using the wavelet method. Especially for the large acoustic mismatch systems and the mixed solid-liquid systems where the PWE fails, the surface defect modes can be efficiently calculated by the present method. The band structures of the quasi-fractal crystals may be calculated in the similar way. We find that the band structures exhibit multi-frequency gaps, and are compressed integrally with the increase of the fractal level.(4) The wave propagation in honeycomb materials can be analyzed by viewing them as special phononic crystals. It is found that the structure of the materials has significant effects on the wave propagation behavior, while that the component materials have almost no influence. All three structures—square, triangle and hexagon don't exhibit complete band gaps, but the square and triangular structures may exhibit directional band gaps in certain region of the propagating direction. However, the hexagonal structure doesn't exhibit directional band gaps in any direction. Compared with the square structures, the triangular structures with the same smaller porosity can generate wider directional band gaps in wider propagating directions at lower frequencies. The introducing of the rubber layer is helpful to produce the elastic band gaps and absorb the enery.