【作者】 闫志忠；

【导师】 汪越胜；

【作者基本信息】 北京交通大学 ， 固体力学， 2007， 博士

【摘要】 声子晶体(phononic crystal)是一种具有周期性结构并呈现弹性波禁带的功能材料。由于其独特的性能和广阔的应用前景,近年来已经成为凝聚态物理、声学、力学及其它相关学科活跃的研究领域之一。由于弹性波在声子晶体中的物理过程相当复杂,人们很难对其做精确的解析分析,因此理论模拟声子晶体变得十分重要。声子晶体的理论模拟分析通常涉及大规模的数值计算,所以寻求精确快捷的计算方法一直是声子晶体理论研究的主要内容之一。在已提出的声子晶体计算方法中,平面波展开法(PWE)提出得最早、应用最广。但平面波展开法在计算声子晶体带结构、场分布等时其收敛性存在某些问题,如对弹性阻抗失配较大的系统其收敛性很慢,对固-流声子晶体体系则难以给出正确结果。为解决这些问题,本文提出了一种基于小波理论的计算方法,即对弹性常数和弹性波场的小波级数展开,利用变分理论,将一般各向异性弹性波方程化为一般矩阵特征值问题,采用小波积分技术数值求解矩阵特征值得到问题的解。该方法具有较快的收敛速度,并且可以准确地计算固-流声子体系。利用该方法分别研究了如下问题:(1)对一、二维固-固、固-流和流-流声子晶体体系的体波能带结构进行了详细的计算,并与已有结果进行了比较,讨论了小波方法的有效性和优越性;另外计算分析了三种典型蜂窝材料的色散关系。(2)针对固-固、固-流两类二维声子晶体体系计算分析了表面波或伪表面波能带结构特性,系统地探讨了影响表面波的因素。特别是有效地计算了弹性声阻抗失配较大以及固-流混合体系等平面波方法不适用的情形。(3)结合超胞近似的思想,将小波方法由理想声子晶体推广到含点、线缺陷的声子晶体系统,计算分析了包括体波和表面波在内的缺陷态。研究结果表明:(1)小波方法可有效地用于计算一维、二维声子晶体体波的能带结构。与平面波展开法相比,能够大大地提高收敛性和准确性,特别是可以克服平面波展开法的缺点,应用于固-流混合体系,有效地消除非物理平带。(2)小波方法可有效地用于计算声子晶体表面波的能带结构,包括平面波展开法不适用的声阻抗失配较大的情况以及固-流混合体系。计算发现声阻抗比对表面波的影响很大:对声阻抗比在1附近的体系,虽然体波不易产生完全带隙甚至方向带隙,但存在(伪)表面波之间或表面波与伪表面波之间的方向带隙;对声阻抗比在中高值的体系,可能存在表面波和伪表面波转化的填充比临界值;声阻抗比越大越不易出现表面波,只存在伪表面波。晶格排列是影响表面波的重要因素,而散射体形状对表面波带结构影响不大;体波和表面波带隙宽度均随填充比的逐渐增大先增大后减少。(3)结合超胞技术,小波方法可有效地用于计算含点、线缺陷的二维声子晶体体波和表面波的能带结构,但需要引入较多的小波基函数以保证计算精度。对典型声子体系的能带结构计算可以很好地再现平面波展开法获得的缺陷态的特征。特别对声阻抗比较大的体系以及固-流混合体系,平面波法已不适用,而小波方法可以有效地计算得到其表面波的缺陷态。另外对准分形声子晶体也可以借助类似的思想计算其能带结构。结果发现,准分形声子晶体的能带具有多频带隙的特性,且随着级数的增大,能带结构整体地向低频压缩。(4)蜂窝材料可以作为一种特殊的声子晶体结构研究其波传播特性,其频散关系特征主要受排列结构的影响,与制作材料无关;正方、三角与六角这三种典型结构都不存在二维全方向的完全带隙,但正方和三角结构在一定的传播方向范围内存在方向带隙,而六角结构则在任何方向都不存在方向带隙;与正方结构相比,三角结构在相同孔隙率下,在更广的传播方向和更低的频率下,能产生较宽的方向带隙。蜂窝材料中引入橡胶夹层后,有助于带隙的产生,起到较好的吸能作用。更多还原

【Abstract】 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.更多还原