影响声子晶体带隙的材料参数研究
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摘要
声子晶体(phononic crystals)是一种具有周期性人工结构并呈现弹性波(声波)带隙特征的声学功能材料。其丰富的声学特性、超强的可设计性以及广泛的应用前景,使得通过带隙调制,设计符合各种需求的声子晶体结构成为相关领域进一步研究的前沿课题。
影响带隙的参数包括材料参数和结构参数,本文针对影响带隙的材料参数,从基本波动方程出发分析了影响固/固、流/流、固/流体系声子晶体带隙的材料参数；并进一步利用平面波展开法计算了二维固/固体系、以及二维和三维流/流体系声子晶体的带隙,利用Dirichlet-to-Neumann (DtN)映射法计算了二维固/流体系的带隙；讨论了材料参数对能带结构中可能出现的第一条带隙(也是最具有应用价值的带隙)的相对宽度、中心频率和上下边界等带隙特征参量的影响。主要结果有：
1、影响一维固/固、流/流、固/流体系声子晶体带隙的材料参数均为该体系中波(横波或纵波)的阻抗比和波速比(或质量密度比和模量比),其中阻抗比具有决定性的影响。阻抗比为1时,没有带隙。阻抗比偏离1时产生带隙,且其相对宽度随着阻抗比偏离1的程度而增大。
2、影响二维固/固声子晶体反平面剪切波带隙的材料参数是散射体和基体的质量密度比和剪切模量比(或波阻抗比和波速比)；影响二维和三维声子晶体混合波带隙的材料参数除上述比值外,还包括散射体和基体的泊松比。利用平面波展开法对二维体系进行了详细计算,结果表明：对反平面剪切波,质量密度比对带隙起更主要的作用；对平面混合波,密度比和模量比对带隙有同等重要的影响。对任意模态的波,其带隙的相对宽度均在质量密度比和剪切模量比都较大时,有较大的值,且随这两个比值同时增大(即阻抗比增大且波速比接近1)而增大。除此之外,正方点阵和三角点阵均会在密度比较大模量比较小时出现带隙,在两比值都接近于1时均没有带隙。两种点阵不同的是,正方点阵不会在密度比较小模量比较大时出现带隙,而三角点阵不会在密度比和模量比都较小时出现带隙。无论点阵形式和材料参数如何,出现带隙时中心频率均随波速比的减小而降低。泊松比对带隙的影响很小,相对于散射体,基体的泊松比对带隙影响稍大；相对于正方点阵,三角点阵带隙受泊松比的影响稍大。
3、影响流/流声子晶体带隙的材料参数是散射体和基体的质量密度比和体积模量比(或波阻抗比和波速比)。利用平面波展开法对二维和三维流/流声子晶体带隙进行了详细计算,结果表明：体积模量比对带隙有更重要的影响。带隙在质量密度比和体积模量比均较小时,其相对宽度较大,且随着这两个比值同时减小(即阻抗比减小波速比接近1)而增大。对二维流/流声子晶体,正方点阵和三角点阵情况均会在密度比较大模量比较小时出现带隙,在两比值都接近于1时均没有带隙。不同的是,正方点阵不会在密度比较小模量比较大时出现带隙,而三角点阵不会在密度比和模量比都较大时出现带隙；对三维流/流声子晶体,简立方、体心立方、面心立方三种点阵情况均还会在质量密度比较大体积模量比较小时出现带隙；相同填充率下三种点阵形式的带隙相对宽度依次增大。无论点阵形式和材料参数情况,出现带隙时中心频率均随波速比的减小而降低。
4、影响二维固/流声子晶体带隙的材料参数是纵波的阻抗比和波速比(或质量密度比和模量比)、以及散射体泊松比。利用Dirichlet-to-Neumann映射法(DtN方法)以正方点阵为例进行了详细的计算,结果表明：纵波波速比大于1且模量比小于1时,随着模量比减小(即阻抗比和波速比同时减小)带隙相对宽度增大,同时带隙中心频率降低。纵波波速比大于1且质量密度比大于1时,随着纵波阻抗比增大出现了带隙,其相对宽度在很小的材料参数变化范围内迅速增大,达到一定值后即保持不变(出现“平台”)。固体泊松比对带隙的影响很小,泊松比较小时带隙相对宽度稍大。
带隙相对宽度、中心频率和上下边界随材料参数的变化图,可以直观地表征材料参数对带隙的影响,对于根据材料参数设计带隙具有指导意义。
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.
影响带隙的参数包括材料参数和结构参数,本文针对影响带隙的材料参数,从基本波动方程出发分析了影响固/固、流/流、固/流体系声子晶体带隙的材料参数；并进一步利用平面波展开法计算了二维固/固体系、以及二维和三维流/流体系声子晶体的带隙,利用Dirichlet-to-Neumann (DtN)映射法计算了二维固/流体系的带隙；讨论了材料参数对能带结构中可能出现的第一条带隙(也是最具有应用价值的带隙)的相对宽度、中心频率和上下边界等带隙特征参量的影响。主要结果有：
1、影响一维固/固、流/流、固/流体系声子晶体带隙的材料参数均为该体系中波(横波或纵波)的阻抗比和波速比(或质量密度比和模量比),其中阻抗比具有决定性的影响。阻抗比为1时,没有带隙。阻抗比偏离1时产生带隙,且其相对宽度随着阻抗比偏离1的程度而增大。
2、影响二维固/固声子晶体反平面剪切波带隙的材料参数是散射体和基体的质量密度比和剪切模量比(或波阻抗比和波速比)；影响二维和三维声子晶体混合波带隙的材料参数除上述比值外,还包括散射体和基体的泊松比。利用平面波展开法对二维体系进行了详细计算,结果表明：对反平面剪切波,质量密度比对带隙起更主要的作用；对平面混合波,密度比和模量比对带隙有同等重要的影响。对任意模态的波,其带隙的相对宽度均在质量密度比和剪切模量比都较大时,有较大的值,且随这两个比值同时增大(即阻抗比增大且波速比接近1)而增大。除此之外,正方点阵和三角点阵均会在密度比较大模量比较小时出现带隙,在两比值都接近于1时均没有带隙。两种点阵不同的是,正方点阵不会在密度比较小模量比较大时出现带隙,而三角点阵不会在密度比和模量比都较小时出现带隙。无论点阵形式和材料参数如何,出现带隙时中心频率均随波速比的减小而降低。泊松比对带隙的影响很小,相对于散射体,基体的泊松比对带隙影响稍大；相对于正方点阵,三角点阵带隙受泊松比的影响稍大。
3、影响流/流声子晶体带隙的材料参数是散射体和基体的质量密度比和体积模量比(或波阻抗比和波速比)。利用平面波展开法对二维和三维流/流声子晶体带隙进行了详细计算,结果表明：体积模量比对带隙有更重要的影响。带隙在质量密度比和体积模量比均较小时,其相对宽度较大,且随着这两个比值同时减小(即阻抗比减小波速比接近1)而增大。对二维流/流声子晶体,正方点阵和三角点阵情况均会在密度比较大模量比较小时出现带隙,在两比值都接近于1时均没有带隙。不同的是,正方点阵不会在密度比较小模量比较大时出现带隙,而三角点阵不会在密度比和模量比都较大时出现带隙；对三维流/流声子晶体,简立方、体心立方、面心立方三种点阵情况均还会在质量密度比较大体积模量比较小时出现带隙；相同填充率下三种点阵形式的带隙相对宽度依次增大。无论点阵形式和材料参数情况,出现带隙时中心频率均随波速比的减小而降低。
4、影响二维固/流声子晶体带隙的材料参数是纵波的阻抗比和波速比(或质量密度比和模量比)、以及散射体泊松比。利用Dirichlet-to-Neumann映射法(DtN方法)以正方点阵为例进行了详细的计算,结果表明：纵波波速比大于1且模量比小于1时,随着模量比减小(即阻抗比和波速比同时减小)带隙相对宽度增大,同时带隙中心频率降低。纵波波速比大于1且质量密度比大于1时,随着纵波阻抗比增大出现了带隙,其相对宽度在很小的材料参数变化范围内迅速增大,达到一定值后即保持不变(出现“平台”)。固体泊松比对带隙的影响很小,泊松比较小时带隙相对宽度稍大。
带隙相对宽度、中心频率和上下边界随材料参数的变化图,可以直观地表征材料参数对带隙的影响,对于根据材料参数设计带隙具有指导意义。
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.
引文
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