【作者】 陈阿丽；

【导师】 汪越胜；

【作者基本信息】 北京交通大学 ， 固体力学， 2008， 博士

【摘要】 自从Kushwaha提出声子晶体的概念以来,许多科学家开始致力于研究这种具有声带隙特征的人工周期弹性/声晶格结构。声子晶体的本质特征是它的声子带隙(或称声波禁带),即处于声波带隙频率范围内的振动或波被禁止在声子晶体中传播。声子晶体的这一特性在声滤波器、声波导、振动隔离控制,噪音抑制和设计新的传感器等方面有着潜在的工程应用。但是在实际中由于随机参杂的杂质或制作误差往往会使周期结构出现一定程度上的失谐直至完全无序。失谐的存在会引起声子晶体中弹性波传播的局部化现象,这也为我们利用失谐实现对弹性波传播的控制提供思路。而周期性的丧失使问题的数学处理变得困难,某些表征参数也不再有效。本文研究了以下失谐声子晶体中弹性波传播的若干问题:(1)引入局部化因子的概念,利用传递矩阵法计算分析了弹性波在一维谐调和随机失谐声子晶体中的传播和局部化特征,利用局部化因子表征了弹性波以任意角度斜入射时声子晶体的频带结构。结果表明:局部化因子是一个表征一维谐调和随机失谐声子晶体频带结构的有效参量,可以很好地描述其中的弹性波传播和局部化特性。(2)将基于平面波展开的传递矩阵法推广用于计算局部化因子,研究了二维谐调和在某个方向随机失谐的声子晶体的频带结构和局部化特征。利用局部化因子表征了二维声子晶体沿布里渊区域的频带结构。结果表明:局部化因子可以很好地描述二维谐调和在一个方向随机失谐另一个方向周期排列的二维声子晶体的频带结构和局部化现象。上述两问题的研究表明,局部化因子是描述一维失谐系统波动局部化行为的有效表征参量,其计算方法简便,其值在通带区间随着系统失谐度的增加而增加,在带隙区间随着系统失谐度的增加而减小,该局部化行为在高频区间尤为显著。(3)利用超元胞法结合平面波展开法计算了二维随机失谐声子晶体的频带结构,其中随机失谐参量设为散射体的半径或位置。并将计算结果与利用有限元方法得到的响应结果做了比较,两者吻合得很好。研究表明:利用频散曲线表示的频带结构,其通带边缘出现了许多类似于缺陷态的平直带,这些平直带表示弹性波在其对应的频率下出现了局部化现象。(4)准周期是另一类处于周期与无序之间的状态,本文将准周期声子晶体看做是谐调声子晶体以准周期特征失谐而形成的。分别利用传递矩阵法和基于平面波展开的传递矩阵法研究了一维准周期声子晶体(Fibonacci序列)和一个方向上具有Fibonacci排列、另一个方向谐调的二维声子晶体中的弹性波传播特征。并用局部化因子表示上述两种准周期系统的频带结构。计算结果表明:准周期声子晶体中弹性波传播也会表现出局部化现象,利用局部化因子可以很好地描述其中的弹性波传播行为;准周期结构与它对应的周期平均结构和随机失谐结构相比较,带隙数目相对较多但是带隙宽度小;通过向准周期结构中引入平移对称性可以得到真正的通带;通过引入镜面对称可以得到更多的频带;对准周期声子晶体来说在较低的频率区间也会出现带隙。最后利用超元胞结合平面波展开法初步探索了具有8重对称性的二维准周期声子晶体中弹性波的传播特征。另外,本论文还对局部化因子、频散曲线、响应频谱等表征参量的特点进行了对比分析。更多还原

【Abstract】 Since Kushwaha proposed the concept of "phononic crystal"(PNC),an artificial periodic elastic/acoustic structure that exhibits so-called "phononic band gaps",a great deal of attention has been focused on this kind of artificial lattice structures.The essential property of the PNC is its band gaps(or stopbands) which have numerous potential engineering applications such as acoustic filters,control of vibration isolation, noise suppression and design of new transducers.However in practical cases,disorder in certain degree or completely disorder always exists due to randomly distributed material defaults or manufacture errors during production process.Disorder in the PNCs will lead to the wave localization phenomenon which can be used to control the wave propagation.Because of the lack of periodicity the mathematic process becomes more difficult and some parameters used to describe the band structures fail.In this thesis,we study the following problems for elastic waves propagationg in disordered phononic crystals:(1) The concept of the localization factor calculated by the transfer matrix method is introduced to describe the band structures and localization behaviors of elastic waves propagating obliquely in 1D perfect and randomly disordered PNCs.The results show that the localization factor is an effective parameter in characterizing the band structures and localization phenomenon of 1D perfect and randomly disordered PNCs.(2) The plane-wave-expansion based transfer matrix method(PWE-based TMM) is generalized to calculate the localization factor which is introduced to describe the band structures and localization behaviors of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.The band structures in the first Brillouin Zone are discribed by the localization factors.The results show that the localization factor is an effective parameter in characterizing the band structures of 2D perfect PNCs and 2D PNCs randomly disordered in one direction and perfect in the other direction.The studies of the above two problems show that the localization factor which can be calculated simplely is an effective parameter in characterizing the wave localization phenomenon of the 1D disordered systems.As the disorder of the system increases,the value of the localization factor increases in the pass bands and decreases in the band gaps.The localization behavior is more pronounced at higher frequencies.(3) Combined with the plane wave expansion method,the supercell technique is used to calculate the dispersion curves of 2D randomly disordered PNCs in which the radius or locations of the scatterers are disordered.The band structures expressed by the dispersion curves have a good aggrernent with those by the frequency response calculated by the finite element method.And the results show that there appear many flat bands as the defected states in the band structures of the randomly disordered PNCs.This implies that the localization phenomenon exists at the corresponding frequencies of the flat bands.(4) A quasi periodic system is another intermediate case between the perfectly ordered and completely disordered systems.Treating this system as a disordered one deviating quasi-periodically from a periodic one we study the properties of the wave propagating in the 1D(Fibonacci sequency) quasi periodic phononic crystals (QPNCs) and the 2D phononic crystal with Fibonacci sequency in one direction and translational symmetry in the other direction using the transfer matrix and PWE-based TMM,respectively.The band structures of the above two systems are expressed by the localization factors and response.The results show that the localization phenomenon appears when the wave propagates in the QPNCs,and that the localization factor is an effective parameter in characterizing the band structures of the QPNCs.The band gaps of the QPNCs are more and narrower than those of its periodic average structure and randomly disordered systems.Pure passbands appear if we introduce translational symmetry to the QPNCs.More structures in the frequency bands are obtained if the mirror symmetry is further introduced.For the QPNCs band gaps may appear in the low frequencies.At last, the supereell technique combined with the PWE method is used to conduct a primary research on the band structures of the 2D QPNC with 8-fold symmetry.In addition,the parameters of the localization factor,the dispersion curve and the frequency response in charactering the properties of wave propagation and localization phenomenon are discussed and compared in this thesis.更多还原