手性及星形蜂巢狀結構是具有負泊松比特性的週期性結構，亦可稱之為膨脹材料，先前的研究通常為透過有限元素軟體與布洛赫定理分析此類型週期結構之波傳行為，入射一彈性波，其中包含縱波、橫波及其耦合現象等，觀察其不同物理現象，進而發現在某些頻段下的彈性波在此週期結構中無法傳遞，產生能隙的現象，使特定角度與頻率之彈性波無法通過聲子晶體。本文首先計算二次手性、環形二次手性、環形正方蜂巢、反四手手性及星形蜂巢狀結構五種結構之等效材料參數，討論其泊松比及楊氏係數之變化，並將此參數實際代入聲子晶體，討論負泊松比對於能隙之影響，分析由二維聲子晶體基板與具週期柱狀突出物組合成之聲子晶體，參考文獻中原先使用矽當基板材料，本文則用先前計算出之等效材料參數將其替換後，觀察能隙現象之改變情形，並分別改變聲子晶體結構之幾何參數、替換柱子之材料與形狀，同樣比較這些改變對於能隙中心頻率及頻寬之影響。
在應用方面，本文利用此結構設計出含缺陷之聲子晶體與局部共振型聲子晶體。含缺陷聲子晶體方面，藉由有限元素軟體配合超晶胞法計算其色散關係，並分析其缺陷模態與品質因子；局部共振型聲子晶體中，則藉由加入橡膠層作為軟材料，並且使用不同的配置方式，藉由調變其填充比、高度等，讓聲子晶體之完全能隙有不同的工作頻率範圍，可用於設計濾波器或聲波開關等裝置，同時透過COMSOL Multiphysics®有限元素軟體分析進行全波模擬，證明這些設計之可行性。

Chiral and star-shaped honeycomb structures can be considered as periodic phononic structures. These structures can be design to have an effective negative Poisson’s ratio. The materials which have negative Poisson’s ratio are known as auxetic materials. The elastic wave propagations within the periodic honeycomb structures are investigated by using the finite element method and Bloch’s theorem.
In the present thesis, we calculate first the effective Poisson’s ratio and Young’s modulus of tetrachiral structures with quadratic ligament, anti-tetrachiral structures and star-shaped honeycomb structures by finite element static analysis. By changing the geometric parameters, we can design the honeycomb structures which can have effective negative Poisson’s ratio.
The dispersion analysis of two-dimensional phononic crystal plates with periodic stubs are analyzed. By changing different effective material parameters, we investigate the evolution of the complete band-gap for different negative Poisson’s ratio. We find that the first band-gap of tetrachiral structures with quadratic ligament is the lowest. By changing the geometric parameters of the phononic crystal, material and shape of the stubs, respectively, the influence on the center frequency and bandwidth of the first band-gap is also compared. We find that different height of stubs, plate thickness and lattice constants will have different effect on band-gap. And the larger the density of stub’s material is, the lower center frequency and narrower bandwidth we will get.
In the application, we use these structures to design the phononic crystals with defects and locally resonant. In the case of phononic crystal with defects, we analyze its defect modes and quality factor, and find that the higher the resonance frequency is, the better quality factor we will get. In locally resonant phononic crystal, the rubber layer is added as soft material. By changing its filling ratio or height, the complete band-gap will have different operating frequency ranges. The larger the filling ratio is, the higher the center frequency of the first band-gap we will find. But the band-gap frequency does not decrease as the height of rubber layer increases. These features can be used to design different kinds of multi-channel filters. We also demonstrate the feasibility of these designs by full wave simulations of COMSOL Multiphysics®.

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