本論文主要討論低維彈性系統的拓樸相變 (topological phase transition) 與拓樸邊緣態 (topological edge states) 之研究。第一部份主要透過連續調變一維週期繩波系統的單位繩段長度以及繩線密度使繩波系統產生拓樸相變，再藉由數值方法來計算系統的札克相位 (Zak phase) 來探討拓樸相並透過系統波阻抗的邊界條件來預測界面態的存在，最後實際建構一條由兩種不同拓樸相所構成的繩波週期系統來驗證界面態的存在。第二部份是利用COMSOL Multiphysics模擬軟體中的弱形式方程 (weak form) 模組來進行拓樸純剪波邊緣態的研究。此處的二維週期彈性波系統是由釔鐵石榴石 (Yttrium iron garnet, 簡稱 YIG) 以三角晶格方式排列於鎢 (Tungsten, 化學符號為 W) 背景中所構成的聲子晶體。藉由純剪波的頻帶結構圖找到一個狄拉克簡併點 (Dirac point) 後，再透過加入與純剪波振動方向平行的外加磁場來破壞系統的時間反演對稱性 (time-reversal symmetry)，使頻帶結構中的的狄拉克點處開啟一個拓樸非平庸的頻隙，最後透過超晶胞方法 (supercell method) 驗證拓樸純剪波邊緣態的存在，也討論一些未來可進一步探討的方向。

In this thesis, we discuss the topological phase transition and topological pure shear modes in low-dimension elastic systems. First part, we investigate the topological phase transition by varying the length and the linear densities of the segment strings in one unit string. We can distinguish the topological phase of the periodic string system by calculating the Zak phases and predict the existence of the interface state by the boundary condition of the interface impedances in the string system. At the last, we build a string system which consists of two different topological phases of periodic strings at the interface to verify the existence of the interface state numerically. Second part, we use the weak form module in the COMSOL Multiphysics software to show that the topological pure shear modes can exist in a two dimensional triangle lattice phononic crystal which is composed of yttrium iron garnet (YIG) rods embedded in the tungsten (W) background. In the absence of external magnetic field, the shear wave band structure shows the existence of Dirac cones that are well-known in graphene. As a uniform magnetic field is applied to the system, due to the time-reversal symmetry breaking arising from the magnetoelastic interactions in YIG, the Dirac points are spitted to topologically non-trivial band gaps. The nontrivial topological nature is verified by the numerical calculation of Chern numbers for each band, and simulation of the propagation of the edge states. A possible design for realization of such a system is discussed.

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