壓電聲子晶體之平面應力模式

Abstract

聲子晶體的頻帶間隙特性造就許多有效的新裝置。本文研究具平面應力特性的二維壓電聲子晶體的波傳頻譜，基於平面應力忽略厚度方向上位移在動態行為下慣性造成的影響，所以提出了一修正平面應力模式，以期能適用於更大的頻率範圍。 首先利用平面應力條件配合上下表面電性的邊界條件，找出厚度方向上因為泊松效應造成的應變及相對應的厚度方向上的位移，使用漢彌爾頓原理找出此狀況下的統御方程式。再利用平面波展開法推導其波傳理論，使用倒晶格向量配合布洛赫理論對相關的材料常數與位移作傅立葉展開，代入統御方程式，推導出一廣義特徵方程式，利用數值方法可解出特徵值和特徵向量，由此求得二維壓電週期結構之波傳頻譜與位移場形。 使用前述理論推導，針對不同材料組成、厚度以及電性邊界下，探討其對頻帶間隙的影響。發現未修正模式與修正模式下低頻的色散曲線並無太大差異，但在高頻時修正平面應力模式下的色散曲線則會隨著厚度的增加而往低頻移動；並探討各個頻帶的位移場形圖，進一步瞭解各材料的運動情形，提供更多可資應用的特性。

Possessing the properties of band gaps, phononic crystals have led the invention of many new devices. In order to handle this characteristic potential, the spectrum of a plane-stress piezoelectric phononic crystal is studied in this thesis. Base on plane-stress assumption neglect the inertia effect of the displacement in the direction of thickness in dynamic situation. Modified plane-stress model is suitable for use in a wider frequency range. First of all, the plane-stress and electric boundary conditions are employed to derive the strain and relative displacement because of Poisson effect, then we derive the governing equation by Hamilton principle. The plane wave expansion method and the Bloch theorem are used to modify the governing equation into the one fit for periodic structures. The material parameters and displacement fields are expanded with Fourier series with respect to reciprocal lattice vectors. Finally, a generalized eigenvalue problem is formed that is solved with numerical method to obtain the frequency spectrum and the displacement fields. The band gaps are found from the frequency spectrum. A study based on changing the materials, thickness, and electric boundary conditions is performed. We find that there is no difference between the dispersion curve of plane-stress model and modified plane-stress model in low frequency range. But the dispersion curve of modified plane-stress model will move down in high frequency range. Finally, by observing the phase of the displacement field, the dynamic situation of materials in the periodic structure is further understood that also provide useful information.