二維壓電聲子晶體之頻帶間隙研究

Band Gaps Study for Two-Dimensional Piezoelectric Phononic Crystals

紀兆騏(Chou-Chi Chi)

指導教授 ： 周元昉

國立臺灣大學/工學院/機械工程學研究所/碩士(2004年)

https://doi.org/10.6342/NTU.2004.10247

摘要

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，探討影響其頻帶間隙的各種重要參數。首先利用平面應力的條件配合上下表面電性的邊界條件，推導出含壓電行為的等效彈性係數，再利用平面波展開法推導其波傳理論。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，代入波動方程式，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，由此可得二維壓電週期結構之波傳頻譜與位移場形。
使用前述推導的理論，針對不同的材料組成、填充比和填充物形狀等，探討其對頻帶間隙的影響，研究發現週期結構以三種材料組成所得的頻帶間隙會比兩種材料組成的還要高頻且頻帶更寬。且以金屬為核心材料，壓克力為外層包覆材料，壓電材料為基材，圓形正方晶格排列的週期結構，在第六、第七色散曲線間，具有高頻且頻帶很寬的頻帶間隙。另外，探討此材料組成下，上下表面有無鋪設電極、壓電效應等對頻帶間隙的影響，以及各個頻帶的位移場形圖，進一步瞭解各處質點的運動軌跡，提供更多可資應用的特性。

關鍵字

平面應力 ； 聲子晶體 ； 頻帶間隙 ； 三種材料 ； 壓電 ； 平面波展開法

並列摘要

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 photonic crystal is studied in this thesis. First of all the plane-stress and electric boundary conditions are employed to derive the equivalent elastic moduli. The plane wave expansion method and the Bloch theorem are used to modify the wave 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.
In order to handle the frequency span of band gaps, a study based on changing the materials, the filling ratio and the shape of inclusions is performed. Wide band gaps can be achieved by using composite inclusions that is made of two different kinds of materials. When metals are used as the core materials that are surrounded with PMMA to form the inclusions, and piezoelectric material as the host matrix provides very wide band gaps. In addition, the effects of electrodes are also studied. Finally, by observing the phase of the displacement field, the locus of particles in the periodic structure is further understood that also provide useful information.

並列關鍵字

phononic crystal ； piezoelectric ； band gap ； three component ； plane-stress ； plane wave expansion method

參考文獻

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