Phononic Local Resonances of Lamb Waves in a Plate with Periodic Stubbed Surface

本論文主要為分析及討論具週期性柱狀表面平板之板波局部共振現象。本文所使用的計算方法為有限元素法，有限元素法長久以來被廣泛使用在結構問題的探討及模擬，搭配上布拉格（Bloch）週期性邊界條件，有限元素法可以有效的分析及計算二維聲子晶體的頻散曲線圖（dispersion relation）及位移場。本文亦利用算例來比較及分析有限元素法與平面波展開法及時間域有限差分法的差異。 在分析計算二維聲子晶體結構中，首先利用鋼圓柱以正方晶格無窮排列於鋼基材之兩面。由頻散關係圖形，可清楚的發現頻溝現象，經過分析及比對，可以知道由鋼柱高度及鋼板厚度比例即為決定頻溝現象的主要參數。 第二部分，在分析以矽圓柱為無窮週期單面排列於矽基材上時，發現有駐波型式的模態出現在頻散關係曲線圖中，由於駐波模態的出現可知其群速度為零。研究中指出，當圓柱上有凹槽時，可以使駐波型式的模態頻率下降，當調整凹槽的幾何條件，可以使駐波模態座落在頻溝之中，造成所謂的局部共振現象。當局部共振現象發生時，彈性波無法在結構中傳遞，而造成彈性波會被侷限在波源附近，即所謂之波局域化（wave localization）。 最後，根據以上之物理性研究，及特定之材料組合暨合理的幾何條件，在聲子晶體頂端的幾何凹槽內，位移場的變化型態主要在XY平面上，若在此缺陷內滴入微量的液體或液滴，可造成液體或液滴因外圍邊界變形而受到擠壓，局部共振型式的現象可以被設計作為在微分析上液珠混合或高頻震盪加速反應時效之應用。In this thesis, the phononic local resonances of lamb waves in a plate with periodic stubbed surface are simulated and analyzed. The calculations of dispersion curves of these active structures are conducted with the help of the finite element method (FEM). Numerical examples are obtained for a square lattice of columns grown on a steel slab. It is found that several complete band gaps with a variable bandwidth exist for elastic waves. We found that the key parameter for the existence position and width of these complete band gaps is the ratio about the columns height, h, and the slab thickness, d. Secondly, the phenomena of standing wave modes in phononic crystal structures are discussed and analyzed. According to the cavities set on the phononic crystal structures, some standing wave modes are located in the band gap. When the standing wave modes are located in band gaps, local resonances are occurred. Finally, some applications of local resonances are designed because the dispersion curves and displacement fields are simulated. This advantage can be used to mix or vibrate in high frequency. In microanalysis, this method can be used to accelerate the reactions of liquids.謝誌-I 中文摘要-II ABSTRACT-III CONTENTS-IV NOTATIONS-VI LIST OF FIGURES-VIII LIST OF TABLES-XIII CHAPTER 1 INTRODUCTION-1 1-1 Motivation-1 1-2 Literature Review-2 1-3 Overview of the Text-3 CHAPTER 2 THEORY OF PHONONIC CRYSTALS AND FINITE ELEMENT METHOD-7 2-1 Theory of the Wave Propagation in the Phononic Crystals-7 2-1.1 Governing Equation-8 2-1.2 Bloch’s Theorem and Periodic Boundary Condition-9 2-2 Comparison between FEM and PWE in Bulk and Lamb Waves-10 2-3 Comparison between FEM and FDTD in Bulk Waves-13 CHAPTER 3 COMPLETE BAND GAPS IN PHONONIC-CRYSTAL SLABS ACCOMPANIED WITH PERIODIC COLUMNS-23 3-1 Numerical Simulation of Band Structures-23 3-2 Eigenmodes Analysis-28 3-3 Application of Phononic Crystal Slabs with Columns-31 CHAPTER 4 LOCAL RESONANCES OF LAMB WAVES IN A PLATE WITH PHONONIC BAND STRUCTURES ON THE TOP-59 4-1 Characteristics of Local Resonances-59 4-2 Phononic Crystal Structures on Plates-60 4-3 Phononic Crystal Structures with Cavities-62 4-4 Application of Local Resonance-65 CHAPTER 5 CONCLUSIONS AND OUTLOOK-86 5-1 Conclusions-86 5-2 Outlook-88 REFERENCES-8