臺灣博碩士論文加值系統

摘 要
聲子晶體乃是一種複合材料，其結構包含均質的填充物週期性地排列於具有不同物理性質的基底材料中。兩種組成聲子晶體的材料在物理性質上的差異包括質量密度和彈性勁度等。當聲波在聲子晶體中傳播，會因週期性填充物的散射而阻擋特定頻率範圍的聲波通過或將能量侷限在聲子晶體的結構中。藉由這些特性，許多的文獻已經發表了聲子晶體在操控聲波行為的應用，這些應用包括聲波反射鏡、聲波折射元件、高效率的波導、可調式濾波器和雙工器等。此外，在高頻元件的應用及學術研究的興趣上，探討由壓電材料所組成的聲子晶體，以及表面波與板波於聲子晶體結構中的傳播特性具有其重要性。
本論文旨在研究徹體波、表面波和板波於含有壓電材料之聲子晶體結構中傳播之特性及行為。首先，本文簡短的介紹用於描述週期性結構的基本數學定義。接著，整合三維平面波展開法及壓電晶體波傳理論，本文發展一套數學理論作為分析工具，並加以數值的計算結果來分析聲波在壓電聲子晶體中的頻散關係或頻帶結構；且進一步探討如聲子晶體的晶格對稱性、填充材料之填充率、材料性質的對比及壓電效應對於聲波波傳及頻溝之影響。
文中特別針對表面波及板波模態進行分析。其中包含了週期性結構所引致之表面波機電耦合係數之頻散性質和虛擬表面波的產生。並且，於氧化鋅和硫化鎘所組成之二維壓電聲子晶體中亦發現Bleustein-Gulyaev表面波之傳播及其頻帶反折效應；此種表面波僅存在於特定的壓電介質中。再者，針對具有兩個平行表面的平板，本文藉由引入另一邊界條件於三維平面波展開法中，使得平面波展開法得以進一步用計算板波於平板結構的聲子晶體中的傳播特性。分析結果顯示，除了聲子晶體的晶格對稱性、填充材料之填充率、材料性質對比的影響外，板波於聲子晶體中的頻溝大小乃至其存在與否與平板的厚度密切相關。
最後，本文結合Mindlin的板理論和平面波展開法來分析低階的板波模態於聲子晶體平板中的特性。相較於三維平面波展開法，以Mindlin板理論為基礎的平面波展開法在計算板波頻散曲線上所需的時間上有相當大幅的改善；因此，當聲子晶體平板的填充物及基材之間的材料性質差異性大或聲子晶體平板含有如方柱等具不平滑輪廓之填充物時，此方法更適用於這些結構的波傳分析。這些結構在計算上需要較多的平面波展開數量才能得到良好的收斂性，當計算中包含有較大量的平面波展開數量時，此方法可將所需的計算時間縮短至可接受的範圍。利用此方法，本文亦進一步地計算及探討二維聲子晶體平板的局部共振現象及其極低頻區域的頻溝形成。
綜言之，本文發展了用於分析於二維壓電聲子晶體結構中傳播的徹體波、表面波以及板波特性的平面波展開法，並藉此方法計算及探討這些形式的聲波在其結構中的特徵。

ABSTRACT
Phononic crystals are composite materials which consist of homogeneous elastic inclusions distributed periodically in a background medium characterized by different physical properties, such as mass density and elastic stiffness. Thus far, numbers of released researches have demonstrated the possible usage of phononic crystals for acoustic manipulations, such as acoustic mirrors/refractive devices, high-efficiency waveguides with frequency modulation in the transmittivity, tunable filters, and de-multiplexers, etc, based on the localization and the formation of frequency band gap of acoustic waves in such periodic composites. For high-frequency applications and academic interests, phononic crystals comprised of piezoelectric materials and the propagation of guided waves like the surface modes and plate modes in such composites are important to study.
In this study, the propagation of bulk acoustic waves, surface acoustic waves, and Lamb/plate waves in phononic-crystal structures containing piezoelectric constituents is theoretically investigated. First, the basis in the description of periodic structures is briefly introduced. Next, the full three-dimensional plane wave expansion (PWE) method is utilized to develop the mathematical formulation by integrating the method into the governing field equations of waves in piezoelectric solids. Then the nume- rical calculations are presented to analyze the dispersion relations or the frequency band structures of acoustic waves and to discuss the effects of lattice symmetries, filling fractions of inclusions, material contrasts, and piezoelectricity on the complete frequency band gaps.
In particular, the characteristics of surface modes and plate modes in phononic- crystal structures are probed. The periodicity of the structure results in a dispersive property for the electromechanical coupling coefficients of surface waves and the existence of pseudosurface waves. In addition, the Bleustein-Gulyaev surface wave, which has no counterpart in a non-piezoelectric medium, in ZnO/CdS piezoelectric phononic crystal and the folding effect are found. Moreover, propagation of Lamb waves in plate structure created by a phononic crystal is analyzed by introducing boundary conditions for another plane surface into the PWE formulation. In addition to the lattice symmetries, filling fractions, and material contrasts, the existence and the width of complete band gaps of Lamb waves are crucially affected by the ratio of the plate thickness to the lattice period.
Finally, Mindlin’s plate theory is applied to address the problem of lower order Lamb modes in a phononic-crystal plate. Compared to the full three-dimensional PWE method, Mindlin’s theory based PWE formulation has excellent performance in coping with the phononic-crystal plate consisting of constituents with large acoustic mismatch and/or inclusions with a non-smooth contour in their cross section such as square rods that need to sum over numerous plane waves to ensure the convergence by reducing the computation time considerably. The frequency band structures of locally resonant phononic-crystal plates and subfrequency band gaps are calculated as well.
In brief, the PWE methods to analyze the propagation of bulk waves and guided waves in two-dimensional piezoelectric phononic-crystal structures are formulated, and their characteristics are investigated and discussed in this work.

CONTENTS
Acknowledgement......i
Chinese Abstract......iii
Abstract......v
Contents......vii
Symbols......xi
List of Figures......xv
List of Tables......xxi
1. Introduction......1
1.1 Motives......1
1.2 Literature Review......3
1.3 Outline of the Dissertation 7
2. Crystal Lattices and Theory of Piezoelectric Wave Propagation......11
2.1 Bravais Lattice......11
2.2 Wigner-Seitz Primitive Unit Cell......12
2.3 Primitive Lattice and Reciprocal Lattice in Two Dimensions......13
2.4 Brillouin Zones of a Two-dimensional Lattice......14
2.5 Plane Wave Expansion Method......15
2.6 Piezoelectric Waves in a Two-dimensional Periodic Structure......16
2.6.1 Piezoelectric Constitutive Relations......17
2.6.2 Equations of Motion and Poisson’s Equation......19
2.6.3 Eigenvalue Problem 20
2.6.4 Mixed and Transverse Polarization Modes......23
3. Bulk Waves in Two-dimensional Piezoelectric Phononic Crystals......33
3.1 Square, Triangular, and Hexagonal Lattices......33
3.1.1 Square Lattice......33
3.1.2 Triangular Lattice......34
3.1.3 Hexagonal Lattice......35
3.2 Structure Factors......36
3.3 Band Structures of Bulk Piezoelectric Waves......37
3.3.1 Bi12GeO20/Epoxy Piezoelectric Phononic Crystal......38
3.3.2 PZT-4/Epoxy Piezoelectric Phononic Crystal......40
3.3.3 Air/X-Cut LiNbO3 Piezoelectric Phononic Crystal......41
3.3.4 Air/ST-Cut Quartz Piezoelectric Phononic Crystal......43
3.4 Effect of Piezoelectricity on the Band Structures......44
4. Surface Waves in Two-dimensional Piezoelectric Phononic Crystals......61
4.1 PWE Formulation for Surface Waves......61
4.1.1 Eigenvalue Problem of Surface Wave......61
4.1.2 Surface Boundary Conditions......62
4.2 Band Structures of Piezoelectric Surface Waves......64
4.3 Displacement Fields of Piezoelectric Surface Waves......66
4.4 Pseudosurface Waves 66
4.5 Electromechanical Coupling Coefficients 67
4.6 Bleustein-Gulyaev Surface Waves 68
4.6.1 BG Waves in a Homogeneous Piezoelectric Crystal......69
4.6.2 BG Waves in ZnO/CdS Piezoelectric Phononic Crystal......72
5. Lamb Waves in Two-dimensional Phononic-crystal Plates......89
5.1 PWE Formulation for Lamb Waves in a Phononic-crystal Plate......90
5.1.1 General Solution of Lamb Wave......90
5.1.2 Surface Boundary Conditions......90
5.2 Lamb Waves in a Homogeneous Piezoelectric Plates......92
5.2.1 Dispersion Curves......92
5.2.2 Modes......93
5.3 Band Structures of Lamb Waves in Phononic-crystal Plates......94
5.3.1 Elastic Phononic-crystal Plate......94
5.3.2 Piezoelectric Phononic-crystal Plate......96
5.4 Effect of Plate Thickness on the Band Structures......97
6. Mindlin’s Plate Theory Based PWE Method for Lower Order Lamb Modes......109
6.1 Mindlin’s Theory Based PWE Formulation......110
6.1.1 Mindlin’s Plate Theory......110
6.1.2 Application of PWE Method to Mindlin’s Theory......114
6.2 Calculations of Frequency Band Structure ......116
6.2.1 Solid/Solid Phononic-crystal Thin Plate......116
6.2.2 Air/Solid Phononic-crystal Thin Plate......119
6.3 Locally Resonant Phononic-crystal Thin Plate......120
6.3.1 Two-component Phononic-crystal Thin Plate......120
6.3.2 Three-component Phononic-crystal Thin Plate......124
7. Conclusions and Prospects......141
7.1 Conclusions......141
7.2 Prospects......143
Appendix A. Proof of Bloch Theorem for Piezoelectric Phononic Crystal......147
Appendix B. Category of Piezoelectric Materials......149
Appendix C. Matrix Components of the Matrix: M......155
Appendix D. List of Used Material Constants......159
Appendix E. Matrix Components of the Matrix: D......161
References......165
Vita......173

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