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研究生：	徐瑋鴻 Hsu, Wei-Hung
論文名稱：	含嵌入式任意方向裂縫之功能梯度壓電材料面外問題破壞分析 Mode III Fracture Analysis of an Embedded Arbitrarily Oriented Crack in Functionally Graded Piezoelectric Materials
指導教授：	褚晴暉 Chue, Ching-Hwei
學位類別：	博士 Doctor
系所名稱：	工學院 - 機械工程學系 Department of Mechanical Engineering
論文出版年：	2008
畢業學年度：	96
語文別：	英文
論文頁數：	170
中文關鍵詞：	強度因子、任意方向裂縫、奇異積分方程式、功能梯度壓電材料、能量密度因子
外文關鍵詞：	energy density factors, Intensity factors, Arbitrarily oriented crack, Functionally graded piezoelectric material, Singular integral equations
相關次數：	點閱：193 下載：8
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本論文主要目的在於分析含有嵌入式任意方向裂縫之功能梯度壓電材料的破壞問題。壓電材料之極化方向為六方對稱型，材料梯度假設為指數型函數。利用Fourier轉換法，可將問題轉變成一組奇異積分方程式，再藉由Gauss-Chebyshev積分公式進行數值求解，並從數值解中探討邊界條件、裂縫角度及非均質材料參數對於強度因子和能量密度因子的影響。相關的退化問題也於文中有詳細的討論。
研究結果顯示，應力強度因子與電位移強度因子分別僅與其外加之機械負載與電負載、裂縫長度及位置有關，且彼此為非耦合。只要外加機械及電負載為相同形式，非滲透型(impermeable)及滲透型(permeable)裂縫所對應的應力強度因子是相同的。至於較大的強度因子會發生在材料較強的裂縫尖端上，這結論與現有文獻一致。而強度因子隨著裂縫方向角度變大而降低，因此裂縫方向對於強度因子的影響甚劇。邊界條件及接合基底材料的強度也影響著強度因子的大小。
本文使用能量密度因子作為裂縫成長之驅動力，研究結果顯示較大的能量密度因子會發生在材料較弱的裂縫尖端上，然而功能梯度材料的材料參數Scr仍屬未知，故目前尚無法直接判斷裂縫開始成長的方向。

This dissertation investigates the fracture problem of an embedded arbitrarily oriented crack in an FGPM strip bonded to an FGPM layer subjected to anti-plane mechanical and in-plane electric loads. The piezoelectric material has 6mm symmetry and the material gradient is assumed to be in an exponential form. By using the Fourier transform, the problem can be formulated as a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects of edges, crack orientation, and inhomogeneous material parameters on intensity factors and energy density factors are discussed. Some degenerated problems are also discussed.
The results show that stress and electric displacement intensity factors are uncoupled and both depend on the applied mechanical and electric loads, crack length, and crack location. If the applied mechanical and electric loads are the same function, the stress intensity factors are the same for electrically impermeable cracks and electrically permeable cracks. The greater normalized intensity factor occurs at a crack tip with stronger material properties. This result is consistent with existing literature. The results also show that normalized intensity factors decrease with increasing crack orientation. Therefore, the intensity factors are strongly affected by crack orientation.
In this dissertation, the energy density factors are used as the driving force of crack propagation. The results show that the greater normalized energy density factor occurs at a crack tip with softer material properties. However, because the material parameter Scr is unknown, the direction of crack propagation cannot be predicted.

Abstract	i
摘要	ii
誌謝	iii
Table of contents	iv
List of tables	viii
List of figures	x
Nomenclatures	xvii
Chapter 1 Introduction	1
1.1 Introduction	1
1.2 Functionally graded materials	3
1.2.1 Concept	3
1.2.2 History	4
1.2.3 Applications	4
1.2.4 Manufacturing process techniques	5
1.2.5 The types of gradation	6
       1.3 Piezoelectric materials	9
       1.3.1 History	9
       1.3.2 Fundamental piezoelectric relationships	10
       1.3.3 Electric boundary condition on crack surfaces	13
       1.4 Literature survey	15
       1.5 Problem statement	17
       1.6 Outline of the dissertation	17
Chapter 2 Formulation of the problem	19
       2.1 Geometry of the problem	19
       2.2 Solution procedure	20
       2.3 Governing equations	21
       2.4 Superimposition technique	23
       2.4.1 The cracked functionally graded strip	23
       2.4.1.1 Sub-problem 1	24
       2.4.1.2 Sub-Problem 2	26
       2.4.2 The uncracked functionally graded strip	30
       2.5 Boundary conditions	33
Chapter 3 Singular integral equations and intensity factors	35
       3.1 Development of singular integral equations	35
       3.1.1 Condition along the plane of the crack	35
       3.1.2 Boundary and continuous conditions	36
       3.1.3 Mixed boundary conditions	39
       3.2 Asymptotic analysis of Fredholm kernels	42
       3.2.1 Fredholm kernel k2(x, t)	43
       3.2.2 Fredholm kernel k3(x, t)	44
       3.3 Evaluation of singularities and expressions for intensity factors	45
       3.3.1 Internal crack	45
       3.3.2 Crack terminating at the interface	50
       3.4 Degenerated problems	51
       3.4.1 A crack in an FGPM strip bonded to a homogeneous piezoelectric layer	51
       3.4.2 A crack in an FGPM strip bonded to an FGPM half plane	51
       3.4.3 A crack in an FGPM strip bonded to a homogeneous piezoelectric half plane	52
       3.4.4 A crack in an FGPM infinite plane	53
       3.4.5 A crack in an FGPM strip	53
Chapter 4 Numerical procedure and energy density theory	55
       4.1 Numerical procedure for the general case	55
       4.2 Numerical procedure for specific crack configuration	57
       4.2.1 Internal crack	57
       4.2.2 Cracks terminating at the interface	62
       4.3 The intensity factors for electrically impermeable and
permeable cracks	62
       4.4 Energy density theory	65
Chapter 5 Results and discussion	69
       5.1 A crack in an FGPM infinite plane	69
       5.2 A crack in an FGPM strip	77
       5.3 A crack in an FGPM strip bonded to a homogeneous piezoelectric half plane	92
       5.4 A crack in an FGPM strip bonded to an FGPM half plane	105
       5.5 A crack in an FGPM strip bonded to a homogeneous piezoelectric layer	110
       5.6 A crack in an FGPM strip bonded to an FGPM layer	122
Chapter 6 Conclusion and future work	131
       6.1 Conclusion	131
       6.2 Future work	133
       6.2.1 Introduction of type of loads	133
       6.2.2 Arbitrarily oriented crack and material gradient	134
References	135
Appendix A Real parts of the parameters p1 and p2	157
Appendix B Behavior of the dislocation functions gi(t)	158
Appendix C Plemelj formulas	161
Appendix D Chebyshev polynomials	164
Appendix E The weight of Chebyshev polynomials of the first kind	168
Appendix F Derivation of Eq. (4.22)	169
                                    

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