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研究生：	陳衍吉 Chen, Yen-Ji
論文名稱：	雙功能梯度壓電條板之第III型破壞分析 Mode III Fracture Analysis of Embedded Cracks in Two Bonded Functionally Graded Piezoelectric Strips
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：	2009
畢業學年度：	97
語文別：	英文
論文頁數：	148
中文關鍵詞：	奇異積分方程式、功能梯度壓電材料、強度因子、能量密度因子
外文關鍵詞：	Intensity factors, Singular integral equations, Energy density factors, Functionally graded piezoelectric material
相關次數：	點閱：128 下載：4
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本論文主要目的在於分析含裂縫功能梯度壓電條板接合之面外問題。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、裂縫長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，不可滲透型及滲透型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，這結果與現有文獻一致。強度因子將隨著裂縫位置所在處的材料性質梯度遞增減而提高或降低，且邊界條件對於強度因子的影響甚為強烈，裂縫離邊界的距離漸增，邊界的影響愈趨緩，另一方面，強度因子也會因為相鄰材料的梯度變化而有不同的結果，當其非均質材料參數分別趨近於正負極限值時，可以分別退化成為簡單的不同邊界問題。而強度因子隨著裂縫中心距離變大而降低，因此裂縫中心距離,板條的邊界條件及接合鄰近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 studies the fracture behavior of two bonded functionally graded piezoelectric strips with internal cracks. The structure is subjected to anti-plane mechanical and in-plane electric loads simultaneously. The poling type of 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 to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects of edge boundary conditions, crack lengths, and the material non-homogeneous parameters on the intensity factors and energy density factors are discussed. Several degenerated problems are also discussed.
The results show that stress and electric displacement intensity factors are uncoupled and depend on the applied mechanical and electric loads, crack lengths, and crack locations. 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 factors occur at the crack tips with stiffer material properties. This result agrees with the existence literatures. The effects of boundary conditions and gradient variations of the adjacent bonded material on the intensity factors are also significant. When one of the material non-homogeneous parameters approaches positive or negative infinity, the corresponding results can be reduced to one non-homogeneous material with different boundary conditions. The results also show that normalized intensity factors decrease with increasing the center distance of two cracks. Therefore, the intensity factors are strongly affected by the crack distance and the edge boundaries of strips.
In this article, the energy density factors are used as the driving force of crack propagation. The results show that a greater normalized energy density factor occurs at a crack tip with softer material properties. However, because the material parameter Scr is unknown, it is unable to predict the direction of crack propagation.

Table of Contents
Abstract	i
摘要	ii
誌謝	iii
Table of Contents	iv
List of Figures	ix
List of Tables	xiii
Nomenclatures	xv
Chapter 1 Introduction	1
1.1 Background	1
1.2 Literature survey	2
1.2.1 Functionally graded materials	2
1.2.2 Manufacturing process of functionally graded materials	5
1.2.3 Four-type assumptions of Gradient Material	6
1.2.4 Piezoelectric materials	9
1.2.5 The constitutive equations of piezoelectric materials	11
1.2.6 Boundary conditions on crack surfaces of piezoelectric materials	14
1.2.7 Functionally graded piezoelectric material	17
1.3 Problem statement	18
1.4 Outline of the dissertation	20
Chapter 2 Formulation of the Problem	22
2.1 Geometry of the problem	22
  2.2 Solution procedure	23
  2.3 Governing equations	24
  2.4 Superposition technique	25
  2.5 Impermeable and permeable crack problems	28
  2.6 Boundary Conditions	28
       2.6.1 Continuity conditions on the interface	29
       2.6.2 Boundary conditions on the crack surfaces	31
       2.6.3 Boundary conditions on the edges surfaces	32
  2.7 Degenerated problems	40
       2.7.1 A FGPM strip bounded to a FGPM medium with an internal crack, respectively	40
       2.7.2 A homogeneous piezoelectric half-plane containing two internal cracks	45
       2.7.3 Two cracked FGPM half-planes are bonded together	45
       2.7.4 A cracked FGPM surface layer bonded to a FGPM surface layer (crack 2 vanishes)	46
       2.7.5 A cracked FGPM strip bonded to a FGPM half-plane (crack 2 vanishes)	47
       2.7.6 A crack in a FGPM strip	48
       2.7.7 A cracked FGPM half-plane bonded to a FGPM strip (crack 2 vanishes)	50
       2.7.8 A crack in one of two bonded FGPM half-planes	51
       2.7.9 An infinite FGPM medium contains an internal crack	52
       2.7.10 A FGPM half-plane contains a crack (i.e. h1 → ∞, h2 → 0, crack 2 vanish)	52
  2.8 Permeable crack problem	54
Chapter 3 Numerical Procedure and Fracture Criterion	56
  3.1 Solutions of singular integral equations	56
  3.2 Stress and electric displacement intensity factors	61
  3.3 Comparison the intensity factors of electrically impermeable and permeable cracks	65
  3.4 Energy density theory	68
Chapter 4 Results and Discussions	73
  4.1 Two bonded FGPM cracked-strips	73
       4.1.1 A homogeneous piezoelectric strip containing two cracks	74
       4.1.2 Two bonded FGPM strips with symmetric crack locations	76
       4.1.3 Effects of two bonded FGPM strips with non-symmetric crack locations	78
       4.1.4 A center crack of a FGPM strip bonded to a center crack of another FGPM strip 	81
       4.1.5 A crack of a FGPM strip bonded to another cracked FGPM strip when the length of crack 1 is changed	85
       4.1.6 Effects of edge boundary conditions	88
       4.1.7 A FGPM strip containing two cracks with different non-homogeneous parameters	89
       4.1.8 Effects of the different electric loads	91
  4.2 A cracked FGPM strip bounded to a cracked FGPM substrate	93
       4.2.1 A homogeneous piezoelectric half plane containing two internal cracks	93
       4.2.2 A cracked FGPM surface layer bonded to a cracked piezoelectric homogeneous half-plane	96
       4.2.3 Two lined cracks in a FGPM half plane	98
       4.2.4 A cracked FGPM surface layer bonded to a cracked FGPM half-plane	100
  4.3 Two cracked FGPM half-planes bonded together	103
       4.3.1 A homogeneous piezoelectric medium containing two collinear cracks	103
       4.3.2 Two bonded FGPM cracked media with the same non-homogeneous parameters	105
  4.4 A crack in an FGPM strip bonded to a FGPM strip	107
  4.5 A crack in a FGPM strip	109
  4.6 A cracked FGPM substrate bonded to a FGPM strip	111
  4.7 A crack in one of two bonded FGPM half-planes	113
  4.8 The convergence of number of the collocation points	113
Chapter 5 Conclusions and Further Research	115
  5.1 Conclusions	115
  5.2 Further Research	118
       5.2.1 Introduction of external loading types	118
       5.2.2 Two bonded FGPM strips with an embedded arbitrarily oriented crack, respectively.	118
References	119
Appendix A:  Behavior of the dislocation functions gj (t)	132
Appendix B:  Plemelj formulas	134
Appendix C:  Kernels kiA-A (x,t) in Eqs. (2.85, 2.86)	136
Appendix D:  Kernels kiA-A (x,t) in Eqs. (2.85, 2.86)	138
Appendix E:  Kernels kiA (x,t) in Eqs. (2.115, 2.116)	140
Appendix F:  Kernels kiB (x,t) in Eqs. (2.115, 2.116)	141
Appendix G:  Chebyshev polynomials	142
Appendix H:  The weight of the Chebyshev polynomials of the first kind	146
Appendix I:  The residue theory used in this dissertation	147
                                    

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校內：2010-07-29公開
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