臺灣博碩士論文加值系統

本研究主要以彈性力學的觀念探討功能梯度材料(FGM)之力學行為。研究中分為兩大部分：第一部分是以有限元素之數值方法探討FGM濺鍍層與多層梯度濺鍍層之破壞特性，第二部分則推導FGM板分別在受到橫向載重以及純彎矩載重下之理論解，並以有限元素法驗證。在兩部分之分析當中，所用之材料為冪次函數、S型函數及指數函數之FGM。
在第一部分之FGM破壞研究中，考慮濺鍍層-基材之複合材料，基材為均質材料，但濺鍍層則使用FGM材料或多層梯度材料。分析裂縫由濺鍍層表面產生後，以垂直界面的方向延伸時之破壞行為。研究發現濺鍍層使用FGM可降低複合材料自由邊界上，由材料互相約束所產生之內應力。對於濺鍍層勁度高於基材勁度之複合材料，若濺鍍層使用FGM，則材料之分佈消除了兩材料界面上之應力強度異點之現象，當裂縫尖端接近兩材料之界面時，應力強度因子快速下降，因此阻抗裂縫之延伸。對於濺鍍層勁度低於基材勁度之複合材料而言，FGM濺鍍層則像一座橋樑將濺鍍層與基材之材料差異連接起來，幫助裂縫往基材內部延伸。
第二部分FGM板之研究，主要推導簡支矩形FGM板受到橫向載重以及純彎矩載重下之級數解，假設板為中等厚度，且厚度為均勻之板。藉著將外力化為Fourier級數，FGM板之全解則以三角函數表之。此級數解並以有限元素法之數值解比較之。結果顯示：FGM板中性面之位置隨著修正Dunders參數之增加而向受壓側移動。另外，由FGM板之應力分佈曲線顯示，FGM可將板邊緣上所產生之最大撓曲張應力轉移至板內部，部分由內部材料承受，此結果表示FGM相對提高材料之抗拉性，而三種不同FGM中又以S型FGM有最佳之效果。

The object of this thesis mainly investigates the mechanics behaviors of the functionally graded materials (FGM) based on the elastic concept. There are two parts in this thesis. Part one is to study the fracture behaviors of cracked coating-substrate composite with FGM and multi-layered coatings by the finite element method. Part two is to derive the analytical solutions for FGM plates subjected to transverse loads and the pure bending. The analytical solutions are proved by the numerical solutions obtained by finite element method. In both part one and part two, the variation of the material properties in FGMs is defined by the volume fraction of constitute material based on power-law function, sigmoid function, or exponential function.
In part one, cracking in a caoting-substrate composite is considered. The material of the substrate of the composite is assumed to be homogeneous, while the coating uses the FGMs or multi-layered materials. The fracture behaviors of a crack appearing at the surface of the coating and propagating with the direction perpendicular to the interface between the coating and substrate are analyzed. Results showed that the use of FGMs coating can reduce the stresses due to the material mismatch on the free edge of the composite. For the case that the coating is stiffer than the substrate, if the coating uses FGMs, the continuous material on the interface eliminates the stress singularity and hence the stress intensity factors rapidly decrease when the crack tip arrives at the interface of the coating and substrate. Consequently, the crack propagation is arrested. However, for the case that the coating is softer than the substrate, the FGM coating behaves like a bridge to connect the mismatch of the coating and substrate and helps the crack further propagating into the substrate.
In the analysis of FGM plates, the complete solutions of rectangular, simple-supported, FGM plates under transverse loads and pure bending are derived. The FGM plates are assumed to have middle thickness and uniform thickness. By expressing the external load into Fourier series, the complete solutions of the FGM plates are found in terms of trigonometric functions. The complete solutions are compared with the numerical results of finite element method. Results revealed that the location of the neutral surface of the FGM plate moves to the compressive part of the plate as the modified Dundurs’ parameters increase. Moreover, the maximum bending stresses are inside the FGM plates, instead of in the upper or lower edges of the homogeneous isotropic plates. This phenomenon indicated that the maximum tensile stresses can be born impartially by the material inside the FGM plates and hence the bending resistance is increased. Among the P-, S-, and E-FGM plates, the S-FGM performs better than the other two FGMs.

第一章 緒論 ………………………………………………………………………1
1.1 研究目的 ……………………………………………………………………...2
1.2 研究內容 ……………………………………………………………………...3
第二章 功能梯度複合材料 ……………………………………………………5
2.1 功能梯度材料設計基礎理論 ………………………………………………...6
2.1.1 混合法則(Rule of Mixture Theories) …………….……………………….7
2.1.2 平均場理論(Mean Field Theories) ……………….………………………7
2.2 功能梯度材料之熱機械力學及基礎破裂理論 …………………………….10
2.2.1梯度材料製備所引入之應力 …………………………………………10
2.2.2邊界效應與應力之奇異場 ……………………………………………...13
2.2.3非均質固體破壞力學 …………………………………………………...16
2.2.4 梯度材料中裂縫成長之驅動力 ………………………………………..18
2.3 功能梯度材料之製備 ……………………………………………………….21
2.3.1 功能梯度材料之構造法製備 …………………………………………..23
2.3.1.1 固態粉末致密化製備 ……………………………………………...23
(A) 傳統粉末固結 ………………………………………………………23
(i) 粉末預製塊之製備 ………………………………………………..25
(ii) 粉末預製塊之致密化 …………………………………………….27
(B) 自蔓延高溫燃燒合成法(Self-propagating Hhigh-temperature Synthesis, SHS)(粉末反應法) ……………………………………….27
2.3.1.2 塗層法 ……………………………………………………………...28
(A) 等離子噴塗成型(Plasma Spray Forming) …………………………..28
(B) 雷射熔覆法(Laser Cladding) ………………………………………..30
(C) 薄膜沉積法(Film Deposition Processes) ……………………………30
(i) 物理氣相沉積(PVD法) ……………………………………………31
(ii) 化學氣相沉積(CVD法) …………………………………………..32
2.3.2 功能梯度材料之傳輸製備方法 ………………………………………..33
2.3.2.1 物質傳輸製備(Mass Transport Processes) …………………………34
2.3.2.2離心分離法和沉降法(Centrifugal Separation and Settling) ………..34
2.3.2.3宏觀偏析原理(Macrosegregation Processes) ……………………….35
2.3.3 小結 ……………………………………………………………………..36
2.4 FGM之力學相關研究文獻 ……………………………………………….36
2.5 結語 …………………………………………………………………………39
第三章 功能梯度材料板承受橫向載重之力學分析 …………………...43
3.1 FGM板之基本理論 ………………………………………………………….43
3.1.1平衡方程式 ………………………………………………………….......45
3.1.2 FGM板之應變－位移關係方程式 …………………………………….46
3.1.3 FGM板之應力－應變方程式及協和方程式 …………………………47
3.2 簡支矩形FGM板之理論解 ……………………………………………….51
3.3 冪次方函數型FGM板 …………………………………………………….56
3.3.1冪次方函數型FGM板之理論解 ………………………………………56
3.3.2 冪次函數型FGM板之數值分析 ………………………………………60
3.3.2.1 收斂測試 …………………………………………………………….62
3.3.2.2 修正Dundurs’參數 ＞0， =0 之問題 ………………………65
(A) 均佈載重 ……………………………………………………………65
(B) 線載重 ………………………………………………………………..67
(C) 點載重 ………………………………………………………………..71
3.3.2.3 修正Dundurs’參數 ＞0， 之問題 …………………..74
(A) 均佈載重 …………………………………………………………….74
(B) 線載重 ……………………………………………………………….76
(C) 點載重 ………………………………………………………………..79
3.4 S型函數FGM板 ……………………………………………………………83
3.4.1 S型函數FGM板之理論解 …………………………………………….83
3.4.2 S-FGM板之數值分析 …………………………………………………86
4.4.2.1 修正Dundurs’參數 ＞0， =0之問題 ………………………..87
(A) 均佈載重 …………………………………………………………..87
(B) 線載重 ……………………………………………………………..90
(C) 點載重 ……………………………………………………………..93
3.4.2.2 修正Dundurs’參數 ＞0， 之問題 ……………………95
(A) 均佈載重 …………………………………………………………..95
(B) 線載重 ……………………………………………………………..97
(C) 點載重 ……………………………………………………………...100
3.5指數函數型FGM板(E-FGM) ……………………………………………..103
3.5.1 E-FGM板之理論推導 …………………………………………………103
3.5.2 E-FGM板之數值分析 …………………………………………………107
3.5.2.1 修正Dundurs’參數 ＞0， =0之問題 ………………………...107
(A) 均佈載重 …………………………………………………………..108
(B) 線載重 ……………………………………………………………...110
(C) 點載重 ……………………………………………………………..114
3.5.2.2 修正Dundurs’參數 ＞0， 之問題 …………………..115
(A) 均佈載重 …………………………………………………………..115
(B) 線載重 ……………………………………………………………..117
(C) 點載重 ……………………………………………………………..120
3.6 S-FGM、S-FGM與S-FGM之力學行為比較 ……………………………123
(A) 均佈載重 …………………………………………………………..123
(B) 線載重 ……………………………………………………………..124
(C) 點載重 ………………………………………………………….…..127
第四章 功能梯度材料板於純彎矩作用之力學分析 ……………….....129
4.1 FGM板受純彎矩作用之理論解 …………………………………………..129
4.1.1 基本解 …………………………………………………………………132
4.1.2 一般彎矩載重之解 ……………………………………………………136
4.2 FGM板受純彎矩作用之數值分析 ………………………………………137
4.2.1 修正Dundurs參數 ， 之問題 …………………………..137
4.2.2 修正Dundurs參數 ， 之問題 ……………………..148
4.3 綜合討論 …………………………………………………………………...159
第五章 功能梯度材料及多層材料之破壞分析 ………………………161
5.1 理論基礎 …………………………………………………………………162
5.1.1 相異材料界面之應力異點現象 ………………………………………164
5.1.2 多層材料含裂縫之應力強度因子及能量釋放率 ……………………169
5.1.3 功能梯度材料之基本理論 ……………………………………………..71
(A) S-FGM …………………………………………………………………172
(B) P-FGM …………………………………………………………………173
(C) E-FGM …………………………………………………………………175
5.1.4 功能梯度材料含裂縫之應力強度因子及能量釋放率 ………………176
5.1.5 修正Dundurs參數 ……………………………………………..177
5.2 數值分析 …………………………………………………………………180
5.2.1 問題描述 ………………………………………………………………180
5.2.2 收斂測試及使用公式之測試 …………………………………………181
(A) 收斂測試 ……………………………………………………………181
(B) 使用公式之測試 ……………………………………………………182
5.2.2 濺鍍層使用多層材料之裂縫問題分析 ………………………………183
(A) 單層濺鍍層含垂直裂縫之問題 ……………………………………184
(B) 多層濺鍍層含垂直裂縫之問題 ……………………………………186
(C) 單層材料濺鍍層及多層材料濺鍍層之SIF行為比較 ………………189
5.2.3 S-FGM含裂縫問題之分析 ………………………………………….191
5.2.3.1 裂縫尖端位在濺鍍層內 ………………………………………….192
(A) 修正Dundurs參數 >0， =0 ……………………………………192
(B) 修正Dundurs參數 <0， =0 ……………………………………194
5.2.3.2 裂縫貫穿濺鍍層，並延伸入基材內，且外力均佈作用在裂縫表面………………………………………………………………….195
(A) 修正Dundurs參數 >0， =0 ……………………………………196
(B) 修正Dundurs參數 ＜0， ＝0 ………………………………….196
5.2.4 裂縫尖端位於基材內，外力只均佈作用在濺鍍層之裂縫表面 ……197
(A) 修正Dundurs參數 ＞0， ＝0 ………………………………….197
(B) 修正Dundurs參數 ＜0， =0 …………………………………..200
5.2.5 P-FGM含裂縫問題之分析 ………………………………………….200
(A) 修正Dundurs參數 ＞0， ＝0 ………………………………….201
(B) 修正Dundurs參數 ＜0， ＝0 ………………………………….203
5.2.5 E-FGM含裂縫問題之分析 ………………………………………….205
(A) 修正Dundurs參數 ＞0， ＝0 ………………………………….206
(B) 修正Dundurs參數 ＜0， ＝0 ………………………………….207
5.3 討論與比較 ………………………………………………………………209
第六章 結論與建議 …………………………………………………………..213
6.1 結論 ………………………………………………………………………...213
6.2 建議 ………………………………………………………………………...214
參考文獻 ………………………………………………………………………...215

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