本論文使用零場邊界積分方程法（null-field boundary integral equation method）求解含圓孔洞之功能梯度材料的反平面問題，本文所採用的功能梯度材料其剪力模數為水平向指數變化，因此控制方程式並非為Laplace方程式，但可藉由變數變換將控制方程式轉換成修正型Helmholtz方程式。針對單圓邊界問題，將考慮三種邊界條件:1.圓孔洞問題、2.剛性置入物、3.Eshelby的置入物問題，其中前2個問題在無窮遠處受一剪切應力作用。因考慮圓形邊界可透過搭配退化核函數（degenerate kernel）與傅立葉級數（Fourier series）取代閉合型基本解與邊界密度，可得到其半解析解。本文更延伸至無限域中含多圓孔洞之功能梯度材料的反平面問題，藉由自適性座標系統(局部座標系統)的使用，可充分使用三角函數的正交性，因此無須數值積分的方式求解邊界弧長積。最後將本文方法的數值結果與Matlab R2019a的工具箱PDE Toolbox，也就是有限元素法（finite element method）的數值結果做對比，針對不同非均勻空間變換參數（non-homogeneous parameter）對場解的影響，除了全場位移的比較之外，也針對圓形孔洞邊界上的位移與應力集中因子（stress concentration factor）做比較，其兩種方法的結果都一致吻合，且當β=0時，也與均質的結果一致吻合。

The null-field boundary integral equation method is employed to solve anti-plane problems containing circular holes of a functionally graded material (FGM). The shear modulus of the present FGM is an exponential variation. Therefore, the governing equation isn’t a typical Laplace equation. By using the change of variable, the governing equation can be transform into the modified Helmholtz equation. In this thesis, we consider three kinds of boundary condition, for the problem containing a single circular boundary. For the former two kinds, one is a traction free boundary condition and the other is rigid inclusion. Both are subject to a remote shear. The third one is an Eshelby inclusion problem. By using the degenerate kernels and Fourier series expansions, the semi-analytical solution can be obtained. We also extend the problem containing a single circular hole to multiple circular holes by using the adaptive observer system, while the orthogonality of angular function can be fully utilized. In this way, using the numerical quadrature to calculate the boundary arc length integral is free. Finally, all numerical results are compared with those results by using the finite element method (FEM). The displacement field and the stress concentration factor along the circular hole are considered to discuss the effect of the non-homogeneous parameter of materials. The results by using the present method and the FEM are consistent. For the special case of non-homogeneous, the present results are the same with the results of homogeneous case.

目錄 I
圖目錄 III
表目錄 V
第一章 緒論 1
1.1 研究動機及文獻回顧 1
1.1.1 功能梯度材料 1
1.1.2 無窮遠處受剪切應力(Remote shear) 3
1.1.3 零場邊界積分方程法 3
1.2 論文架構 6
第二章 無限域中含有圓洞之問題 8
2.1 問題描述 8
2.2 控制方程式 9
2.3 邊界條件 13
2.3.1 孔洞邊界(Traction free)-----曳引力為零 13
2.3.2 剛性置入物 (Rigid inclusion)---位移為零 18
2.3.3 Eshelby圓形置入物 (Eshelby circular inclusion)的界面問題 18
2.4 零場邊界積分方程法 20
2.5 含單圓洞之無限域問題求解過程 21
2.5.1 曳引力為零 (Traction free) 23
2.5.2 剛性置入物 (Rigid inclusion) 26
2.5.3 Eshelby圓形置入物 (Eshelby circular inclusion) 27
2.6 應力集中因子 28
2.7 數值結果 29
2.7.1 曳引力為零 30
2.7.2 剛性置入物 32
2.7.3 Eshelby圓形置入物 33
第三章	無限域中含有多圓洞之問題 48
3.1 問題描述 48
3.2 控制方程式與邊界條件 49
3.3 零場邊界積分方程法 50
3.4 應力集中因子 55
3.5 數值結果 56
3.5.1 兩顆圓孔洞的反平面問題 56
3.5.2 三顆圓孔洞的反平面問題 75
第四章	結論與未來展望 79
4.1 結論 79
4.2 未來展望 80
參考文獻 81
圖1-1 論文架構圖	7
圖2-1 無限域中含有單一圓孔洞的反平面力場問題示意圖	8
圖2-2 無限域中含有單一圓孔洞之座標幾何關係	9
圖2-3 分離核	12
圖2-4 圓形孔洞問題疊加示意圖	15
圖2-5 圓形置入物問題疊加示意圖	19
圖2-6 單顆圓洞問題的均勻配置邊界點位示意圖	24
圖2-7 有限元素法的網格分佈圖	30
圖2-8 剪切應力在x方向時 本方法之位移等高線(孔洞)	34
圖2-9 剪切應力在x方向時 有限元素法之位移等高線(孔洞)	34
圖2-10 剪切應力在y方向時 本方法之位移等高線(孔洞)	35
圖2-11 剪切應力在y方向時 有限元素法之位移等高線(孔洞)	35
圖2-12 剪切應力在x方向時的 邊界位移分佈(孔洞)	36
圖2-13 剪切應力在y方向時的 邊界位移分佈(孔洞)	36
圖2-14 剪切應力在x方向時圓孔洞邊界上位移對β的影響(孔洞)	37
圖2-15 剪切應力在y方向時圓孔洞邊界上位移對β的影響(孔洞)	37
圖2-16 剪切應力在x方向時應力集中因子對項數n_c的收斂圖(孔洞)	38
圖2-17 剪切應力在y方向時應力集中因子對項數n_c的收斂圖(孔洞)	38
圖2-18 剪切應力在x方向時的 應力集中因子分佈圖(孔洞)	39
圖2-19 剪切應力在y方向時的 應力集中因子分佈圖(孔洞)	39
圖2-20 剪切應力在x方向時β對圓孔邊上應力集中因子的影響(孔洞)	40
圖2-21 剪切應力在y方向時β對圓孔邊上應力集中因子的影響(孔洞)	40
圖2-22 剪切應力在x方向時 本論文方法之位移等高線(剛性)	41
圖2-23 剪切應力在x方向時 有限元素法之位移等高線(剛性)	41
圖2-24 剪切應力在y方向時 本論文方法之位移等高線(剛性)	42
圖2-25 剪切應力在y方向時 有限元素法之位移等高線(剛性)	42
圖2-26 剪切應力在x方向時的 應力集中因子(剛性)	43
圖2-27 剪切應力在x方向時的 應力集中因子(剛性)	43
圖2-28 剪切應力在x方向時β對圓孔洞邊上應力集中因子的影響(剛性)	44
圖2-29 剪切應力在y方向時β對圓孔洞邊上應力集中因子的影響(剛性)	44
圖2-30 本文方法之位移結果	45
圖2-31 論文[8]解析解之位移結果	45
圖2-32 本文方法之應變結果	46
圖2-33 論文[8]解析解之應變結果	46
圖2-34 本文方法之應力結果	47
圖2-35 論文[8]解析解之應力結果	47
圖3-1 無限域中含有兩顆圓孔洞的反平面力場問題示意圖	48
圖3-2 無限域中含有兩顆圓孔洞之座標幾何關係	49
圖3-3 兩顆圓洞問題的均勻配置邊界點位示意圖(以N=41為例)	52
圖3-4 下標示意圖	53
圖3-5 剪切應力在x方向 β=0 本文方法之等高線結果	59
圖3-6 剪切應力在x方向 β=0 有限元素法之等高線結果	59
圖3-7 剪切應力在x方向 β=0.2 本論文方法之等高線結果	60
圖3-8 剪切應力在x方向 β=0.2 有限元素法之等高線結果	60
圖3-9 剪切應力在y方向 β=0 本論文方法之等高線結果	61
圖3-10 剪切應力在y方向 β=0 有限元素法之等高線結果	61
圖3-11 剪切應力在y方向 β=0.2 本論文方法之等高線結果	62
圖3-12 剪切應力在y方向 β=0.2 有限元素法之等高線結果	62
圖3-13 孔洞1的應力集中因子 (剪切應力在x方向 β=0)	63
圖3-14 孔洞2的應力集中因子 (剪切應力在x方向 β=0)	64
圖3-15 孔洞1的應力集中因子 (剪切應力在x方向 β=0.2)	65
圖3-16 孔洞2的應力集中因子 (剪切應力在x方向 β=0.2)	65
圖3-17 孔洞1的應力集中因子 (剪切應力在y方向 β=0)	66
圖3-18 孔洞2的應力集中因子 (剪切應力在y方向 β=0)	67
圖3-19 孔洞1的應力集中因子 (剪切應力在y方向 β=0.2)	68
圖3-20 孔洞1的應力集中因子 (剪切應力在y方向 β=0.2)	68
圖3-21 剪切應力在x方向時應力集中因子對項數n_c的收斂圖(孔洞1)	69
圖3-22 剪切應力在y方向時應力集中因子對項數n_c的收斂圖(孔洞1)	69
圖3-23 孔洞1的應力集中因子 (剪切應力在x方向 β=0)	70
圖3-24 孔洞1的應力集中因子 (剪切應力在x方向 β=0)	70
圖3-25 孔洞1的應力集中因子 (剪切應力在x方向 β=0.2)	71
圖3-26 孔洞2的應力集中因子 (剪切應力在x方向 β=0.2)	71
圖3-27 孔洞1的應力集中因子 (剪切應力在y方向 β=0)	72
圖3-28 孔洞2的應力集中因子 (剪切應力在y方向 β=0)	73
圖3-29 孔洞1的應力集中因子 (剪切應力在y方向 β=0.2)	74
圖3-30 孔洞2的應力集中因子 (剪切應力在y方向 β=0.2)	74
圖3-31 無限域中含有三顆圓孔洞的反平面力場問題示意圖	75
圖3-32 剪切應力在x方向 β=0 應力集中因子的結果	77
圖3-33 剪切應力在y方向 β=0 應力集中因子的結果	77
圖3-34 剪切應力在x方向 β=0.2 應力集中因子的結果	78
圖3-35 剪切應力在y方向 β=0.2 應力集中因子的結果	78
表2-1 變數變換前後的數學模型差異比較	14

[1]	王保林, 韩杰才, 张幸红, “非均匀材料力学”, 科学出版社, 北京市,(2003)

[2]	新野正之, 平井敏雄, 渡辺龙三, “倾斜机能材料―宇宙机用超耐热材料を目指して”, 日本複合材料学会志, (10), 1-8, (1987)

[3]	G.I. Popov, “Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion”, Journal of Applied Mathematics and Mechanics, 37, 1109-1116, (1973)

[4]	Y.Y. Yang, “Stress analysis in a joint with a functionally graded material under a thermal loading by using the Mellin transform method”, International Journal Solids and Structures, 35, 1261-1287, (1998)

[5]	H. Li, J. Lambros, B.A. Cheeseman and M.H. Santare, “Experimental investigation of the quasi-static fracture of functionally graded materials”, International Journal of Solids and Structures, 37, 3715-3732, (2000)

[6]	M. Ozturk and F. Erdogan, “Anti-plane shear crack problem in bonded materials with a graded interfacial zone”, International Journal of Engineering Science 31, 1641-1657, (1993)

[7]	J. Chen, “Anti-plane problem of periodic cracks in a functionally graded coating-substrate structure” Archive of Applied Mechanics, 75, 138-152, (2006)

[8]	X. Wang, E. Pan, A. K. Roy, “A functionally graded plane with a circular inclusion under uniform anti-plane eigenstrain”, Journal of Applied Mechanics, 75, 014501, (2008)

[9]	D.V. Kubair, “Stress concentration factor in functionally graded plates with circular holes subjected to anti-plane shear loading”, Journal of Elasticity, 114, 179-196, (2013)

[10]	D.V. Kubair, “Stress concentration factors and stress-gradients due to circular holes in radially functionally graded panels subjected to anti-plane shear loading”, Acta Mechanica, 224, 2845-2862, (2013)

[11]	P.P. Shi, “Stress field of a radially functionally graded panel with a circular elastic inclusion under static anti-plane shear loading”, Journal of Mechanical Science and Technology, 29, 1163-1173, (2015)

[12]	J. Li, T. Huang, J.H. Yue, C. Shi and P.H. Wen, “Anti-plane fundamental solutions of functionally graded materials and applications to fracture mechanics”, Journal of Strain Analysis, 00, 0, (2017)

[13]	W.C. Shen, “Null-field approach for Laplace problems with circular boundaries using degenerate kernels”, Thesis supervised by Prof. Jeng Tzong Chen, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan, (2005)

[14]	E. Honein, T. Honein, and G. Herrmann, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics, 3, 479-499, (1992)

[15]	K.H. Chen, J.T. Chen and J.H. Kao, “Regularized meshless method for anti-plane shear problems with multiple inclusions”, International Journal for Numerical Methods in Engineering, 73, 1251-1273, (2007)

[16]	V.A. Lubarda, “On the circumferential shear stress around circular and elliptical holes”, Archive of Applied Mechanics, 85, 223-235, (2014)

[17]	J.T. Chen, J.H. Kao, Y.L. Huang, S.K. Kao, “On the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernels”, Summitted, (2020)

[18]	J.T. Chen, W.C. Shen, A.C. Wu, “Null-field integral equations for stress field around circular holes under anti-plane shear”, Engineering Analysis with Boundary Elements, 30, 205-217, (2006)

[19]	J.T. Chen, A.C. Wu, “Null-field approach for piezoelectricity problems with arbitrary circular inclusions”, Engineering Analysis with Boundary Elements, 30, 971-993, (2006)

[20]	J.T. Chen, A.C. Wu, Null-field approach for the multi-inclusion problem under anti-plane shears”, Journal of Applied Mechanics, 74, 496-487, (2007)

[21]	J.T. Chen, C.C. Hsiao, “Null-field integral equation approach for plate problems with circular boundaries”, Journal of Applied Mechanics, 73, 679-693, (2006)

[22]	J.T. Chen, H.Z. Liao, W.M. Lee, “An analytical approach for the green's functions of biharmonic problems with circular and annular domains”, Journal of Mechanics, 25, 59-74, (2009)

[23]	J.T. Chen, C.T. Chen, P.Y. Chen, I.L. Chen, “A semi-analytical approach for radiation and scattering problems with circular boundaries”, Computer Methods in Applied Mechanics and Engineering, 196, 2751-2764, (2007)

[24]	J.T. Chen, P.Y. Chen, C.T. Chen, “Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach”, Soil Dynamics and Earthquake Engineering, 28, 58-72, (2008)

[25]	J.T. Chen, J.W. Lee, C.F. Wua, I.L. Chen, SH-wave diffraction by a semi-circular hill revisited a null-field boundary integral equation method using degener”, Soil Dynamics and Earthquake Engineering, 31, 729-736, (2011)

[26]	J.T. Chen, J.W. Lee and W.S Shyu, “SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid met”, Geophysical Journal International, 188, 177-194, (2012)

[27]	J.T. Chen, Y.T. Lee, K.H. Chou, “Revisit of two classical elasticity problems by using the null-field integral equations”, Journal of Mechanics, 26, 393-401, (2010)

[28]	J.T. Chen, W.M. Lee, “Scattering of flexural wave in a thin plate with multiple circular inclusions by using the null-field integral equation approach”, Journal of Sound and Vibration, 329, 1042-1061, (2010)

[29]	J.T. Chen, J.W. Lee, “A semi-analytical method for near-trapped mode and fictitious frequencies of multiple scattering by an array of elliptical cylinders in water waves”, Physics of Fluids, 25, 097103, (2013)

[30]	J.T. Chen, J.W. Lee and Y.C. Tu, “Focusing phenomenon and near-trapped modes of SH waves”, Earthquake Engineering and Engineering Vibration, 15, 477-486, (2016)

[31]	J.T. Chen, S.K. Kao, Y.H. Hsu and Y. Fan, “Scattering problems of the SH wave by using the null-field boundary integral equation method”, Journal of Earthquake Engineering, 22, 1-35, (2017)

[32]	Y.T. Shih, “SH-wave scattering by a circular hole in a functionally graded material using the null-field boundary integral equation method”, Thesis supervised by Prof. Jia Wei Lee, Department of Civil Engineering, Tamkang University, New Taipei City, Taiwan, (2019)