本論文使用零場邊界積分方程法求解含有多顆圓形孔洞的功能梯度材料受水平剪力波(SH-wave)的散射問題，本文考慮的功能梯度特性其剪力模數與密度都為沿水平方向的指數變化，且透過變數變換的方法，可將控制方程式轉成一般的二維Helmholtz方程式，且孔洞邊界條件原為曳引力為零但同樣透過變換成Robin型態。搭配極座標系統下的退化核函數(degenerate kernels)並假設邊界密度為傅立葉級數(Fourier series)展開，可充分發揮三角函數的正交性，因此每個沿著圓形邊界的弧長積分都可解析求得，最後透過配點法滿足邊界條件建構線性代數方程式求得未知係數。數值結果包含全場位移振幅、圓形孔洞邊界上位移振幅與動態應力集中因子(dynamic stress concentration factor)，並探討不同波數k與非均勻空間變換參數β對位移振幅與動態應力集中因子的影響，此外也使用有限元素法(finite element method)做數值結果比對，其結果都一致吻合。最後本論文更將此問題延伸至含單一圓孔洞的雙介質SH波散射問題，其左半平面為均質介質，右半平面為含圓孔洞的功能梯度介質，透過疊加原理將全場拆解成輻射場與入射場，再將輻射場拆解成兩個半平面問題，最後將兩個半平面問題各自埋藏成全平面問題，並考慮其孔洞邊界條件、界面位移連續性與應力平衡等條件求解。數值結果包含界面位移振幅與圓孔洞邊界上的動態應力集中因子，並與文獻上的數值結果做比對，且在β=0時可退回全均質的情況，並也跟解析解結果比對一致吻合。

In this thesis, the null-field boundary integral equation method is used to solve the SH-wave scattering the problem of functionally graded material (FGM) containing multiple circular holes. For the present FGM, the shear modulus and the density are the form of exponential variation along the horizontal direction. By using the method of change variable, the governing equation can be transformed into a general two-dimensional Helmholtz equation. The traction free (Neumann) boundary condition is transformed into the Robin boundary condition. Adopting the degenerate kernel in terms of the polar coordinates to replace the closed-form fundamental solution and using the Fourier series to expand the boundary densities, the orthogonality of the trigonometric function can be fully utilized. Therefore, each boundary arc length integral along the circular boundary can be analytically determined. By employing the collocation method to construct a linear algebraic equation, all unknown Fourier coefficients can be straightforwardly determined. The numerical results include the full-field displacement amplitude, displacement amplitude and dynamic stress concentration factor along the boundary of the circular holes. The influences of those results versus the wave number k and nonhomogeneous parameter β are also discussed. In addition, all numerical results are compared with those results by using the finite element method and good agreements are made. Finally, this thesis is extended to the SH-wave scattering problem of bimaterial containing a circular hole. The half-plane of the left-hand side is a homogeneous material, the other half-plane of the right-hand side is a FGM with a circular hole. Both the boundary condition on a circular hole and continuous conditions on the interface are considered. The numerical results are compared with the results of the literature. When β is equal to zero, it can reduce to the case of homogeneous material. After comparing the analytical solution for β=0, the present result is consistent with the analytical solution.

目錄 I
圖目錄 III
表目錄 V
第一章 緒論 1
1.1研究動機與目的 1
1.2文獻回顧 3
1.2.1 水平剪力波散射問題 3
1.2.2 零場邊界積分方程法 4
1.3 論文架構 6
第二章 無限域中含多顆圓孔洞之SH波散射問題 8
2.1 問題描述 8
2.2 控制方程式與邊界條件 9
2.2.1 控制方程式 9
2.2.2 邊界條件 11
2.3 零場邊界積分方程法 13
2.3.1 域內點邊界積分公式 13
2.3.2 使用極座標將退化核用於閉合型的核函數 15
2.3.3 邊界密度 17
2.4 問題求解過程 18
2.5 動態應力集中因子 24
2.6 數值結果探討 26
2.6.1 無限域的功能梯度介質中含有兩顆圓形孔洞問題 27
2.6.2 無限域的功能梯度介質中含有三顆圓形孔洞問題 31
第三章 雙材料中含圓形孔洞之問題 43
3.1 問題描述 43
3.2 控制方程式與邊界條件 44
3.2.1 控制方程式 44
3.2.2 邊界條件 45
3.3 全場位移 46
3.4 問題求解過程 48
3.4.1 功能梯度材料全平面透過邊界積分方程求解 50
3.4.2 邊界條件 52
3.4.3 均質全平面透過零場邊界積分方程求解 53
3.4.4 界面條件位移連續 54
3.4.5 界面力平衡條件 56
3.5 動態應力集中因子 58
3.6 數值結果探討 59
第四章 結論與未來展望 70
4.1 結論 70
4.2 未來展望 71
參考文獻 72

圖目錄
圖1-1 FGM示意圖 2
圖1-2論文架構圖 7
圖2-1無限域中含多顆圓形孔洞問題示意圖 8
圖2-2問題拆解示意圖 9
圖2-3局部座標系統 20
圖2-4邊界節點分佈圖(N=41) 21
圖2-5有限元素法節點與元素數目示意圖 27
圖2-6 ρx=1且ϕx=π之位移對傅立葉項數的收斂情形 28
圖2-7兩顆圓孔洞圓心的直線距離示意圖 29
圖2-8 ka=1.0時零場邊界積分方程法之全場位移振幅 34
圖2-9 ka=1.0時有限元素法 之全場位移振幅 34
圖2-10 ka=1.0時零場邊界積分方程法之圓形孔洞位移振幅分佈 35
圖2-11 ka=1.0時有限元素法 之圓形孔洞位移振幅分佈 35
圖2-12零場邊界積分方程法(ka=1)的動態應力集中因子 36
圖2-13零場邊界積分方程法與有限元素法之動態應力集中因子比較 36
圖2-14波數k對圓形孔洞邊界上ϕx=π的位移振幅影響 37
圖2-15非均勻空間變化參數β對圓形孔洞邊界上ϕx=π的位移振幅影響 37
圖2-16波數k對圓形孔洞邊界上ϕx=π/2 的動態應力集中因子影響 38
圖2-17非均勻空間變化參數β對圓形孔洞邊界上ϕx=π/2 的動態應力集中因子影響 38
圖2-18圓孔洞間距對邊界位移振幅的影響 39
圖2-19 ka=1.0時零場邊界積分方程法之等高線結果 40
圖2-20 ka=1.0時有限元素法之等高線結果 40
圖2-21 ka=1.0時零場邊界積分方程法之圓形孔洞位移振幅分佈 41
圖2-22 ka=1.0時有限元素法之圓形孔洞位移振幅分佈 41
圖2-23零場邊界積分方程法 (ka=1) 的動態應力集中因子 42
圖2-24本論文使用方法與有限元素法之動態應力集中因子比較圖 42
圖3-1均質-功能梯度材料中含單顆圓形孔洞示意圖 43
圖3-2全場問題拆解示意圖 44
圖3-3輻射場拆解示意圖 48
圖3-4均質全平面與功能梯度材料全平面 49
圖3-5 kR=2.0,h/R=1.5，x1軸範圍-5~5的界面位移 62
圖3-6 βR=0.2,h/R=1.5，x1軸範圍-5~5的界面位移 62
圖3-7 βR=0.2,kR=2.0，x1軸範圍-5~5的界面位移 63
圖3-8 kR=2.0,h/R=1.5，x1軸範圍-20~20的界面位移 63
圖3-9 βR=0.2,h/R=1.5，x1軸範圍-20~20的界面位移 64
圖3-10 βR=0.2,kR=2.0，x1軸範圍-20~20的界面位移 64
圖3-11解析解界面位移(kR=2,β=0,h/R=1.5) 65
圖3-12不同非均勻空間參數β的圓孔洞 動態應力集中因子 (kR=2.0,h/R=1.5) 66
圖3-13 h/R=1.1(solid)與h/R=5.0(dash line)在不同波數k的圓孔洞 動態應力集中因子 (βR=0) 67
圖3-14不同波數k的圓孔洞 動態應力集中因子 (βR=0.2,h/R=1.5) 68
圖3-15 不同的h/R的圓孔洞 動態應力集中因子 (βR=0.2,kR=2.0) 69 

表目錄
表2-1 閉合型基本解與退化核之等高線圖 15
表2-2零場邊界積分方程法與有限元素法之比較 28

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