本論文使用零場場邊界積分方程法（null-field BIEM）求解含單一圓孔洞之功能梯度材料中水平剪力波（SH-wave）的散射問題，本文所採用的功能梯度材料參數呈指數變化，因此控制方程式並為典型的Helmholtz方程式，藉由使用變數變換將控制方程式轉換成Helmholtz方程式，且曳引力為零的Neumann邊界則轉換成Robin邊界。如此操作即可利用零場邊界積分方程法求解水平剪力波在功能梯度材料中的散射問題，搭配退化核函數（degenerate kernel）與傅立葉級數（Fourier series）取代基本解（fundamental solution）與邊界密度（boundary densities）可得到半解析解；本研究更延伸至水平剪力波在半無限域中含單一圓孔洞之功能梯度材料的散射問題，藉由引入映射法將半平面含單一圓孔洞問題轉換成全平面含兩個相等圓孔洞問題，其中也的另一關鍵則是剪力模數函數與材料密度函數的映射是關鍵；最後將本文究方法之數值結果與傳統邊界元素法（boundary element method BEM）使用常數元素的數值結果做對比，其結果都一致吻合，除位移場的比較之外，也對圓形孔洞邊界上的動態應力集中因子（dynamic stress concentration factor）做比較，針對不同非均勻空間變換參數（non-homogeneous parameter）對其場量的影響。

In this thesis, the problem of SH-wave scattering by a circular hole buried in infinite functionally graded materials (FGM) is solved by using the null-field boundary integral equation method (null-field BIEM). For the considered FGM, the patterns of the shear modulus and the density are the form of exponential variation. Therefore, the governing equation for the time-harmonic motion is not a typical Helmholtz equation. By using the change of variables, the original governing equation can be transformed into the Helmholtz equation. The Neumann boundary condition due to the traction free condition is transformed into the Robin boundary condition. Therefore, the null-field BIEM can be straightforward employed to solve the problem of SH-wave scattering in the FGM. Using the degenerate kernel and the Fourier series to substitute for the closed-form fundamental solution and boundary densities, the semi-analytical solution can be obtained. In addition, the problem of SH-wave scattering by a circular hole buried in semi-infinite FGM is also considered. By using the image method, the semi-infinite plane problem containing a circular hole is transformed into the infinite plane problem containing two identical circular holes. The other key point is that the functions of the shear modulus and the density are also imaged. Finally, all numerical results are compared well with those of numerical results by using the conventional boundary element method (BEM) with the constant element scheme. Not only the displacement field of the whole domain but also the dynamic stress concentration factor along the circular hole is presented. The effect of the non-homogeneous parameter of materials is also considered.

目錄
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論文架構	3
第二章 文獻回顧	5
2.1 水平剪力波的散射問題	5
2.2 零場邊界積分方程法	6
第三章 無限域中含單一圓洞之問題	8
3.1問題描述	8
3.2控制方程式與邊界條件	9
3.3邊界積分方程法	14
3.4基本解的展開形式-退化核	16
3.5無限域中含單一圓洞問題的求解過程	18
3.6動態應力集中因子	22
第四章 半無限域中含單一圓洞之問題	25
4.1問題描述	25
4.2控制方程式與邊界條件	28
4.3零場邊界積分方程法	31
4.4動態應力集中因子	37
第五章 數值結果探討	39
5.1無限域中含單一圓洞問題	39
5.1-1位移場	39
5.1-2動態應力集中因子	41
5.2半無限域中含單一圓洞問題	42
5.2-1位移場	42
5.2-2動態應力集中因子	44
第六章 結論與未來展望	66
6.1結論	66
6.2未來展望	67
參考文獻	69
 
圖目錄
圖3-1無限域中含單一圓形孔洞問題示意圖	8
圖3-2無限域中含單一圓形孔洞問題之數學模型	9
圖4-1半無限域含單一圓形孔洞的水平剪力波散射問題示意圖	25
圖4-2半使用映射法後轉成的全平面問題示意圖	26
圖4-3無限域問題的極座標系統示意圖	27
圖5-1 ka=1.0時傳統邊界元素法之等高線結果	46
圖5-2 ka=1.0時 本論文方法之等高線結果	46
圖5-3 ka=2.0時傳統邊界元素法之等高線結果	47
圖5-4 ka=2.0 時本文方法之等高線結果	47
圖5-5位移場在圓孔洞邊界ϕ=π/2的位置沿半徑向外變化	48
圖5-6動態應力集中因子在圓孔洞邊界ϕ=π/2的位置沿半徑向外變化	48
圖5-7在距離圓形孔洞圓心ρ=1.1,ϕ=π/2位移與傅立葉級數項數	49
圖5-8 βa=0時圓孔洞邊界上的位移	50
圖5-9 βa=0.2時圓孔洞邊界上的位移	50
圖5-10 βa=-0.2時圓孔洞邊界上的位移	51
圖5-11 βa對圓孔洞邊界上ϕ=π的位移影響	52
圖5-12 ka對圓孔洞邊界上ϕ=π的位移影響	52
圖5-13 βa對圓孔洞邊界上ϕ=π/2的位移影響	53
圖5-14 ka對圓孔洞邊界上ϕ=π/2的位移影響	53
圖5-15 βa=0時圓孔洞邊界上的動態應力集中因子分布	54
圖5-16 βa=0.2時圓孔洞邊界上的動態應力集中因子分佈	54
圖5-17 βa=-0.2時圓孔洞邊界上的動態應力集中因子分佈	55
圖5-18 不同βa對圓孔洞邊界ϕ=π/2的動態應力集中因子的影響	56
圖5-19 不同ka對圓孔洞邊界ϕ=π/2的動態應力集中因子的影響	56
圖5-20 ka=1.0,b/a=2 時傳統邊界元素法的等高線結果	57
圖5-21 ka=1.0,b/a=2 時本文的等高線結果	57
圖5-22 ka=2.0,b/a=2 時傳統邊界元素法的高線結果	58
圖5-23 ka=2.0,b/a=2 時本文的等高線結果	58
圖5-24 b/a=2 與 βa=0時圓孔洞邊界上的位移分布	59
圖5-25 b/a=2 與 βa=0.2時圓孔洞邊界上的位移分布	59
圖5-26 b/a=2 與 βa=-0.2時圓孔洞邊界上的位移分布	60
圖5-27 b/a=2 時 βa對圓孔洞邊界上ϕ=π的位移影響	61
圖5-28 b/a=2 時ka對圓孔洞邊界上ϕ=π的位移影響	61
圖5-29 b/a=2 時 βa對圓孔洞邊界上ϕ=π/2的位移影響	62
圖5-30 b/a=2 時 ka對圓孔洞邊界上ϕ=π/2的位移影響	62
圖5-31 b/a=2且βa=0時圓孔洞邊界上的動態應力集中因子分布	63
圖5-32 b/a=2且βa=0.2時圓孔洞邊界上的動態應力集中因子分布	63
圖5-33 b/a=2且βa=-0.2時圓孔洞邊界上的動態應力集中因子分布	64
圖5-34 b/a=2 時βa對圓邊界ϕ=π/2的動態應力集中因子的影響	65
圖5-35 b/a=2 時 ka對圓邊界ϕ=π/2的動態應力集中因子的影響	65
 
表目錄
表6-1零場邊界積分方程法與邊界元素法之比較	68

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