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系統識別號 U0002-1607201913364100
中文論文名稱 零場邊界積分方程法求解含圓形孔洞功能梯度介質引致的SH波散問題
英文論文名稱 SH-wave scattering by a circular hole in a functionally graded material using the null-field boundary integral equation method
校院名稱 淡江大學
系所名稱(中) 土木工程學系碩士班
系所名稱(英) Department of Civil Engineering
學年度 107
學期 2
出版年 108
研究生中文姓名 施奕廷
研究生英文姓名 Yi-Ting Shih
學號 606380318
學位類別 碩士
語文別 中文
第二語文別 英文
口試日期 2019-07-05
論文頁數 71頁
口試委員 指導教授-李家瑋
委員-陳正宗
委員-郭世榮
中文關鍵字 SH波的散射  功能梯度材料  邊界元素法  零場邊界積分方程法 
英文關鍵字 SH-wave scattering  Functionally graded materials  Boundary element method  Null-field boundary integral equation method 
學科別分類 學科別應用科學土木工程及建築
中文摘要   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。
英文摘要   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.
論文目次 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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