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研究生:施博仁
研究生(外文):Po-Jen Shih
論文名稱:向量多極之傅氏譜表示式及其在彈性和孔隙彈性半空間散射問題分析之應用
論文名稱(外文):Fourier Spectrum Representation of Vector Multipoles with Application to Analysis Wave Scattering in an Elastic and/or Poroelastic Half-space
指導教授:葉超雄葉超雄引用關係
指導教授(外文):C. S. Yeh
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:198
中文關鍵詞:半空間彈性散射問題孔隙彈性
外文關鍵詞:ElasticityPoroelasticityHalf-spaceScattering problem
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  • 被引用被引用:5
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彈性波在介質中傳播,當其遇到障礙物時,將與障礙物發生交互作用,形成一個新的波源並發出次生波(或稱散射波)。伴隨著入射波(縱波或橫波),其反射波均耦合著縱波和橫波,稱為邊界耦合效應。Biot(1956)提出孔隙彈性介質波傳理論,並揭示三種體波的存在,即為第一縱波 、第二縱波 ,及向量橫波。該理論因考慮固體骨架和液體之間兩相的相對交互作用,建構起質量慣性耦合與粘性耦合交互作用機制,合併稱之為本質耦合。本文中首先將探討彈性介質半無限域中埋入球形散射體,藉此建立具有邊界耦合效應之環境,並推廣至孔隙彈性介質合併考慮邊界耦合與本質耦合效應,藉此探討雙重耦合情形下之散射現象。
本研究提出有別於傳統的角度譜積分(angular spectrum representation)表示式之傅氏譜表示式(Fourier spectrum representation),該表示式具備卡氏座標及雙重無限積分形式之特徵,且將向量波函數之各分量分別寫成純量波函數之遞迴關係,其最大的優點在於建構經由自由平面反射的反射波。另外,T矩陣公式是解決散射問題之利器,文中應用Betti第三恆等式及相關的三組正交條件,建立起無限域中彈性與孔隙彈性介質之T矩陣演算公式,並推廣至半空間的散射問題。再者,為建立對地震工程有實質幫助之算例,本文提出接近大地應力行為之Goodier-Bishop應力駐波並設定其為入射自由場域,此場域具單向張應力之特徵,當入射波頻率極小即為波長較長時,在較深處局部的環境可達到近似均勻張力,而較淺處因受地表自由邊界之影響而呈遞減至零。
最後,為求所提出傅氏譜表示式之適用性與正確性,本文藉由T矩陣理論和改良型最速陡降路徑積分法的數值方法,將其分別實踐於無限域和半空間中具空球穴的散射問題,並分別探討彈性介質與孔隙彈性介質中空穴周圍邊界上的箍應力(hoop stress)之應力集中情形。
第一章 緒論 1
1.1 前言 1
1.2 研究動機 1
1.3 散射問題解法之分類 3
1.4 論文架構 5
第二章 向量多極場之傅氏譜表示式及其運用於彈性半無限域之散射問題 7
2.1 文獻回顧與本章介紹 7
2.2 向量多極基函數 10
2.2.1 向量基函數的角度譜表示式 10
2.2.2 向量基函數的傅氏譜表示式 13
2.2.3 驗證傅氏譜表示式的正確性 18
2.3 向量多極場的傅氏譜表示式之應用 20
2.3.1 自由表面反射之反射波推導 20
2.3.2 T矩陣法運用於再散射現象 23
2.3.3 針對二次源形成反射波部分之正交性驗證 26
2.4 數值算例 27
2.4.1 入射自由場:GOODIER-BISHOP應力駐波 28
2.4.2 對深埋置空球穴之擬靜態近似 30
2.4.3 淺空球穴之動態應力集中與自由地表之交互影響 32
2.5 本章結語 33
第三章 T矩陣法運用於無限域中孔隙彈性介質之散射問題 35
3.1 文獻回顧與本章介紹 35
3.2 基函數與正交條件 37
3.2.1 孔隙彈性介質中的Betti第三恆等式 37
3.2.2 運動方程式之解耦與正交化 38
3.2.3 球座標系統下的基底函數 40
3.2.4 正交條件 43
3.3 孔隙彈性介質的T矩陣公式 44
3.3.1 入射波、傳射波和散射波之級數展開 44
3.3.2 孔隙彈性介質之T矩陣公式應用於任意形狀之埋置物 46
3.3.3 針對球型孔隙彈性介質埋置物的T 矩陣公式 48
3.3.4 針對球型彈性介質埋置物的T矩陣公式 49
3.4 數值結果 52
3.4.1 核實數值演算 52
3.4.2 平面縱波照射在球型埋置物上 54
3.5 本章結語 55
第四章 向量多極場之傅氏譜表示式及其運用於孔隙彈性半無限
域之散射問題 57
4.1 文獻回顧與本章介紹 57
4.2 向量多極基函數 59
4.2.1 向量多極基函數的傅氏譜表示式 59
4.2.2 驗證傅氏譜表示式的正確性 64
4.3 向量多極場的傅氏譜表示式之應用 65
4.3.1 自由表面反射之反射波推導 66
4.3.2 T矩陣法運用於再散射現象 71
4.3.3 針對二次源形成反射波部分之正交性驗證 75
4.4 數值算例 77
4.4.1 入射自由場:GOODIER-BISHOP應力駐波 77
4.4.2 淺空球穴之動態應力集中與自由地表之交互影響 80
4.5 本章結語 82
第五章 綜合討論與結論展望 83
5.1 綜合討論 83
5.2 結論 86
5.3 研究展望 88
參考文獻 91
附表 99
附圖 105
附錄A 相互垂直之三組角向量 之轉換 139
附錄B 球諧函數之遞迴關係式 143
附錄C 彈性介質中波源反射部之相關傅氏譜振幅 145
附錄D 簡介改良型最速陡降路徑積分法(一) 148
附錄 E 彈性無限域中球型空球穴的T矩陣公式 152
附錄F 彈性介質中GOODIER和BISHOP應力駐波之相關勢能場 154
附錄G 彈性介質中GOODIER和BISHOP應力駐波自由場之基函數
展開 156
附錄 H 孔隙彈性介質中耦合縱波的解耦過程 161
附錄 I 孔隙彈性介質中埋置孔隙彈性材質球所構成 矩陣之子元素
暨相關正交條件證明 163
附錄 J 孔隙彈性介質中埋置彈性材質球所構成 矩陣之子元素 169
附錄K 孔隙彈性介質中波源反射部之相關傅氏譜振幅 173
附錄L 簡介改良型最速陡降路徑積分法(二) 179
附錄M 孔隙彈性介質中球型空球穴所構成 矩陣之子元素 183
附錄N 孔隙彈性介質中GOODIER和BISHOP應力駐波
之相關勢能場 186
附錄O 孔隙彈性介質中GOODIER和BISHOP應力駐波
之自由場基函數展開表示式 189
附錄P 符號表 195
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