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研究生:曾宏量
研究生(外文):Hung-Liang-Tseng
論文名稱:二維垂直橫觀等向性彈性圓柱山谷承受時間諧和震波之散射研究
論文名稱(外文):The Scattering of a Vertical Transverse Isotropic Cylindrical Canyon Subjected to Time-Harmonic Elastic Wave
指導教授:葉超雄葉超雄引用關係
口試委員:陳東陽鄧崇任廖文義
口試日期:2015-07-27
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:應用力學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:中文
論文頁數:80
中文關鍵詞:橫觀等向性散射問題Lamb問題最速陡降路徑-駐相法
外文關鍵詞:transversely isotropicscattering problemLamb problemsteepest descent-stationary phase method.
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  • 被引用被引用:1
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本研究旨在探討頻率域中在不同角度的平面波入射之下,一個橫觀等向性材料半平面山谷之散射問題。此一問題之解可以分解成自由場以及散射場所組成,其中自由場由入射以及反射波所構成,而散射場則是由Lamb問題各階奇異解所展開而成之級數所構成。首先將散射問題拆成反平面與共平面散射問題分別進行討論,利用水平波數複數平面中之最速陡降路徑-駐相法解決Lamb問題各階奇異解之水平波數域積分中,由於含有震盪項以至於積分收斂速度緩慢的困擾。對於共平面散射問題而言,其滿足控制方程式之外傳散射波場僅需由不同的分支切割與分支點所切割出來之四個複數平面黎曼面中的其中兩個黎曼面就可完全描述。在每個黎曼面上,根據材料的特性可訂定出各個分支切割與分支點以確保讓多值函數在該黎曼面上變成單值。最後運用最小平方法解得待定係數以滿足半平面山谷之邊界條件,則整個山谷之應力場與位移場均能得知。

The objective of this research is to study the scattering of a vertically transversely isotropic cylindrical canyon subjected to the incidence of time harmonic plane elastic wave. The total displacement field of either the anti-plane or in-plane scattering problem can be decomposed into two parts, namely, free field as well as scattering filed part. The known free field part can be further separated into incident wave and reflected wave in order to satisfy the ground surface condition. While the unknown scattering field part is expanded into a series of n-th order outgoing singular solutions of Lamb’s problem with unknown amplitude which can be determined by boundary condition of canyon itself. The displacement field and stress field of each n-th order outgoing singular solutions of Lamb’s problem can only be expressed into a form of horizontal wave-number integral which can be evaluated efficiently in complex wave-number domain by using the so called steepest descend-stationary phase method. For in-plane scattering problem, the outgoing scattering field contains two kinds of wave field, namely, P wave and S wave, only two sheets of the four Riemann Surface are sufficient to describe the outgoing scattering field. In order to ensure the single value of a multi-value radical function in each Riemann sheet, the branch points and the associated branch cuts are carefully chosen according to the material considered. Least Square method is employed to solve the unknown coefficients of the expansion series of the scattering field. Once the coefficients are determined, the complete displacement field and stress field can be obtained.

目錄
中文摘要 i
Abstract ii
目錄 iii
圖目錄 v
表目錄 vii
第1章 導論 1
1.1 研究動機 1
1.2 彈性波散射問題之歷史演進回顧 1
1.3 散射問題之求解方法概述 4
1.4 本文之研究方法與研究架構 5
第2章 基本原理與材料性質 7
2.1材料基本性質 7
2.2 基本理論推導 9
2.3 材料限制與輻射條件限制 11
2.4 波數與慢度及其無因次化 12
2.5 慢度圖與波前圖 13
第二章附圖 15
第3章 反平面散射問題 16
3.1 基本理論與材料常數定義 16
3.2 散射場中之Lamb奇異解 17
3.3 無因次化之散射位移場與應力場表示式 19
3.3 最速陡降路徑-駐相法(Steepest-descent Path-stationary Phase Method) 20
3.4 轉換後之高斯積分(Gauss Quadrature) 24
3.5 自由場 25
3.6 散射場待定係數之求解 27
第3章附圖 30
第4章 共平面散射問題 39
4.1 位移場與應力場之求解 39
4.1.1散射場之位移與應力場求解 39
4.1.2散射場待定係數之求解 47
4.2 水平波數域複數平面上的性質 49
4.2.1分支切割與分支點 49
4.2.2 雷利極點(Rayleigh Pole) 52
4.3 最速陡降路徑(Steepest-descent Path) 54
4.3.1 SDP路徑參數式 54
4.3.2鞍點位置之迭代求法 56
4.3.3山谷各角度位置之積分路徑 59
第四章附圖 62
第5章 結論與展望 67
5.1 結論 67
5.2 未來展望 68
參考文獻 69
附錄A改善級數收斂性之作法 71
附錄B 等向性材料山谷散射問題之解析解 74
附錄C 實係數三次方程式標準式之卡丹公式 77
附錄D 利用Ferrari法解四次方程式 79


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