臺灣博碩士論文加值系統

本文宗旨在探討於無窮域垂直橫觀等向性介質中之埋設無窮長特殊扁橢圓形孔穴承受時間諧和平面波場作用下之散射問題以及所引致之動應力集中現象。
本論文探討一入射平面波場通過三種不同材質之外域射入一個長短軸比值為某一特定值之特殊扁橢圓形中空孔穴中之散射問題，根據材料之不同而有不同的解法。其中鎂(Magnesium)材料的慢度面為圓形，故原散射問題在幾何上不需要做轉換，直接使用傳統角度圓柱波函數進行求解。而鈹(Beryl)及鋅(Zn)材料其無因次材料常數比值不是1，因此這兩種材料之慢度圓為一橢圓，在解題上須先將垂直橫觀等向性的橢圓形慢度面轉換成等向性的圓形慢度面，同時在計算上需先將原散射問題的特殊扁橢圓形拉伸成圓形孔穴。其中本論文採用兩種方法進行求解，分別為分離變數法以及離散邊界配點法。分離變數法將轉換後控制方程式之傳統角度圓柱波函數解經變數分離後寫成Hankel函數與傳統三角函數的乘積，再透過變換後之中空圓形孔穴散射問題之邊界條件可求解出散射係數。而離散邊界配點法則是將垂直橫觀等向性介質經轉換後之散射位移場傳統角度圓柱波函數解表示成核函數為非零平面波解之三角函數相位角度譜之複數路徑積分表示式。所有在區域內每一場點上，每一純量波函數所對應之位移場及應力場皆可用同樣複數路徑積分式表示。然後將三角函數相位角度譜之複數路徑積分形式轉換成水平慢度域之傅立葉積分形式後，進一步可在水平慢度域複數平面中，利用所發展之最速陡降路徑-駐相積分法，可精確積分求得分佈在區域內每一場點上用波函數表示之對應位移場或應力場之場值。然後再利用離散邊界配點法以最小平方誤差之方式求解級數展開之待定散射係數以滿足變換後圓形孔穴之法向剪應力為零之邊界條件。最後兩種方法均可求得沿特殊扁橢圓形孔穴表面分布之切向剪應力幅值大小。

The objectives of this thesis is aim to study the scattering as well as the dynamic stress concentration phenomenon of a vertically transversely isotropic special oblate elliptic cylindrical cavity subjected to the obliquely incidence of time harmonic plane elastic wave. An incident plane wave field traveling through three different types of exterior medium then impinging onto a special oblate elliptic cylindrical cavity with certain specific aspect ratio. There will have different solution strategies according to different types of material property. Since the slowness surface for Magnesium is a circle, therefore, the original scattering problem needs not be converted in geometry. We use the classical cylindrical wave function to solve the corresponding scattering problem, directly. However, for Beryllium and Zinc , since their dimensionless material constant are not equal to 1, therefore, both the slowness surface of these two types of material are ellipse. In order to solve the corresponding scattering problem, firstly, we convert the original elliptical slowness surface for a transverse isotropic material into a circular slowness surface for an isotropic material. At the same time, the geometry of the original problem have been converted from a special oblate elliptical cavity into a circular cavity. In this thesis, two methods are used to solve the corresponding problem, namely, the separation of variable method as well as the discrete boundary collocation point method. We first use the separation of variable method to separate the classical wave function into the product of a Hankel function and a trigonometric function in classical cylindrical coordinate system, and then use the boundary condition of the circular cavity to solve the unknown scattering coefficients. Another alternative method is boundary collocation point method, we propose that after the transformation, the unknown scattering field part can be expanded into a series of n-th order wave function. Each wave function is defined by a trigonometric function angular spectrum along a complex contour integral path with a kernel function which is non-trivial plane wave solution of the corresponding wave equation. The trigonometric angular spectrum of each n-th order wave function can be further converted into an infinite horizontal slowness integral which can be evaluated efficiently in complex slowness domain by employing the steepest descend-stationary phase method. In order to satisfy the boundary condition at each boundary collocation point which allocate along the cavity surface, Least Square method is employed to obtain the unknown coefficient of the expansion series of the scattering field. Thus, the dynamic stress concentration phenomenon of a vertically transversely isotropic special oblate elliptic cylindrical cavity subjected to the obliquely incidence of time harmonic plane elastic wave is thoroughly studied by both of the proposed methods for three different typical materials.

致謝 i
中文摘要 iv
Abstract v
目錄 vii
圖目錄 ix
表目錄 xiii
第1章 問題溯源與演進 1
1.1 研究動機 1
1.2 彈性波散射問題之歷史演進回顧 3
1.3 散射問題研究方法與研究架構 6
第2章 基本原理與材料性質 7
2.1 垂直橫觀等向性介質之三維控制方程式拆解為共平面散射問題與反平面散射問題 7
2.2 垂直橫觀等向性介質反平面問題之變換為等向性散射問題 12
2.3 垂直橫觀等向性分析及其拉伸變換後之慢度面與波前面 16
2.4 三種典型材料中特殊扁橢圓型孔穴散射問題在拉伸變換前後之邊界條件改變 18
第3章 圓柱與橢圓柱波函數路徑積分表示式 30
3.1 非傳統與傳統圓柱波函數相位角度譜路徑積分表示式 30
[A] cos2m''φ'',z*≥ 0 30
[B] cos2m''φ'',z* < 0 34
[C] cos2m''+1φ'',z* ≥ 0 36
[D] cos2m''+1φ'',z* < 0 37
[E] sin2m''+1φ'',z* ≥ 0 38
[F] sin2m''+1φ'',z*< 0 40
[G] sin2m''+2φ'',z*≥0 41
[H] sin2m''+2φ'',z*<0 42
[I] 將[A]、[B]、[C]、[D]、[E]、[F]、[G]與[H]合併 44
[J] 由非傳統圓柱波函數相位角度譜至傳統圓柱波相位角度譜 45
3.2 傳統圓柱波函數散射位移與應力場之水平慢度域積分表示式 49
3.2.1 傳統型式圓柱波函數散射位移場之水平慢度域積分表示式 50
3.2.2 傳統圓柱波函數所引致法向剪應力與切向剪應力場之水平慢度域積分表示式 52
第4章 等向性介質中與垂直橫觀等向性介質中特殊扁圓形孔洞散射問題求解 55
4.1 傳統型水平慢度域積分表示式之最速陡降路徑-駐相積分法 55
4.2 鎂材料之圓型孔穴之散射問題求解 65
4.2.1 分離係數法 65
4.2.2 離散配點法 74
4.3 鈹(綠玉)及鋅兩種材料中特殊扁橢圓型孔穴之散射問題求解 78
4.3.1 鈹與鋅材料常數經拉伸變換後之分離係數法求解過程 78
4.3.2 利用離散配點法代入轉換後之邊界條件求解散射係數之過程 96
第5章 結論與未來展望 101
5.1 結論 101
5.2 未來展望 101
參考文獻 103

[1]I.S. Gradshteyn and I.M. Ryzhik-“Table of Integrals, Series, and Products”, Translated from Russian by Scripta Technica, Inc. (1943)
[2]V. T. Buchwald & A. Davis -“Surface Waves in Anisotropic Elastic Media” , The Quarterly Jornal Nature 191, 899 - 900 (1963)
[3]王培宇-“二維垂直橫向等向性彈性半平面頻率域Lamb問題之解題方法研究”國立台灣大學應用力學研究所碩士論文(2014)
[4]曾宏量-“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” 國立台灣大學應用力學研究所碩士論文(2015)
[5]聶祥宇-“二維垂直等向性彈性孔洞承受時間諧和震波之動應力集中研究” 國立台灣大學應用力學研究所碩士論文(2016)
[6]駱弘杰-“含剛性圓形夾域之平板受簡諧力之反應”國立台灣大學土木工程研究所碩士論文(2003)
[7]Milton Abramowitz and Irene A, Stegun -“Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, Dover Publications, ncorporated (1974)
[8]Mow Chao-Chow & Pao Yih-Hsing -“The Diffraction Of Elastic Waves And Dynamic Stress Concentrations”, Communications Department(1971)
[9]Fresmel. A, J. -”Memoire sir la Diffraction de la Lumiere ”, Annales de Chimie et de Physique (1826)
[10]Young, Thomas. -“Miscellaneous Works”, London, J. Murray (1855)
[11]Todhunter, Isaac & Karl Pearson-“A History of the Theory of Elasticity”, Cambridge, University Press (1960)
[12]D.K. Arrowsmith, C.M. Place,-“An Introduction to Dynamical Systems “Chapman & Hall, London, Section 3.3 , (1992)
[13]Robert W. Scharstein & Anthony M. J. Davis-“Acoustic scattering by a rigid elliptic cylinder in a slightly viscous medium”, The Journal of the Acoustical Society of America ,Volume 121, Issue 6 , 10.1121/1.2727332 (2007)
[14]St´ephane Ancey, Emmanuelle Bazzali, Paul Gabrielli, Micha¨el Mercier-“Acoustic scattering by elastic cylinders of elliptical cross-section and splitting up of resonances”, Journal of Applied Physics , Volume 115, Issue 19 ,10.1063/1.4876678 (2014)
[15]Arfken, G. "Bessel Functions of the First Kind" and "Orthogonality." §11.1 and 11.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-591 and 591-596(1985)
[16]K.Wright –“Chebyshev collocation methods for ordinary differential equations”
, The Computer Journal, Volume 6, Issue 4, 1 January 1964, Pages 358–365 (1964)
[17]José M. Carcione -“Wave Fields in Real Media”, Journal of Applied Geophysics , Volume 101, pp. 1-144 (2014)
[18]Horace Lamb-” On the Propagation of Tremors over the Surface of an Elastic Solid”, Published 1 January 1904.DOI: 10.1098/rsta.1904.0013,The Royal Society