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

The Scattering of a Vertical Transverse Isotropic Cylindrical Canyon Subjected to Time-Harmonic Elastic Wave

曾宏量(Hung-Liang-Tseng)

指導教授 ： 葉超雄

國立臺灣大學/工學院/應用力學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.00332

摘要

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。

關鍵字

橫觀等向性 ； 散射問題 ； Lamb問題 ； 最速陡降路徑－駐相法

並列摘要

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.

並列關鍵字

transversely isotropic ； scattering problem ； Lamb problem ； steepest descent-stationary phase method.

參考文獻

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