二維垂直橫觀等向性彈性孔洞承受時間諧和震波之動應力集中研究

Abstract

本論文旨在探討於無窮域垂直橫觀等向性介質中之埋置無窮長圓形柱面孔穴在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態應力集中現象。 本論文所建議採用之散射問題求解方法為將所欲求解之未知散射位移場或勢能場進行以複數路徑積分定義之純量波函數級數展開，而每一項以積分定義之級數展開純量波函數皆滿足垂直橫觀等向性介質之位移或勢能控制方程式以及遠場幅射條件並可以表示成核函數為非零解平面波其慢度係隨相位角而變之三角函數相位角度譜之複數路徑積分形式。所有在域內每一場點上，每一純量波函數所對應之位移場或應力場皆可用同樣之複數路徑積分形式表示，其差別僅在於積分中之被積函數不同而已。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Fourier積分形式後，進一步可在水平慢度複數平面中，利用所發展之最速陡降路徑－駐相積分法，可精確積分求得分布在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。接著；利用邊界離散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孔穴之法向曳引力為零之邊界條件。 特別地；針對垂直橫觀等向性介質中之反平面散射問題，可以說明經過座標拉伸變換之後成為一個相速度或慢度為定值之等向性介質反平面散射問題。而原先所建議採用以複數路徑積分定義之純量波函數展開級數，再經過散射係數重新組合定義並利用某一特定橢圓柱波函數之核函數為平面波之馬修函數相位角度譜複數路徑積分表示式之後，在新的座標系當中完可以證明全等價於相速度或慢度為定值之某一特定之橢圓柱波函數展開法。 最後；限於篇幅及研究時間為有限之條件下，本論文只針對足以代表垂直橫觀等向性材料特性之兩種代表性介質之反平面散射問題之散射問題進行求解。分別探討無窮長圓形柱面孔穴在承受時間諧和平面波場在不同入射角作用下，沿孔穴表面之切向箍應力分布以及動態應力集中現象。

The objective of this thesis is to study the scattering as well as dynamic stress concentration phenomenon of a vertically transversely isotropic circular cylindrical cavity subjected to the obliquely 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, the incidence field as well as the scattering field part. We propose that 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 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 phase velocity of the plane wave kernel function varies with the phase angle. The trigonometric angular spectrum of each n-th order wave function can be further converted to 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 collocation point which allocate along the cavity surface, 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. Thus, the dynamic stress concentration phenomenon of a vertically transversely isotropic circular cylindrical cavity subjected to the obliquely incidence of time harmonic plane elastic wave is thoroughly studied. Specially, for the anti-plane scattering problem of a circular cylindrical cavity embedded in a vertically transversely isotropic medium, In order to demonstrate the above proposed procedure is valid theoretically, through a coordinate transformation technique and recombination of the original expansion series, it can be shown that the original expansion series is identical to elliptic cylindrical wave function expansion for an isotropic medium, However, the original circular cylindrical cavity is transformed into an elliptic cylindrical cavity whose scattering problem can be solved analytically. From which the dynamic stress concentration phenomenon is thoroughly studied.