傾斜橫向等向性材料承受三向度點荷重的三維位移與應力基本解

Abstract

本論文主要探討橫向等向性材料在橫向等向面與水平面呈傾斜狀況下，承受三向度點荷重在無限或半無限空間中，三維位移與應力的基本解。一般而言，在無限空間或半無限空間中，運動或力平衡方程式為偏微分方程式（Partial Differential Equations）。傅立業轉換(Fourier Transform)與拉普拉司轉換(Laplace Transform)，是常用來解決無限空間與半無限空間邊界值問題(Boundary Value Problems)的有效方法。先針對變數x與y部分進行二維傅立業轉換(Double Fourier Transform)，將偏微分方程式轉換成常微分方程式(Ordinary Differential Equations)。本論文提出三種方法來求解前述常微分方程式，以獲得無限空間與半無限空間的應力及位移解析解。第一種方法，利用待定係數法及分離變數法直接求解受點荷重後的非齊性(Nonhomogeneous)常微分方程式，在無限或半無限空間中之齊性解(Homogeneous Solution)及特解(Particular Solution)。第二種方法是將無限空間區分為三個區域 (區域2-上半平面) 、 (虛擬空間) 及 (區域1-下半平面)，或是半無限空間區分為兩個區域 (虛擬空間) 及 (區域1-下半平面)，而作用之點荷重在無限空間是作用在 ，半無限空間是作用在 。在區域1及區域2內，力平衡方程式的右邊並無力量作用，故可視為齊性方程式。接著分別考量無限空間中區域1、區域2及虛擬空間或半無限空間中區域1及虛擬空間的組合邊界值條件。第三種方法，在無限空間中針對變數z進行傅立業轉換，能將前面所求出之常微分方程式轉換成多項式方程式。這種方法同時針對變數x, y和z進行傅立業轉換，故也可以稱之為三維傅立業轉方法（Triple Fourier Transforms）。換句話說，在無限空間中針對x, y和z進行三維傅立業轉換，可以將偏微分方程式轉換成多項式方程式。在半無限空間中，利用前面二維傅立業轉換得到的常微分方程式，再針對變數z進行拉普拉司轉換，同樣可得到多項式方程式。因此可求得在三維傅立業轉換域的無限空間位移解( )及在二維傅立業及拉普拉司轉換域半無限空間位移解( )。接著將前面所求得不同轉換域的解進行逆轉換分別為三維傅立業逆轉換（Inverse Triple Fourier Transforms）或二維傅立業及拉普拉司逆轉換（Inverse Double Fourier and Laplace Transforms）。利用這種轉換方式可明確的將作用在傾斜的橫向等向性材料三維點荷重的應力及位移解析解求出。本解析解的主要影響參數包括（1）橫向等向面的旋轉角度（2）各個材料參數的異向度（3）幾何位置參數（4）三維的點荷重的形式。 最後本研究比較王承德與廖志中（1991）的解析解，並針對影響參數對位移與應力影響加以探討，發現，在無限空間中，當材料是均質、線彈性及橫向等向面平行水平方向時，所求得的解有一致的結果。在半無限空間中，利用本方法所求出承受點荷重的解與王承德與廖志中的結果有明顯差異。

Three-dimensional fundamental solutions of displacements and stresses due to three-dimensional point loads in a transversely isotropic material, where the planes of transverse isotropy are inclined with respect to the horizontal loading surface, are presented in this thesis. Generally, the governing equations for infinite or semi-infinite solids are partial differential equations. The Fourier and Laplace integral transforms are commonly two efficient methods for solving the corresponding boundary value problems of full or half space. Employing the Fourier transform, the partial differential equations can be simplified as ordinary differential equations (ODE). Then, three distinct approaches were used to solve the ODE and the solutions were presented for both infinite and semi-infinite solids in this thesis. Firstly, we solve traditionally the nonhomogeneous ordinary differential equations by the methods of undetermined coefficients and separate variables Secondly, the method of an imaginary space was proposed for deriving the solutions of the problems. Thirdly, the method of algebraic is adopted for deriving the solutions for both full space and half space problems. Finally, the present fundamental solutions are derived by performing the required triple inverse Fourier transforms, or double inverse Fourier and Laplace transforms. These transformations are powerful to generate the displacements and stresses resulting from the three-dimensional point loads, acting in an inclined transversely isotropic material. The yielded solutions demonstrate that the displacements and stresses are profoundly influenced by: (1) the rotation of the transversely isotropic planes (□), (2) the type and degree of material anisotropy (E/E□, □/□□, G/G□), (3) the geometric position (r, □, □), and (4). the types of three-dimensional loading (Px, Py, Pz). The proposed solutions are exactly the same as those of Wang and Liao (1999) if the full-space is homogeneous, linearly elastic, and the planes of transversely isotropy are parallel to the horizontal loading surface. Additionally, a parametric study is conducted to elucidate the influence of the above-mentioned factors on the displacements and stresses. Computed results reveal that the induced displacements and stresses in the planes of transversely isotropic are parallel to the horizontal loading surface of isotropic/transversely isotropic rocks by a vertical point load are quite different from those from Wang and Liao (1999). Therefore, in the fields of practical engineering, the dip at an angle of inclination should be taken into account in estimating the displacements and stresses in a transversely isotropic rock subjected to applied loads.