臺灣博碩士論文加值系統

　　從1882年V. Cerruti[3]發表集中切力作用在二分之一空間固體表面的解析解後，就很少有學者對此問題做正確性的探討；但此問題乃是三維彈性力學的基本題型，其正確與否將攸關後續三維彈性力學的研究。
　　本文以使用不同的拍伯克維支－紐伯－布斯尼斯克函數(Bx, By, Bz, β)來推導新的解析解，其中方法(一)是吾人依施力方向為x方向所假設的Bx、β之推導方法，而方法(二)是以假設函數為Bx, Bz, β之推導方法，此法為V. Cerruti[3]的原始解析方法；另外本文引用A. S. Saada[17]所記載的Cerruti's Problem解析解作為本文的方法(三)，此法是以合併Galerkin vector及Lame strain potential來解此題。而在數值分析上本文使用COSMOS/M套裝軟體，並以8個節點元素進行有限元素的數值分析。
　　在經由三種解析法與數值分析的結果一一的比較後，發現只有方法(一)在假設拍伯維克維支－紐伯－布斯尼斯克函數為Bx, β 時，解析解的位移、應力值及位移、應力的圖形，較接近數值分析結果的物理意義，但此法還是將所有的積分函數省略，因此還不能算完全正確的解。
　　由本文及以往學者的分析結果，可獲得欲計算二分之一空間固體受內部集中力(Mindlin's Problem)、外部集中力(Boussinesq's Problem)以及外部集中切力(Cerruti's Problem)作用時的解析解，均只需假設兩個獨立拍伯維克維支－紐伯－布斯尼斯克函數即可獲得正確的解析解，以供後續學者做研究參考。

　　In 1882, V. Cerruti [3] published the analytical solution of concentrated tangential force acting on the surface of a half-space solid. Since then, the correctness of this problem was seldom been explored. Its correctness has great influence on the follow-up research of three-dimensional elasticity because this problem is one of the basic types of problems in three-dimensional elasticity.
　　The analytical solutions of this thesis were obtained by using different functions out of four Papkovitch-Neuber-Boussinesq functions Bx, By, Bz, β . The method (一) of the analytical solution is derived by using Bx,β out of four Papkovitch-Neuber-Boussinesq functions Bx, By, Bz, β . This idea is because the concentrated tangential force is acting in the X direction of the problem. For the method (二) of the analytical solution, three functions Bx, Bz,βwas used. This analytical solution was originally solved by V. Cerruti [3]. The method (三) was quoted from A. S. Saada's book [17] of " Elasticity: Theory and Applications ". This method solved Cerruti's Problem by combining the Galerkin vector and Lame strain potential. In the numerical analysis, this thesis used the finite element analysis software-COSMOS/M and choose 8 Node Solid Element to obtain the numerical solutions.
　　By one by one comparing the results of the three analytical methods and the FEM analysis. We find that when only two functions Bx and β are chosen from four Papkovitch-Neuber-Boussinesq functions Bx, By, Bz, β [method (一)] , the values and the diagrams of the displacement and stress of the analytical solution are closer to the physical meaning of the results of FEM analysis. This method still left out all integral functions in this paper so this analytical solution has a few errors yet.
　　From the results of this paper and former works, we may conclude that when deriving Mindlin's Problem (a concentrated force acting inside the body), Boussinesq's Problem (a concentrated normal force acting on the boundary), or, Cerruti's Problem (a concentrated tangential force acting on the boundary) of a half-space solid, only two Papkovitch-Neuber-Boussinesq functions will be enough for a satisfactory analytical solution.

誌謝 i
中文摘要 ii
英文摘要 iii
目錄 v
表目錄 viii
圖目錄 ix
符號說明.. xix
第一章 緒論 1
1.1 研究動機 1
1.2 文獻回顧 2
1.3 研究方法 4
第二章 基本理論的探討 6
2.1 位移函數與拍伯克維支-紐伯-布斯尼斯克函數
之關係 6
2.2 應力函數與拍伯克維支－紐伯－布斯尼斯克函數
之關係 10
2.3 格林─邵函數基本項導法之回顧 12
第三章 集中切力作用在二分之一空間固體表面邊界條件
的處理 17
3.1 二分之一空間固體的格林公式 17
3.2 集中切力作用在二分之一空間固體表面的解析 19
3.3 分佈切力作用在二分之一空間固體表面的邊界
條件推導 21
第四章 集中切力作用在二分之一空間固體表面解析解
的推導方法 24
4.1 方法(一) 24
4.1.1 重解方法(一) 29
4.2 方法(二) 32
4.3 方法(三) 35
第五章 COSMOS/M電腦數值分析 40
5.1 COSMOS/M應用軟體簡介 40
5.2 有限元素分析的基本學理 41
5.3 建立三維有限元素圖形 44
5.4 分割三維實體元素模型 45
5.5 材料參數及邊界條件設定 47
第六章 結果與討論 51
6.1 結果 51
6.2 數值討論 60
6.2.1 (0,0,4)位置的數值討論 60
6.2.2 (12,12,4)位置的數值論 61
6.2.3 (0,0,0)位置的數值討論 63
6.2.4 極值位置討論 64
6.3 圖形討論 68
6.3.1 Z=4的圖形討論 68
6.3.2 Z=0的圖形討論 71
第七章 總結與展望 185
參考文獻 188
附錄一 方法(一)在Mathematica的輸入值 193
附錄二 方法(二)在Mathematica的輸入值 201
附錄三 方法(三)在Mathematica的輸入值 203
附錄四 模式(一)在COSMOS/M的輸入值 205
附錄五 模式(二)在COSMOS/M的輸入值 206
附錄六 模式(三)在COSMOS/M的輸入值 207
附錄七 模式(四)在COSMOS/M的輸入值 208
附錄八 COSMOS/M對8-20個節點固體元素定義 209
附錄九 重解方法(三)中 的解析解 215

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