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研究生： 黃紹慈
Huang, Shao-Tzu
論文名稱： 多孔洞/裂縫之邊界有限元素分析
A Boundary Finite Element for Anisotropic Materials containing Multiple Holes and Cracks
指導教授： 胡潛濱
Hwu, Chyan-Bin
學位類別： 碩士
Master
系所名稱： 工學院 - 航空太空工程學系
Department of Aeronautics & Astronautics
論文出版年： 2016
畢業學年度： 104
語文別： 中文
論文頁數： 77
中文關鍵詞： 邊界元素法 、有限元素法 、多孔洞 、多裂縫
外文關鍵詞： boundary element, finite element, multiple holes, multiple cracks
相關次數： 點閱：96  下載：3 
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    • 二維線彈性異向性平板含有孔洞或裂縫的格林函數已經透過史磋方程式取得。將此格林函數作為邊界元素法的基本解，就不需要在孔洞或裂縫的邊界上切割元素，因為孔洞或裂縫邊界上的邊界條件已經自動滿足了。同時，孔洞的應力集中因子和裂縫的應力強度因子只需經由外部的平板邊界節點上的資料就可算出，不必另外計算裂縫附近的數值。為了擴展一個洞或裂縫至多個洞或裂縫，以往將一個孔洞或裂縫視為一個次區域，利用組合區域法，藉由考慮每個次區域交界面的諧和與平衡條件來結合每個次區域。但是，遇到孔洞或裂縫過多時，諧和與平衡條件會使組合區域法變得很複雜。因此，本文利用邊界元素法的曳引力與有限元素法的節點力之間的關係式，將邊界元素轉換為有限元素，因此一個次區域可視為有限元素法中的一個元素，再透過有限元素法將多個次區域結合。此外，在邊界元素法中，不同於以往使用線性的內插形狀函數，本文建立二次式的內插形狀函數。
    最後，經由文獻和有限元素軟體ANSYS提供的解析解或數值解比對驗證，證明本方法的可行性和數值精確度。

    In this study, a special boundary element for a two-dimensional anisotropic elastic solid containing a single elliptical hole or crack is applied. The main feature of this special boundary element is that no meshes are needed along the boundary of holes. Besides, the stress intensity factors of the crack or the stress concentrated factor of hole can be evaluated by using only the remote boundary displacements and tractions.To extend this special boundary element to the problems with multiple holes, a boundary finite element is established further by using the relation between element nodal force of finite element and surface traction of boundary element. The accuracy and efficiency of the proposed boundary finite element were then verified through the numerical examples compared with the solutions obtained by analytical solutions and the commercial finite element software.

    目錄----------------------------------------------------------------------------------i 表目錄-----------------------------------------------------------------------------iii 圖目錄-----------------------------------------------------------------------------iv 符號--------------------------------------------------------------------------------vi 第一章 緒論-----------------------------------------------------------------------1 1.1 文獻回顧-------------------------------------------------------------------1 1.2 研究目的-------------------------------------------------------------------3 第二章 邊界元素法--------------------------------------------------------------4 2.1邊界積分式-----------------------------------------------------------------4 2.2 基本解----------------------------------------------------------------------8 2.3 節點應變和應力的計算-------------------------------------------------9 2.4 內部點的計算-----------------------------------------------------------12 2.5 座標轉換-----------------------------------------------------------------13 2.6 孔洞應力集中因子-----------------------------------------------------16 2.7 裂縫應力強度因子-----------------------------------------------------17 第三章 邊界有限元素---------------------------------------------------------20 3.1轉換邊界元素為有限元素---------------------------------------------21 3.2節點曳引力---------------------------------------------------------------23 第四章 討論---------------------------------------------------------------------25 4.1多邊形孔洞基本解的不連續問題------------------------------------25 4.2以邊界元素法分析三角形洞問題------------------------------------31 第五章 數值範例與結果------------------------------------------------------35 5.1 材料性質-----------------------------------------------------------------35 5.2 單一橢圓洞或裂縫-----------------------------------------------------36 5.2.1 單一水平橢圓洞---------------------------------------------------36 5.2.2 單一傾斜橢圓洞---------------------------------------------------39 5.2.3 單一裂縫------------------------------------------------------------40 5.3 雙孔洞或裂縫-----------------------------------------------------------42 5.3.1 雙橢圓洞------------------------------------------------------------43 5.3.2 雙水平裂縫---------------------------------------------------------46 5.3.3 一個水平裂縫和一個傾斜裂縫---------------------------------48 5.3.4 一個橢圓洞和一個傾斜裂縫------------------------------------50 5.4 多孔洞或多裂縫--------------------------------------------------------51 5.4.1 四個橢圓洞---------------------------------------------------------52 5.4.2 四個裂縫------------------------------------------------------------54 5.4.3 九個傾斜裂縫------------------------------------------------------56 5.4.4 三個橢圓洞和兩個裂縫------------------------------------------58 第六章 結論---------------------------------------------------------------------61 參考文獻--------------------------------------------------------------------------62 附錄 A----------------------------------------------------------------------------65 表目錄 表4.1 處的 ( )--------------------------------28 表4.2 處的 ( )--------------------------------30 表5.1 裂縫右尖端應力強度因子--------------------------------------------41 表5.2 橢圓洞邊緣環應力-----------------------------------------------------45 表5.3 裂縫各尖端應力強度因子--------------------------------------------47 表5.4 裂縫尖端B應力強度因子比較表----------------------------------48 表5.5 裂縫左尖端的應力強度因子 、 ------------------------------51 表5.6 8個裂縫尖端的應力強度因子-------------------------------------55 表5.7 9個裂縫尖端的應力強度因子-------------------------------------57 表5.8 橢圓洞/裂縫的應力強度因子/應力集中因子-------------------60   圖目錄 圖2.1 節點座標系統關係圖--------------------------------------------------10 圖2.2 全域座標與局部座標關係圖-----------------------------------------13 圖3.1 次區域介面處的節點--------------------------------------------------21 圖4.1 時，三角洞在 平面的映射值---------------------27 圖4.2 時，三角洞在 平面的映射值---------------------39 圖4.3 邊界節點與不連續點位置圖-----------------------------------------31 圖4.4 三角形孔洞一端受均佈力一端固定示意圖-----------------------31 圖4.5 三角洞環應力分佈圖--------------------------------------------------32 圖4.6 三角洞環應力分佈圖(20個節點) ---------------------------------33 圖4.7 三角洞環應力分佈圖(84個節點) ---------------------------------33 圖4.8 三角洞環應力分佈圖(164個節點) --------------------------------34 圖5.1 單一橢圓洞受垂直均佈力示意圖-----------------------------------37 圖5.2 單一橢圓洞應變分佈圖-----------------------------------------------38 圖5.3 單一橢圓洞應力分佈圖-----------------------------------------------38 圖5.4 單一橢圓洞環應力分佈圖--------------------------------------------39 圖5.5 單一裂縫受垂直均佈力示意圖--------------------------------------40 圖5.6 裂縫右尖端應力強度因子比較圖-----------------------------------42 圖5.7 雙橢圓洞受垂直均佈力示意圖--------------------------------------43 圖5.8 環應力值收斂性分析--------------------------------------------------44 圖5.9 雙裂縫受垂直均佈力示意圖-----------------------------------------46 圖5.10 裂縫尖端D的應力強度因子 收斂圖( )----------------47 圖5.11 一水平裂縫和一傾斜裂縫受四邊均佈力示意圖---------------48 圖5.12 一橢圓洞和一傾斜裂縫受垂直均佈力示意圖------------------50 圖5.13 四個橢圓洞一端受均佈力一端固定示意圖---------------------52 圖5.14 等向性橢圓洞環應力分佈圖---------------------------------------53 圖5.15 異向性橢圓洞環應力分佈圖---------------------------------------53 圖5.16 四個裂縫受垂直均佈力示意圖------------------------------------54 圖5.17 九個傾斜裂縫受垂直均佈力示意圖------------------------------56 圖5.18 三個橢圓洞和兩個裂縫受垂直均佈力示意圖------------------58 圖5.19 次區域和元素切割示意圖------------------------------------------58 圖5.20 橢圓洞環應力分佈圖 ----------------------------------------------60 圖A.1 程式架構圖-------------------------------------------------------------65 圖A.2 兩橢圓洞問題的節點分佈示意圖----------------------------------67 圖A.3 兩相鄰元素示意圖----------------------------------------------------70 圖A.4 兩角點元素示意圖------------------------------71 圖A.5 雙點系統和單點系統的節點編號----------------------------------74 圖A.6 單點系統節點編號取得過程----------------------------------------76

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