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系統識別號 U0026-2001201711381900
論文名稱(中文) 含角點非對稱複合材料疊層板之邊界元素分析
論文名稱(英文) Boundary Element Analysis of Unsymmetric Laminates with Corners
校院名稱 成功大學
系所名稱(中) 航空太空工程學系
系所名稱(英) Department of Aeronautics & Astronautics
學年度 105
學期 1
出版年 106
研究生(中文) 張瀚文
研究生(英文) Han-Wen Chang
學號 P48961031
學位類別 博士
語文別 英文
論文頁數 113頁
口試委員 指導教授-胡潛濱
口試委員-夏育群
口試委員-楊文彬
口試委員-江達雲
口試委員-吳光鐘
口試委員-馬劍清
口試委員-趙振綱
口試委員-陳正宗
中文關鍵字 邊界元素法  複合材料疊層板  耦合問題  角點不連續  孔洞  史磋公式 
英文關鍵字 Boundary Element Method  Coupled Stretching-bending Analysis  Unsymmetric Composite Laminates  Corner Discontinuities  Rectangular Hole  Complex Variable Formalism  Stroh Formalism 
學科別分類
中文摘要 邊界元素法在現今彈性力學領域中的發展已愈趨成熟,並且呈現出更多元的面向。文獻裡,相關的研究與討論同時關注在其數學上的特性以及對於各種不同材料之彈性體的應用;另一方面,學者對於其發展要素譬如基本解的求得、奇異積分的計算,以及在角點不連續處的處理也有詳盡的討論。然而,這些研究多侷限於求解等向性或者金屬材料彈性體之二維問題、板的純彎矩問題,或是三維問題。為了符合工業設計上對於輕量化的要求和結構體中在受力方向上的結構加強需要,設計者或工程師會更傾向於採用複合材料疊層板來建立結構本體或其部件。近二十年來,以此為基礎和目標的研究多致力於正交性材料、對稱性疊層板和反對稱疊層板的應用;吾人若需考慮非對稱疊層板做為結構材料時,就需應對其更為複雜之力學行為,也就是平面內與出平面之變形/拉伸或彎矩的耦合問題。然而,此類問題在文獻中很少藉由邊界元素法得到較完善的分析和解決。
為了顧及所有不同類型疊層(對稱、反對稱、非對稱)所造成的板之不同力學行為,以及伴隨著角點存在之不連續性所產生的影響,在本研究中,藉由相應之邊界積分式和經由史磋公式推導而來之基本解,非對稱複材疊層板之邊界元分析受到了適當的建立和處理。在離散化邊界積分式的過程中,為準確地計算奇異積分以及解決在角點上因幾何不連續所衍生之線性相依方程式的問題,各種不同的計算和處理方式經過了實際的測試和驗證,並提出了奇異積分之解析顯示解以提供更準確的結果和更有效率的運算,以及四條輔助方程式用以取代相依方程式。另一方面,在傳統的邊界元分析中,節點上或靠近節點處之完整的應變與應力分量還需透過類似於後處理的方式來求得;因此在文獻裡,學者對此問題的處理採取了各種不同的方法與手段。在本研究中,由於邊界上節點之位移和曳引力結果,與稍微遠離邊界之內點上的位移、應變、曲率、合應力和彎矩力分量皆可被正確地求出與計算,這些正確的數值結果均被用以內插手法計算節點上或者鄰近邊界處之近似值。因此,在這過程裡,我們不需要再去執行包括超奇異積分與強奇異積分的計算。
綜合以上各階段的求解方式與流程,此非對稱複材疊層板之邊界元素分析可順利地提供疊層板的全域解,並且其整體的結果能保持一致的穩定性與正確性。
英文摘要 For the boundary element analysis of the elastic bodies in the practical engineering problems, researchers in this field have explored into the many aspects of this numerical method when it is engaged with different applications and different kinds of materials. In their studies, some critical concerns had been paid attention to such as the need of an adequate fundamental solution, the singular integrals, and the corner discontinuities. All of these problems can be regarded as the “classical” topics in these days, and many applications had indeed been well treated in the literature. However, most of these studies were confined to the two dimensional problems, plate bending problems or three dimensional problems utilizing isotropic or metallic materials. In the industry, in order to meet some structural designs such as the criteria of the light weight, or the strengthening of the materials in the directions of applying force, today engineers are more willing to take advantages of the designable characteristics of the composite materials or laminates. With this understanding, over the past few decades some researches had been conducted for the types of laminates such as specially orthotropic materials, symmetric or antisymmetric laminates. Nevertheless, if an unsymmetric laminate is considered, the mechanical behaviors of the plates will become more complex in such a way that the coupling between the in-plane and out-of-plane bending problems will be unavoidable, and this problem was seldom received a thorough solution via the use of boundary elements.
In this dissertation, to cover the complex mechanical behaviors of the composite laminates in response to all the possibility of symmetric, anti-symmetric, or unsymmetric stacking sequences and the corner discontinuities of a laminated plate, the coupled stretching-bending analysis of the general composite laminates via boundary elements has been developed with the help of the associated boundary integral equation, and the fundamental solution obtained via the Green’s function written in the form of Stroh-like complex variable formalism. To effectively treat the singular problem and the corner discontinuities which may result in dependent equations in the system of equations established via the discretization of the boundary integral equation, various methodologies were investigated to see their adequacies for the present application. And, the explicit solutions of the weakly and strongly singular integrals and the four auxiliary equations employed to replace the dependent equations are proposed in this study to solve the nodal displacements and tractions accurately and promptly.
Besides, similar to the needs in the traditional boundary element analysis, the post-processing for the calculations of the complete components of the strains and stress resultants at or near the boundary nodes was also implemented and carried out in the present study. In order to obtain these results, we can make useful the already known nodal displacements through the method of finite difference, and all the other correct results of strains and stress resultants calculated via the derivatives of boundary integral equation at the points not so close to the boundary, and the use of the constitutive equation of laminates. Finally, by utilizing the moving least square method with these results, we can further approximate the solution in the vicinity of boundary nodes with good accuracy. In this process, we don’t need to tackle again the singular problems which involve hyper-singular and strongly singular integrals. Hence, based on all the works required at different stages, the full-domain solution can be obtained for the coupling analysis of composite laminated plates via boundary elements with accuracy and efficiency.
論文目次 CONTENTS

ABSTRACT.................................................i
CONTENTS.................................................v
LIST OF TABLES.........................................vii
LIST OF FIGURES.......................................viii
NOMENCLATURE.............................................x
CHAPTER I INTRODUCTION.............................1
CHAPTER II ANALYSIS OF COMPOSITE LAMINATES..........8
2.1 Classical Lamination Theory..........................8
2.2 Stroh-like Formalism................................12
CHAPTER III BOUNDARY ELEMENT ANALYSIS...............16
3.1 Boundary Integral Equation..........................16
3.2 The Fundamental Solution............................18
3.3 Boundary Element Formulation........................20
3.4 Displacements, Strains and Stresses at Internal Points..................................................26
3.5 Locations of Source Points Related to the corner..................................................27
CHAPTER IV AUXILIARY RELATIONS FOR CORNERS.........35
CHAPTER V SINGULAR INTEGRALS......................43
5.1 Numerical Integration by Finite Part Integrals......44
5.2 Explicit Closed-Form Solutions of the Singular Integrals...............................................47
5.3 Indirect Calculation of the Singular Integrals...............................................54
5.4 Discussion..........................................56
CHAPTER VI COMPLETE SOLUTION AT OR NEAR THE BOUNDARY NODES...................................................59
6.1 Strains and Stresses at Boundary Nodes..............60
6.2 Regularization for the Points Near the Boundary Nodes...................................................64
6.3 Discussion..........................................65
CHAPTER VII NUMERICAL EXAMPLES......................67
7.1 Calculation of the Singular Integrals...............67
7.2 Location of Source Points Related to Corner Nodes with Various Boundary Conditions.............................69
7.3 Calculation of the Strains and Stresses at or near the boundary nodes..........................................76
CHAPTER VIII CONCLUSIONS.............................81
REFERENCES..............................................85
TABLES..................................................90
FIGURES.................................................97
PUBLICATION LIST.......................................114
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