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研究生:許信翔
研究生(外文):Hsin-HsiangHsu
論文名稱:無元素法求解缺角及裂縫尖端附近應力之研究
論文名稱(外文):Study on Stress Field Near the Notch and Crack Tip Solved by Element-Free Method
指導教授:朱聖浩
指導教授(外文):Shen-Haw Ju
學位類別:博士
校院名稱:國立成功大學
系所名稱:土木工程學系
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:英文
論文頁數:122
中文關鍵詞:複合材料裂縫無元素法有限元素法缺角主要解次要解應力強度因子三維
外文關鍵詞:Composite materialCrackElement-free methodFinite element methodNotchPrimary solutionShadow solutionStress intensity factorThree-dimensional
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一般而言,應力在裂縫及缺角尖端會產生無限大之現象,對於二維的裂縫及缺角問題,目前已有理論方法可完整求解,但對三維的裂縫及缺角問題,例如一平版含有缺角,則目前還沒有合理的理論解。本論文將利用無元素法求解材料裂縫及缺角的力學問題,最主要原因是無元素法可容易的將應力或變位函數加到其要推導之勁度矩陣中,目前無元素法已大量的應用在裂縫開裂問題上,但目前卻沒有相關文獻用在缺角的力學問題上。
本論文首先將從無元素法的基礎理論作進一步延伸,以解決目前無元素法在數值分析上的一些缺點;接著使用理論公式推導二維非均質材料之缺角應力與變位場,再將此理論結果應用在無元素法上,並建立一奇異性缺角元素,如此可用較疏的節點求得較精確的分析結果。
另外,本論文也將提出三維彈性體材料含無限裂縫之理論公式,此理論公式包含主要解( primary solution )及次要解( shadow solution ),其中主要解即為傳統平面應變理論下的解析解,而次要解則是根據三維彈性力學理論平衡方程式來求得,最小二乘法結合有限元素法的數值解則被使用來確認未定係數;本論文研究的結果顯示,只要包含足夠的主要解及次要解項次,則可以得到非常準確的三維彈性體裂縫問題的變位場,本方法最主要的優點在於可以求得裂縫尖端附近三維變位場的奇異影響,而不需要有材料特性、邊界條件及施加載重的任何限制。
本論文最後會將無元素法初步應用到三維破壞力學解析理論中,其分析結果將與正常但極密網格之有限元素法之分析結果相比較,此法最大之優點為可得到裂縫的應力奇異性質,此是有限元素法所忽略且無法求得之項目。
Generally speaking, the stress in crack and notch tip will produce infinitely great phenomenon. For the two-dimensional crack and notch problems, stress near the crack and notch tip can solved exactly by the theoretical analysis, but for the three-dimensional crack and notch ones, for example a plate with a notch, there is no rational solutions at present. In this thesis, the element-free method will be used to solve the mechanics problem of crack and notch, and the main reason is that the element-free method can add stress and strain functions in the strength matrix easily. The element-free method has a wild application on crack problem, but there are no relevant documents using this method on the notch mechanics problem at present.
First, the basic theory of the element-free method will be further extended to solve the numerical problems associated with this method in this thesis. Second, the theoretical formula will be used to derive the stress and displacement fields on in-plan notch problems with anisotropic materials, and then this theory results will be applied to the element-free method. Finally, a singularity notch element is set up in this thesis, and this method can obtain more accurate analysis result with the node that is relatively dredged.
Furthermore, this thesis proposes the theoretical solution for the three-dimensional problem containing a semi-infinite crack with primary and shadow solutions, where the primary solution is the traditional plane-strain solution and the shadow ones can be obtained from the 3D equilibrium equation of the theory of elasticity. A least-squares method incorporating the finite element results was used to determine these factors in this theoretical solution. The results of this research show that if enough primary and shadow solutions are included, numerical simulations indicate that the proposed method can obtain an accurate displacement field for 3D crack problems. The major advantage of this method is that a 3D whole displacement field with the analytic singular effect near the crack tip can be obtained, without any limitation with regard to the material properties, boundary conditions, and applied loads.
Finally, the element-free method is applied to the three-dimensional fracture mechanics preliminarily, and the result was compared with the normal finite element method result using a fine mesh. The major advantage is that the crack stress singularity, which is ignored in the finite element method, can be obtained in the proposed method.
CONTENTS
摘 要(CHINESE ABSTRACT)……………………………………………………I
ABSTRACT…………………………………………………………………………III
誌 謝(CHINESE ACKNOWLEDGEMENTS)……………………………………V
CONTENTS………………………………………………………………………VI
LIST OF TABLES…………………………………………………………………X
LIST OF FIGURES………………………………………………………………XI
Chapter 1 INTRODUCTION………………………………………………………1
1.1 Background…………………………………………………………………1
1.2 Objective and Scope of Research…………………………………………2
1.3 Organization of Dissertation………………………………………………3
Chapter 2 LITERATURE REVIEW………………………………………………7
2.1 Development of element-free method……………………………………7
2.2 Error analysis of element-free method……………………………………8
2.3 Element-free method applied to crack and notch problem……………10
2.4 Analysis solution investigation for 3D semi-infinite crack problem……12
Chapter 3 THEORY AND APPLICATION OF ELEMENT-FREE METHOD……15
3.1 Solutions to decrease numerical errors in EFM………………………15
3.2 OpenMP parallel scheme for EFM………………………………………17
3.2.1 Parallel calculation of stiffness matrices and selected nodes for Gaussian points………………………………………………………18
3.2.2 Parallel generation and solution of global stiffness matrix………20
3.3 Numerical examples………………………………………………………23
3.3.1 Investigation of the truncation error in EFM……………………23
3.3.2 Study of efficiency in EFM………………………27
3.4 Chapter summary…………….30
Chapter 4 ANALYTIC AND NUMERICAL SOLUTION FOR CRACK PROBLEM……33
4.1 Relation between stress and strain in an anisotropic elastic material…33
4.2 Introduction of Lekhnitskii formulation and Stroh formulation employ to crack problem…………………………………………………… 34
4.3 Introduction of William’s solution for crack problem…………………40
4.4 Introduction of current element-free method for crack problem………47
Chapter 5 ELEMENT-FREE METHOD FOR IN-PLANE NOTCH PROBLEM WITH ANISOTROPIC MATERIAL……50
5.1 In-plane displacement and stress fields near notches…………………… 50
5.2 Element-free Galerkin method with MLS formulation for Sharp-V notches……53
5.2.1 Shape functions and their derivatives in the notch coordinates……53
5.2.2 Transformation of matrices from local directions to global direction………………………………………………………………55
5.2.3 Stiffness matrix based on the element-free Galerkin method……57
5.3 Illustration of numerical validation and tested cases……………………58
5.3.1 H-integral and least-squares schemes to evaluate notch SIFs……58
5.3.2 Details of V-notches examples………………………59
5.4 Numerical results and comparisons……………63
5.4.1 Effect of the number of Gaussian quadrature points and displacement terms…………………………………………………… 63
5.4.2 Comparison between FEM and EFGM with the mesh size effect… 65
5.5 Chapter summary……………68
Chapter 6 DISPLACEMENT FIELD ANALYSIS SOLUTION OF 3D BODY WITH A SEMI-INFINITE CRACK………………………………69
6.1 Recursive solutions of 3D elasticity bodies…………………… 69
6.2 Displacement fields of a 3D body with a semi-infinite crack……………70
6.2.1 Primary solutions……………………………………………………71
6.2.2 Shadow solutions……………………………………………………72
6.2.2.1 Solving the Mode-II shadow solutions {U1} under {t}={1,0,0}T
for {U0}……………………………………………………............ 72
6.2.2.2 Solving the Mode-II shadow solutions {U2} under {t}={1,0,0}T for {U0}……………………………………………………72
6.3 Least-squares method to fit the 3D finite element results………………74
6.4 Numerical simulations……………76
6.4.1 A plate with a central horizontal crack………………………… 77
6.4.2 A plate with a central slant crack…………………………………… 81
6.5 Chapter summary……………84
Chapter 7 PRELIMINARY STUDY OF ELEMENT-FREE METHOD FOR 3D CRACK PROBLEMS……………………………………………86
7.1 Element-free method applied for 3D crack problem with analytic solutions…….86
7.2 Numerical simulations……96
7.3 Chapter summary………… 98
Chapter 8 CONCLUSIONS AND RECOMMENDATIONS……………………100
8.1 Conclusions………………………………………………………………100
8.2 Recommendations for further research…………………………………102
REFERENCES……………………………………………………………………103
APPENDIX………………………………………………………………………116
自 述(CHINESE VITA)…………………………………………………………122
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