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研究生:薛涵穎
研究生(外文):Han-Ying Hsueh
論文名稱:以延伸有限元素法模擬介面裂縫:設置與驗證
論文名稱(外文):Simulation of interface cracking using XFEM: Implementation and benchmarking
指導教授:余念一
指導教授(外文):Yu.
學位類別:碩士
校院名稱:元智大學
系所名稱:機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:英文
論文頁數:107
中文關鍵詞:延伸有限元素法斷裂力學應力強度因子薄膜
外文關鍵詞:XFEMdiscontinuitiessingularitiesstress intensity factor
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延伸有限元素法(Extended Finite Element Method; XFEM)是由傳統有限元素法延伸而來,當在處裡斷裂力學中裂縫問題時,它不需要定義一個明確的裂縫表面,網格不需要沿著裂縫設置,且在裂縫成長的過程中,不需要重新網格化。利用延伸有限元素法在分析裂縫問題時,可以更快速更有效地求得收斂的數值解。
本論文發展延伸限元素法的程式,模擬一含雙單邊裂縫材料受到單軸拉伸後的位移及應力分佈,並且求得應力強度因子(stress intensity factor),將其模擬結果與斷裂力學之理論解與傳統有限元素法的數值解相互比較。結果顯示其模擬結果與理論解的誤差值小於百分之ㄧ。
並且探討介面裂縫在一半無限大之基材與薄膜間,整個薄膜/基材受到施加在薄膜上單軸拉伸應力時,薄膜與基材不同楊氏係數的效應,以及基材深度變化的效應,模擬結果與參考文獻的理論解相互比較。
The extended finite element method (XFEM) is able to analyze the problems that standard FEM cannot efficiently solve, for example, discontinuities and singularities. XFEM allows modeling a crack without explicitly defining the crack surface and needs no remeshing when being used to solve the crack propagation problem.
In the present work, the code for XFEM is implemented and benchmarked for two problems. Double edge cracks in a plate subjected to axial loading are firstly considered. The results of XFEM are benchmarked by those of traditional FEM and the analytic estimates provided by fracture mechanics. The error between the numerical stress intensity factor and the theoretical one could be less than one percent.
The interface crack between a semi-infinite substrate and film under general edge loading conditions is also considered. Some important effects are studied, for example, the effect of material mismatch and the depth of the substrate. The results are benchmarked with reported theoretical predictions.
Table of Content
Abstract i
Acknowledgments iii
Table of Contents iv
List of Tables vi
List of Figures vii
Chapter 1 Introduction
1.1 Background 1
1.2 Fracture Mechanics…………………………………………………….…..1
1.3 Literature Review 3
1.4 Objective 7
Chapter 2 Fundamental Theorems of XFEM
2.1 Finite element approximations 9
2.2 Finite element approximations with local enrichment 9
2.3 Discontinuous enrichment 10
2.4 Near-tip enrichment 13
2.5 Stress intensity factors computation using the interaction integral 15

Chapter 3 Benchmarking
3.1 Benchmarking problem statement 34
3.2 Matlab code implementation of stiffness matrix and Gauss quadrature 34
3.3 Matlab code implementation of boundary and loading conditions 37
3.4 Matlab code implementation of stress intensity factor 39
3.5 Mode I double edge cracks 39
Chapter 4 Interface cracking in a thin film/substrate system
4.1 Cracking of pre-tensioned films 50
4.2 Matlab code implementation of interface crack 50
4.3 Benchmarking 60
4.4 Energy release rate and stress intensity factors 62
4.5 Results and discussions 64
Chapter 5 Conclusions and Discussions 78
References 79
Appendix 84
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