Extended Finite Element Method （擴展有限單元法）

《Extended Finite Element Method （擴展有限單元法）》是2015年清華大學出版社出版的圖書，作者是Zhuo Zhuang、Zhanli Liu,、Binbin Cheng,、Jianhui Liao。

基本介紹

• 中文名：Extended Finite Element Method （擴展有限單元法）
• 作者：Zhuo Zhuang,、Zhanli Liu、Binbin Cheng,、Jianhui Liao
• 出版社：清華大學出版社
• 出版時間：2015年5月19日
• 定價：79 元
• ISBN：9787302394822

內容簡介

本書為《擴展有限單元法》英文版。擴展有限單元法是為了解決複雜斷裂問題應運而生的一種新的有限元方法，對於各種形狀、沿任意路徑擴展的裂紋具有明顯的精確性與高效性。

本書較為系統地介紹了擴展有限元的理論、套用及最新研究進展，可供力學、土木、機械、航空、航天等專業的科研人員閱讀參考。

PREFACE

The extended .nite element method (X-FEM) is a novel numerical methodology, which was .rst proposed by Belytschko et al. in 1999. It has subsequently been developed very quickly in the mechanics .eld worldwide. Based on the .nite element method and fracture mechanics theory, X-FEM can be applied to solve complicated discontinuity issues including fracture, interface, and damage problems with great potential for use in multi-scale computation and multi-phase coupling problems. The fundamental concept and formula of X-FEM are introduced in this book, as well as the technical process of program implementation. The expressions for enriched shape functions of the elements are provided, which include the displacement discontinued crack, termed as “strong discontinuity”, and strain discontinued interface, termed as “weak discontinuity”, like heterogeneous materials with voids and inclusions, interfaces of bimaterial and two-phase .ows. X-FEM can be used to simulate element-crossed cracks and element-embedded cracks. Cracks with complex geometry can be modeled by structured meshes and can propagate along arbitrary route in the elements without the need for a re-meshing process, which provides considerable savings in computation cost whilst achieving precision.

In the early 1990s, the .rst author of this book, Prof. Zhuo Zhuang, was working on Ph.D. research under Prof. Patraic O’Donoghue at University Col-lege Dublin, Ireland, and completed a thesis on the development of the .nite element method for dynamic crack propagation in gas pipelines. In 1995, he returned to China and took an academic position at Tsinghua University. He has the privilege of learning from and working with Prof. Keh-Chih Hwang, and is striving to simulate arbitrary crack growth in three-dimensional continuities and curved shells. This is a natural choice for crack propagation, in which the original failure behavior of the structures reappears. In 2011, this aspiration was released by Dr. Binbin Cheng, who is the third author of this book. From his Ph.D. thesis work at Tsinghua University, they have developed an X-FEM code, named SAFRAC, with its own properties. The second author, Dr. Zhanli Liu, obtained a doctoral degree in 2009. The research thesis “The Investigation of Crystal Plasticity at Microscale by Discrete Dislocation and Nonlocal Theory” was nominated and achieved a national excellent doctor degree thesis in 2011. After graduation from Tsinghua University, he went to Northwestern University, USA to conduct postdoctoral research under Prof. Ted Belytschko. He continued to develop the X-FEM method for dynamic crack propagation and applications in heterogeneous materials, like ultrasonic wave propagation in three-dimensional polymer matrices enhanced by particles and short .bers. He returned to Tsinghua University and took an academic position in 2012. The fourth author, Dr. Jianhui Liao, obtained a doctoral degree in 2011 at Tsinghua University. His thesis work focuses on the application of X-FEM simulation in two-phase .ows. One pro-fessor and three former Ph.D. students are working together to write this book in order to demonstrate the research achievements on X-FEM in the last decade.

In this book, Chapter 1 reviews the development history, reference summa-rization, and research actuality of X-FEM. Chapters 2 and 3 provide an intro-duction to fracture mechanics, considering the essential concepts of static and dynamic linear elastic fracture mechanics, such as the crack propagation crite-rion, the calculation of stress intensity factor by interaction integral, the nodal force release technique to simulate crack propagation in conventional FEM, and so on. These two chapters provide essential knowledge of fracture mechanics essential for study of the subsequent chapters. Readers who are familiar with fracture mechanics can skip these two chapters. Chapters 4 and 5 contain the basic ideas and formulations of X-FEM. Chapter 4 focuses on the theoretical foundation, mathematical description of the enrichment shape function, discrete formulation, etc. In Chapter 5, based on the program developed by the authors and their co-workers, numerical studies of two-dimensional fracture problems are provided to demonstrate the capability and ef.ciency of the algorithm and the X-FEM program in applications of strong and weak discontinuity problems. In Chapters 6e9, scienti.c research conducted by the author’s group is given as examples to introduce the applications of X-FEM. In Chapter 6, a novel theory formula and computational method of X-FEM is developed for three-dimen-sional (3D) continuum-based (CB) shell elements to simulate arbitrary crack growth in shells using the concept of enriched shape functions. In Chapter 7, the algorithm is discussed and a program is developed based on X-FEM for simu-lating subinterfacial crack growth in bimaterials. Numerical analyses of the crack growth in bimaterials provide a clear description of the effect on fracture of the interface and loading. In Chapter 8, a method for representing discontinuous material properties in a heterogeneous domain by X-FEM is applied to study ultrasonic wave propagation in polymer matrix particulate/.brous composites. In Chapter 9, a simulation method of transient immiscible and incompressible two-phase .ows is proposed, which demonstrates how to deal with multi-phase .ow problems by applying X-FEM methodology. Based on the scienti.c research in the author’s group, Chapter 10 gives the applications of X-FEM in other frontiers of mechanics, e.g. nano-mechanics, multi-scale simulations, crack branches, and so on.

This book was published in a Chinese version in 2012, and was the .rst book on X-FEM published in China. At that time, Dr. Zhanli Liu was a postdoctoral fellow working at Northwestern University in the USA. He presented a copy of the book to Prof. Ted Belytschko to express our respect for him. Ted was very happy to see it and made complimentary remarks about the book, although he could not follow the Chinese characters but only the equations and .gures. He encouraged us to publish this book in an English version.

Regarding the English version, we would like to thank Mr. Lei Shi, Ms. Qiuling Zhang, and Ms. Hongmian Zhao at Tsinghua University Press. Without their encouragement and help, we could not have completed this book. We are also grateful to the Ph.D. candidates Ms. Dandan Xu and Mr. Qinglei Zeng for the computational examples that they provided.

This book is suitable for teachers, engineers, and graduate students on the disciplines of mechanics, civil engineering, mechanical engineering, and aero-space engineering. It can also be referenced by X-FEM program developers.

Zhuo Zhuang Zhanli Liu Binbin Cheng Jianhui Liao October 2013

CONTENTS

1. Overview of Extended Finite Element Method 1

1.1 Signi.cance of Studying Computational Fracture Mechanics 1

1.2 Introduction to X-FEM 2

1.3 Research Status and Development of X-FEM 8

1.3.1 The Development of X-FEM Theory 8

1.3.2 Development of 3D X-FEM 10

1.4 Organization of this Book 11

2. Fundamental Linear Elastic Fracture Mechanics 13

2.1 Introduction 13

2.2 Two-Dimensional Linear Elastic Fracture Mechanics 15

2.3 Material Fracture Toughness 19

2.4 Fracture Criterion of Linear Elastic Material 20

2.5 Complex Fracture Criterion 22

2.5.1 Maximum Circumference Tension Stress Intensity Factor Theory 22

2.5.2 Minimum Strain Energy Density Stress Intensity Fac-tor Theory 24

2.5.3 Maximum Energy Release Rate Theory 27

2.6 Interaction Integral 29

2.7 Summary 31

3. Dynamic Crack Propagation 33

3.1 Introduction to Dynamic Fracture Mechanics 33

3.2 Linear Elastic Dynamic Fracture Theory 35

3.2.1 Dynamic Stress Field at Crack Tip Position 35

3.2.2 Dynamic Stress Intensity Factor 37

3.2.3 Dynamic Crack Propagating Condition and Velocity 38

3.3 Crack Driving Force Computation 41

3.3.1 Solution Based on Nodal Force Release 41

3.3.2 Solution Based on Energy Balance 43

3.4 Crack Propagation in Steady State 44

3.5 Engineering Applications of Dynamic Fracture Mechanics 45

3.6 Summary 49

4. Fundamental Concept and Formula of X-FEM 51

4.1 X-FEM Based on the Partition of Unity 51

4.2 Level Set Method 53

4.3 Enriched Shape Function 55

4.3.1 Description of a Strong Discontinuity Surface 55

4.3.2 Description of a Weak Discontinuity Surface 58

4.4 Governing Equation and Weak Form 60

4.5 Integration on Spatial Discontinuity Field 64

4.6 Time Integration and Lumped Mass Matrix 67

4.7 Postprocessing Demonstration 68

4.8 One-Dimensional X-FEM 68

4.8.1 Enriched Displacement 68

4.8.2 Mass Matrix 72

4.9 Summary 72

5. Numerical Study of Two-Dimensional Fracture Problems with X-FEM 75

5.1 Numerical Study and Precision Analysis of X-FEM 76

5.1.1 A Half Static Crack in a Finite Plate 76

5.1.2 A Beam with Stationary Crack under Dynamic Loading 77

5.1.3 Simulation of Complex Crack Propagation 77

5.1.4 Simulation of the Interface 79

5.1.5 Interaction Between Crack and Holes 81

5.1.6 Interfacial Crack Growth in Bimaterials 83

5.2 T wo-Dimensional High-Order X-FEM 84

5.2.1 Spectral Element-Based X-FEM 84

5.2.2 Mixed-Mode Static Crack 87

5.2.3 Kalthoff’s Experiment 89

5.2.4 Mode I Moving Crack 91

5.3 Crack Branching Simulation 93

5.3.1 Crack Branching Enrichment 94

5.3.2 Branch Criteria 95

5.3.3 Numerical Examples 97

5.4 Summary 101

6. X-FEM on Continuum-Based Shell Elements 103

6.1 Introduction 104

6.2 Overview of Plate and Shell Fracture Mechanics 104

6.2.1 Kirchhoff Plate and Shell Bending Fracture Theory 105

6.2.2 Reissner Plate and Shell Bending Fracture Theory 109

6.3 Plate and Shell Theory Applied In Finite Element Analysis 113

6.4 Brief Introduction to General Shell Elements 115

6.4.1 BelytschkoeLineTsay Shell Element 115

6.4.2 Continuum-Based Shell Element 116

6.5 X-FEM on CB Shell Elements 119

6.5.1 Shape Function of a Crack Perpendicular to the Mid-Surface 119

6.5.2 Shape Function of a Crack Not Perpendicular to the Mid-Surface 123

6.5.3 Total Lagrangian Formulation 125

6.5.4 Time Integration Scheme and Linearization 127

6.5.5 Continuum Element Transformed to Shell 128

6.6 Crack Propagation Criterion 129

6.6.1 Stress Intensity Factor Computation 129

6.6.2 Maximum Energy Release Rate Criterion 133

6.7 Numerical Examples 136

6.7.1 Mode I Central Through-Crack in a Finite Plate 136

6.7.2 Mode III Crack Growth in a Plate 137

6.7.3 Steady Crack in a Bending Pipe 137

6.7.4 Crack Propagation Along a Given Path in a Pipe 139

6.7.5 Arbitrary Crack Growth in a Pipe 140

6.8 Summary 140

7. Subinterfacial Crack Growth in Bimaterials 143

7.1 Introduction 143

7.2 Theoretical Solutions of Subinterfacial Fracture 144

7.2.1 Complex Variable Function Solution for Sub-interfacial Cracks 144

7.2.2 Solution Considering the Crack Surface Affected Area 147

7.2.3 Analytical Solution of a Finite Dimension Structure 149

7.3 Simulation of Subinterfacial Cracks Based On X-FEM 153

7.3.1 Experiments on Subinterfacial Crack Growth 153

7.3.2 X-FEM Simulation of Subinterfacial Crack Growth 155

7.4 Equilibrium State of Subinterfacial Mode I Cracks 158

7.4.1 Effect on Fracture Mixed Level by Crack Initial Position 158

7.4.2 Effect on Material Inhomogeneity and Load Asymmetry 159

7.5 Effect on Subinterfacial Crack Growth from a Tilted Interface 163

7.6 Summary 165

8. X-FEM Modeling of Polymer Matrix Particulate/ Fibrous Composites 167

8.1 Introduction 167

8.2 Level Set Method for Composite Materials 169

8.2.1 Level Set Representation 169

8.2.2 Enrichment Function 172

8.2.3 Lumped Mass Matrix 173

8.3 Microstructure Generation 175

8.4 Material Constitutive Model 176

8.5 Numerical Examples 177

8.5.1 Static Analysis 177

8.5.2 Dynamic Analysis 181

8.6 Summary 187

9. X-FEM Simulation of Two-Phase Flows 189

9.1 Governing Equations and Interfacial Conditions 189

9.2 Interfacial Description of Two-Phase Flows 192

9.3 X-FEM and Unknown Parameters Discretization 194

9.4 Discretization of Governing Equations 200

9.5 Numerical Integral Method 205

9.6 Examples and Analyses 207

9.7 Summary 211

10. Research Progress and Challenges of X-FEM 213

10.1 Research on Micro-Scale Crystal Plasticity 213

10.1.1 Discrete Dislocation Plasticity Modeling 215

10.1.2 X-FEM Simulation of Dislocations 219

10.2 Application of Multi-Scale Simulation 223

10.3 Modeling of Deformation Localization 224

10.4 Summary 228

Appendix A: Westergaard Stress Function Method 229

Appendix B: J Integration 245

References 259

Index 269