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XFEM Fracture Analysis of Composites

ISBN: 978-1-119-97406-2
400 pages
October 2012
XFEM Fracture Analysis of Composites (1119974062) cover image

This book describes the basics and developments of the new XFEM approach to fracture analysis of composite structures and materials. It provides state of the art techniques and algorithms for fracture analysis of structures including numeric examples at the end of each chapter as well as an accompanying website which will include MATLAB resources, executables, data files, and simulation procedures of XFEM.

  • The first reference text for the extended finite element method (XFEM) for fracture analysis of structures and materials
  • Includes theory and applications, with worked numerical problems and solutions, and MATLAB examples on an accompanying website with further XFEM resources
  • Provides a comprehensive overview of this new area of research, including a review of Fracture Mechanics, basic through to advanced XFEM theory, as well as current problems and applications
  • Includes a chapter on the future developments in the field, new research areas and possible future applications of the method
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Preface xiii

Nomenclature xvii

1 Introduction 1

1.1 Composite Structures 1

1.2 Failures of Composites 2

1.2.1 Matrix Cracking 2

1.2.2 Delamination 2

1.2.3 Fibre/Matrix Debonding 2

1.2.4 Fibre Breakage 3

1.2.5 Macro Models of Cracking in Composites 3

1.3 Crack Analysis 3

1.3.1 Local and Non-Local Formulations 3

1.3.2 Theoretical Methods for Failure Analysis 5

1.4 Analytical Solutions for Composites 6

1.4.1 Continuum Models 6

1.4.2 Fracture Mechanics of Composites 6

1.5 Numerical Techniques 8

1.5.1 Boundary Element Method 8

1.5.2 Finite Element Method 8

1.5.3 Adaptive Finite/Discrete Element Method 10

1.5.4 Meshless Methods 10

1.5.5 Extended Finite Element Method 11

1.5.6 Extended Isogeometric Analysis 12

1.5.7 Multiscale Analysis 13

1.6 Scope of the Book 13

2 Fracture Mechanics, A Review 17

2.1 Introduction 17

2.2 Basics of Elasticity 20

2.2.1 Stress–Strain Relations 20

2.2.2 Airy Stress Function 22

2.2.3 Complex Stress Functions 22

2.3 Basics of LEFM 23

2.3.1 Fracture Mechanics 23

2.3.2 Infinite Tensile Plate with a Circular Hole 24

2.3.3 Infinite Tensile Plate with an Elliptical Hole 26

2.3.4 Westergaard Analysis of a Line Crack 28

2.3.5 Williams Solution of a Wedge Corner 29

2.4 Stress Intensity Factor, K 30

2.4.1 Definition of the Stress Intensity Factor 30

2.4.2 Examples of Stress Intensity Factors for LEFM 33

2.4.3 Griffith Energy Theories 35

2.4.4 Mixed Mode Crack Propagation 38

2.5 Classical Solution Procedures for K and G 41

2.5.1 Displacement Extrapolation/Correlation Method 41

2.5.2 Mode I Energy Release Rate 41

2.5.3 Mode I Stiffness Derivative/Virtual Crack Model 42

2.5.4 Two Virtual Crack Extensions for Mixed Mode Cases 42

2.5.5 Single Virtual Crack Extension Based on Displacement Decomposition 43

2.6 Quarter Point Singular Elements 44

2.7 J Integral 47

2.7.1 Generalization of J 48

2.7.2 Effect of Crack Surface Traction 48

2.7.3 Effect of Body Force 49

2.7.4 Equivalent Domain Integral (EDI) Method 49

2.7.5 Interaction Integral Method 49

2.8 Elastoplastic Fracture Mechanics (EPFM) 51

2.8.1 Plastic Zone 51

2.8.2 Crack-Tip Opening Displacements (CTOD) 53

2.8.3 J Integral for EPFM 55

3 Extended Finite Element Method 57

3.1 Introduction 57

3.2 Historic Development of XFEM 58

3.2.1 A Review of XFEM Development 58

3.2.2 A Review of XFEM Composite Analysis 62

3.3 Enriched Approximations 62

3.3.1 Partition of Unity 62

3.3.2 Intrinsic and Extrinsic Enrichments 63

3.3.3 Partition of Unity Finite Element Method 66

3.3.4 MLS Enrichment 66

3.3.5 Generalized Finite Element Method 67

3.3.6 Extended Finite Element Method 67

3.3.7 Generalized PU Enrichment 67

3.4 XFEM Formulation 67

3.4.1 Basic XFEM Approximation 68

3.4.2 Signed Distance Function 69

3.4.3 Modelling the Crack 70

3.4.4 Governing Equation 71

3.4.5 XFEM Discretization 72

3.4.6 Evaluation of Derivatives of Enrichment Functions 73

3.4.7 Selection of Nodes for Discontinuity Enrichment 75

3.4.8 Numerical Integration 77

3.5 XFEM Strong Discontinuity Enrichments 79

3.5.1 A Modified FE Shape Function 79

3.5.2 The Heaviside Function 81

3.5.3 The Sign Function 84

3.5.4 Strong Tangential Discontinuity 85

3.5.5 Crack Intersection 85

3.6 XFEM Weak Discontinuity Enrichments 86

3.7 XFEM Crack-Tip Enrichments 87

3.7.1 Isotropic Enrichment 87

3.7.2 Orthotropic Enrichment Functions 88

3.7.3 Bimaterial Enrichments 88

3.7.4 Orthotropic Bimaterial Enrichments 89

3.7.5 Dynamic Enrichment 89

3.7.6 Orthotropic Dynamic Enrichments for Moving Cracks 90

3.7.7 Bending Plates 91

3.7.8 Crack-Tip Enrichments in Shells 91

3.7.9 Electro-Mechanical Enrichment 92

3.7.10 Dislocation Enrichment 93

3.7.11 Hydraulic Fracture Enrichment 94

3.7.12 Plastic Enrichment 94

3.7.13 Viscoelastic Enrichment 95

3.7.14 Contact Corner Enrichment 96

3.7.15 Modification for Large Deformation Problems 97

3.7.16 Automatic Enrichment 99

3.8 Transition from Standard to Enriched Approximation 99

3.8.1 Linear Blending 100

3.8.2 Hierarchical Transition Domain 100

3.9 Tracking Moving Boundaries 103

3.9.1 Level Set Method 103

3.9.2 Alternative Methods 106

3.10 Numerical Simulations 107

3.10.1 A Central Crack in an Infinite Tensile Plate 107

3.10.2 An Edge Crack in a Finite Plate 109

3.10.3 Tensile Plate with a Central Inclined Crack 110

3.10.4 A Bending Plate in Fracture Mode III 111

3.10.5 Crack Propagation in a Shell 112

3.10.6 Shear Band Simulation 115

3.10.7 Fault Simulation 116

3.10.8 Sliding Contact Stress Singularity by PUFEM 119

3.10.9 Hydraulic Fracture 122

3.10.10 Dislocation Dynamics 126

4 Static Fracture Analysis of Composites 131

4.1 Introduction 131

4.2 Anisotropic Elasticity 134

4.2.1 Elasticity Solution 134

4.2.2 Anisotropic Stress Functions 136

4.3 Analytical Solutions for Near Crack Tip 137

4.3.1 The General Solution 137

4.3.2 Special Solutions for Different Types of Composites 140

4.4 Orthotropic Mixed Mode Fracture 142

4.4.1 Energy Release Rate for Anisotropic Materials 142

4.4.2 Anisotropic Singular Elements 142

4.4.3 SIF Calculation by Interaction Integral 143

4.4.4 Orthotropic Crack Propagation Criteria 147

4.5 Anisotropic XFEM 149

4.5.1 Governing Equation 149

4.5.2 XFEM Discretization 150

4.5.3 Orthotropic Enrichment Functions 151

4.6 Numerical Simulations 152

4.6.1 Plate with a Crack Parallel to the Material Axis of Orthotropy 152

4.6.2 Edge Crack with Several Orientations of the Axes of Orthotropy 155

4.6.3 Inclined Edge Notched Tensile Specimen 156

4.6.4 Central Slanted Crack 160

4.6.5 An Inclined Centre Crack in a Disk Subjected to Point Loads 164

4.6.6 Crack Propagation in an Orthotropic Beam 166

5 Dynamic Fracture Analysis of Composites 169

5.1 Introduction 169

5.1.1 Dynamic Fracture Mechanics 169

5.1.2 Dynamic Fracture Mechanics of Composites 170

5.1.3 Dynamic Fracture by XFEM 172

5.2 Analytical Solutions for Near Crack Tips in Dynamic States 173

5.2.1 Analytical Solution for a Propagating Crack in Isotropic Material 174

5.2.2 Asymptotic Solution for a Stationary Crack in Orthotropic Media 175

5.2.3 Analytical Solution for Near Crack Tip of a Propagating Crack in Orthotropic Material 176

5.3 Dynamic Stress Intensity Factors 178

5.3.1 Stationary and Moving Crack Dynamic Stress Intensity Factors 178

5.3.2 Dynamic Fracture Criteria 179

5.3.3 J Integral for Dynamic Problems 180

5.3.4 Domain Integral for Orthotropic Media 181

5.3.5 Interaction Integral 182

5.3.6 Crack-Axis Component of the Dynamic J Integral 183

5.3.7 Field Decomposition Technique 185

5.4 Dynamic XFEM 185

5.4.1 Dynamic Equations of Motion 185

5.4.2 XFEM Discretization 185

5.4.3 XFEM Enrichment Functions 187

5.4.4 Time Integration Schemes 191

5.5 Numerical Simulations 195

5.5.1 Plate with a Stationary Central Crack 195

5.5.2 Mode I Plate with an Edge Crack 196

5.5.3 Mixed Mode Edge Crack in Composite Plates 199

5.5.4 A Composite Plate with Double Edge Cracks under Impulsive Loading 210

5.5.5 Pre-Cracked Three Point Bending Beam under Impact Loading 213

5.5.6 Propagating Central Inclined Crack in a Circular Orthotropic Plate 217

6 Fracture Analysis of Functionally Graded Materials (FGMs) 225

6.1 Introduction 225

6.2 Analytical Solution for Near a Crack Tip 227

6.2.1 Average Material Properties 227

6.2.2 Mode I Near Tip Fields in FGM Composites 228

6.2.3 Stress and Displacement Field (Similar to Homogeneous Orthotropic Composites) 233

6.3 Stress Intensity Factor 235

6.3.1 J Integral 235

6.3.2 Interaction Integral 236

6.3.3 FGM Auxillary Fields 236

6.3.4 Isoparametric FGM 240

6.4 Crack Propagation in FGM Composites 240

6.5 Inhomogeneous XFEM 241

6.5.1 Governing Equation 241

6.5.2 XFEM Approximation 241

6.5.3 XFEM Discretization 243

6.6 Numerical Examples 244

6.6.1 Plate with a Centre Crack Parallel to the Material Gradient 244

6.6.2 Proportional FGM Plate with an Inclined Central Crack 247

6.6.3 Non-Proportional FGM Plate with a Fixed Inclined Central Crack 250

6.6.4 Rectangular Plate with an Inclined Crack (Non-Proportional Distribution) 251

6.6.5 Crack Propagation in a Four-Point FGM Beam 253

7 Delamination/Interlaminar Crack Analysis 261

7.1 Introduction 261

7.2 Fracture Mechanics for Bimaterial Interface Cracks 264

7.2.1 Isotropic Bimaterial Interfaces 265

7.2.2 Orthotropic Bimaterial Interface Cracks 266

7.2.3 Stress Contours for a Crack between Two Dissimilar Orthotropic Materials 270

7.3 Stress Intensity Factors for Interlaminar Cracks 271

7.4 Delamination Propagation 273

7.4.1 Fracture Energy-Based Criteria 273

7.4.2 Stress-Based Criteria 273

7.4.3 Contact-Based Criteria 274

7.5 Bimaterial XFEM 275

7.5.1 Governing Equation 275

7.5.2 XFEM Discretization 276

7.5.3 XFEM Enrichment Functions for Bimaterial Problems 278

7.5.4 Discretization and Integration 280

7.6 Numerical Examples 280

7.6.1 Central Crack in an Infinite Bimaterial Plate 280

7.6.2 Isotropic-Orthotropic Bimaterial Crack 289

7.6.3 Orthotropic Double Cantilever Beam 291

7.6.4 Concrete Beams Strengthened with Fully Bonded GFRP 294

7.6.5 FRP Reinforced Concrete Cantilever Beam Subjected to Edge Loadings 295

7.6.6 Delamination of Metallic I Beams Strengthened by FRP Strips 298

7.6.7 Variable Section Beam Reinforced by FRP 300

8 New Orthotropic Frontiers 303

8.1 Introduction 303

8.2 Orthotropic XIGA 303

8.2.1 NURBS Basis Function 304

8.2.2 Extended Isogeometric Analysis 305

8.2.3 XIGA Simulations 313

8.3 Orthotropic Dislocation Dynamics 321

8.3.1 Straight Dislocations in Anisotropic Materials 321

8.3.2 Edge Dislocations in Anisotropic Materials 322

8.3.3 Curve Dislocations in Anisotropic Materials 324

8.3.4 Anisotropic Dislocation XFEM 324

8.3.5 Plane Strain Anisotropic Solution 329

8.3.6 Individual Sliding Systems s1 and s2 in an Infinite Domain 330

8.3.7 Simultaneous Sliding Systems in an Infinite Domain 330

8.4 Other Anisotropic Applications 333

8.4.1 Biomechanics 333

8.4.2 Piezoelectric 335

References 339

Index 363

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Soheil Mohammadi, Associate Professor, School of Civil Engineering, University of Tehran, Tehran, IRAN
Soheil Mohammdi studied for his PhD at the University of Wales Swansea and is now a lecturer at the University of Tehran where his academic career began. He teaches PhD courses in contact mechanics, mesh generation and adaptivity, meshless methods, and impact and explosive loadings on structures. He research interests are based in computational mechanics and finite element analysis, and XFEM. He has published many papers in these areas as well as a book on discontinuum mechanics in 2003.
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