Fundamentals of Fracture Mechanics / Edition 1

by Tribikram Kundu

Overview

Almost all books available on fracture mechanics cover the majority of topics presented in this book, and often much, much more. While great as references, this makes teaching from them more difficult because the materials are not typically presented in the order that most professors cover them in their lectures and more than half the information presented is not covered in an introductory course at all.

Focusing on the needs of students and professors, Fundamentals of Fracture Mechanics offers an introduction to the discipline through careful editing and mindfulness toward the audience. The book begins with a review of the fundamentals of continuum mechanics and the theory of elasticity relevant to fracture mechanics. The following material has been carefully selected, only including topics important enough to be covered in a first course on fracture mechanics. Except for the last chapter, no advanced topics are covered. Therefore, instructors of elementary fracture mechanics courses can easily cover the entire book in a three-unit graduate-level course without having to spend too much time picking and choosing appropriate topics for the course from the vast knowledge presented in most fracture mechanic books available today.

Drawing on over 20 years of teaching, the author supplies practical and useful resources, including practice exercises designed to facilitate enjoyable learning and reference for further study. His clear, concise coverage of essential information makes the book ideal not only for an introductory course but also for self-study.

Product Details

ISBN-13:	9780367387778
Publisher:	Taylor & Francis
Publication date:	09/19/2019
Pages:	308
Product dimensions:	6.12(w) x 9.19(h) x (d)

Table of Contents

1 Fundamentals of the Theory of Elasticity 1

1.1 Introduction 1

1.2 Fundamentals of Continuum Mechanics and the Theory of Elasticity 1

1.2.1 Deformation and Strain Tensor 1

1.2.1.1 Interpretation of ε_ijand ω_ij for Small Displacement Gradient 3

1.2.2 Traction and Stress Tensor 6

1.2.3 Traction-Stress Relation 8

1.2.4 Equilibrium Equations 9

1.2.4.1 Force Equilibrium 9

1.2.4.2 Moment Equilibrium 11

1.2.5 Stress Transformation 12

1.2.5.1 Kronecker Delta Symbol (δ_ij) and Permutation Symbol (ε_ijk 14

1.2.5.2 Examples of the Application of δ_ij and ε_ijk 14

1.2.6 Definition of Tensor 15

1.2.7 Principal Stresses and Principal Planes 15

1.2.8 Transformation of Displacement and Other Vectors 19

1.2.9 Strain Transformation 20

1.2.10 Definition of Elastic Material and Stress-Strain Relation 20

1.2.11 Number of Independent Material Constants 24

1.2.12 Material Planes of Symmetry 25

1.2.12.1 One Plane of Symmetry 25

1.2.12.2 Two and Three Planes of Symmetry 26

1.2.12.3 Three Planes of Symmetry and One Axis of Symmetry 27

1.2.12.4 Three Planes of Symmetry and Two or Three Axes of Symmetry 28

1.2.13 Stress-Strain Relation for Isotropic Materials-Green's Approach 30

1.2.13.1 Hooke's Law in Terms of Young's Modulus and Poisson's Ratio 32

1.2.14 Navier's Equation of Equilibrium 33

1.2.15 Fundamental Equations of Elasticity in Other Coordinate Systems 36

1.2.16 Time-Dependent Problems or Dynamic Problems 36

1.3 Some Classical Problems in Elasticity 36

1.3.1 In-Plane and Out-of-Plane Problems 38

1.3.2 Plane Stress and Plane Strain Problems 39

1.3.2.1 Compatibility Equations for Plane Stress Problems 41

1.3.2.2 Compatibility Equations for Plane Strain Problems 42

1.3.3 Airy Stress Function 42

1.3.4 Some Classical Elasticity Problems in Two Dimensions 45

1.3.4.1 Plate and Beam Problems 45

1.3.4.2 Half-Plane Problems 51

1.3.4.3 Circular Hole, Disk, and Cylindrical Pressure Vessel Problems 59

1.3.5 Thick Wall Spherical Pressure Vessel 72

1.4 Concluding Remarks 75

References 75

Exercise Problems 75

2 Elastic Crack Model 85

2.1 Introduction 85

2.2 Williams' Method to Compute the Stress Field near a Crack Tip 85

2.2.1 Satisfaction of Boundary Conditions 88

2.2.2 Acceptable Values of n and λ 90

2.2.3 Dominant Term 92

2.2.4 Strain and Displacement Fields 96

2.2.4.1 Plane Stress Problems 96

2.2.4.2 Plane Strain Problems 98

2.3 Stress Intensity Factor and Fracture Toughness 100

2.4 Stress and Displacement Fields for Antiplane Problems 101

2.5 Different Modes of Fracture 102

2.6 Direction of Crack Propagation 102

2.7 Mixed Mode Failure Curve for In-Plane Loading 105

2.8 Stress Singularities for Other Wedge Problems 107

2.9 Concluding Remarks 107

References 108

Exercise Problems 108

3 Energy Balance 113

3.1 Introduction 113

3.2 Griffith's Energy Balance 113

3.3 Energy Criterion of Crack Propagation for Fixed Force and Fixed Grip Conditions 115

3.3.1 Soft Spring Case 118

3.3.2 Hard Spring Case 119

3.3.3 General Case 120

3.4 Experimental Determination of G_c 120

3.4.1 Fixed Force Experiment 122

3.4.2 Fixed Grip Experiment 122

3.4.3 Determination of G_c from One Specimen 123

3.5 Relation between Strain Energy Release Rate (G) and Stress Intensity Factor (K) 123

3.6 Determination of Stress Intensity Factor (K) for Different Problem Geometries 126

3.6.1 Griffith Crack 126

3.6.2 Circular or Penny-Shaped Crack 129

3.6.3 Semi-infinite Crack in a Strip 130

3.6.4 Stack of Parallel Cracks in an Infinite Plate 131

3.6.5 Star-Shaped Cracks 133

3.6.6 Pressurized Star Cracks 135

3.6.7 Longitudinal Cracks in Cylindrical Rods 138

3.7 Concluding Remarks 141

References 142

Exercise Problems 143

4 Effect of Plasticity 147

4.1 Introduction 147

4.2 First Approximation on the Plastic Zone Size Estimation 147

4.2.1 Evaluation of r_p 148

4.2.2 Evaluation of αr_p 149

4.3 Determination of the Plastic Zone Shape in Front of the Crack Tip 150

4.4 Plasticity Correction Factor 155

4.5 Failure Modes under Plane Stress and Plane Strain Conditions 157

4.5.1 Plane Stress Case 157

4.5.2 Plane Strain Case 158

4.6 Dugdale Model 159

4.7 Crack Tip Opening Displacement 161

4.8 Experimental Determination of K_c 164

4.8.1 Compact Tension Specimen 164

4.8.1.1 Step 1: Crack Formation 165

4.8.1.2 Step 2: Loading the Specimen 166

4.8.1.3 Step 3: Checking Crack Geometry in the Failed Specimen 166

4.8.1.4 Step 4: Computation of Stress Intensity Factor at Failure 167

4.8.1.5 Step 5: Final Check 168

4.8.2 Three-Point Bend Specimen 168

4.8.3 Practical Examples 170

4.8.3.1 7075 Aluminum 170

4.8.3.2 A533B Reactor Steel 170

4.9 Concluding Remarks 171

References 172

Exercise Problems 172

5 J-Integral 175

5.1 Introduction 175

5.2 Derivation of J-Integral 175

5.3 J-Integral over a Closed Loop 178

5.4 Path Independence of J-Integral 180

5.5 J-Integral for Dugdale Model 182

5.6 Experimental Evaluation of Critical J-Integral Value, J_c 183

5.7 Concluding Remarks 187

References 188

Exercise Problems 188

6 Fatigue Crack Growth 189

6.1 Introduction 189

6.2 Fatigue Analysis-Mechanics of Materials Approach 189

6.3 Fatigue Analysis-Fracture Mechanics Approach 189

6.3.1 Numerical Example 193

6.4 Fatigue Analysis for Materials Containing Microcracks 193

6.5 Concluding Remarks 195

References 195

Exercise Problems 195

7 Stress Intensity Factors for Some Practical Crack Geometries 197

7.1 Introduction 197

7.2 Slit Crack in a Strip 197

7.3 Crack Intersecting a Free Surface 199

7.4 Strip with a Crack on Its One Boundary 200

7.5 Strip with Two Collinear Identical Cracks on Its Two Boundaries 201

7.6 Two Half Planes Connected over a Finite Region Forming Two Semi-infinite Cracks in a Full Space 202

7.7 Two Cracks Radiating Out from a Circular Hole 203

7.8 Two Collinear Finite Cracks in an Infinite Plate 204

7.9 Cracks with Two Opposing Concentrated Forces on the Surface 206

7.10 Pressurized Crack 206

7.11 Crack in a Wide Strip with a Concentrated Force at Its Midpoint and a Far Field Stress Balancing the Concentrated Force 207

7.12 Circular or Penny-Shaped Crack in a Full Space 209

7.13 Elliptical Crack in a Full Space 212

7.13.1 Special Case 1-Circular Crack 213

7.13.2 Special Case 2-Elliptical Crack with Very Large Major Axis 214

7.13.3 SIF at the End of Major and Minor Axes of Elliptical Cracks 214

7.14 Part-through Surface Crack 214

7.14.1 First Approximation 215

7.14.2 Front Face Correction Factor 215

7.14.3 Plasticity Correction 215

7.14.4 Back Face Correction Factor 216

7.15 Corner Cracks 216

7.15.1 Corner Cracks with Almost Equal Dimensions 217

7.15.2 Corner Cracks at Two Edges of a Circular Hole 218

7.15.3 Corner Crack at One Edge of a Circular Hole 218

7.16 Concluding Remarks 219

References 219

Exercise Problems 220

8 Numerical Analysis 221

8.1 Introduction 221

8.2 Boundary Collocation Technique 221

8.2.1 Circular Plate with a Radial Crack 223

8.2.2 Rectangular Cracked Plate 223

8.3 Conventional Finite Element Methods 224

8.3.1 Stress and Displacement Matching 224

8.3.2 Local Strain Energy Matching 228

8.3.3 Strain Energy Release Rate 229

8.3.4 J-Integral Method 232

8.4 Special Crack Tip Finite Elements 233

8.5 Quarter Point Quadrilateral Finite Element 236

8.6 Concluding Remarks 239

References 239

9 Westergaard Stress Function 241

9.1 Introduction 241

9.2 Background Knowledge 241

9.3 Griffith Crack in Biaxial State of Stress 242

9.3.1 Stress and Displacement Fields in Terms of Westergaard Stress Function 243

9.3.2 Westergaard Stress Function for the Griffith Crack under Biaxial Stress Field 244

9.3.3 Stress Field Close to a Crack Tip 250

9.4 Concentrated Load on a Half Space 252

9.5 Griffith Crack Subjected to Concentrated Crack Opening Loads P 255

9.5.1 Stress Intensity Factor 256

9.6 Griffith Crack Subjected to Nonuniform Internal Pressure 257

9.7 Infinite Number of Equal Length, Equally Spaced Coplanar Cracks 258

9.8 Concluding Remarks 259

References 259

Exercise Problems 260

10 Advanced Topics 261

10.1 Introduction 261

10.2 Stress Singularities at Crack Corners 261

10.3 Fracture Toughness and Strength of Brittle Matrix Composites 263

10.3.1 Experimental Observation of Strength Variations of PRBMCs with Various Fiber Parameters 265

10.3.2 Analysis for Predicting Strength Variations of FRBMCs with Various Fiber Parameters 267

10.3.2.1 Effect of Fiber Volume Fraction 268

10.3.2.2 Effect of Fiber Length 271

10.3.2.3 Effect of Fiber Diameter 274

10.3.3 Effect on Stiffness 276

10.3.4 Experimental Observation of Fracture Toughness Increase in FRBMCs with Fiber Addition 276

10.4 Dynamic Effect 277

10.5 Concluding Remarks 278

References 278

Exercise Problems 280

Index 283