Fracture Mechanics (eBook)

Alan Zehnder joined the faculty at Cornell University in 1988. Since then he has served in a number of leadership roles including Chair of the Department of Theoretical and Applied Mechanics, and Director of the Sibley School of Mechanical and Aerospace Engineering.  He teaches applied mechanics and his research topics focus on fracture, experimental mechanics and nonlinear dynamics of nanomechanical oscillators.  He was awarded the 1988 Rudolf Kingslake Medal and Prize for his Optical Engineering paper on optical methods in dynamic-fracture experimentation.

Fracture Mechanics 4
Preface 6
References 6
Contents 8
Acronyms 12
Chapter 1: Introduction 16
1.1 Notable Fractures 16
1.2 Basic Fracture Mechanics Concepts 18
1.2.1 Small Scale Yielding Model 19
1.2.2 Fracture Criteria 19
1.3 Fracture Unit Conversions 20
1.4 Exercises 20
References 21
Chapter 2: Linear Elastic Stress Analysis of 2D Cracks 22
2.1 Notation 22
2.2 Introduction 22
2.3 Modes of Fracture 23
2.4 Mode III Field 23
2.4.1 Asymptotic Mode III Field 24
2.4.2 Full Field for Finite Crack in an In?nite Body 28
2.4.2.1 Complex Variables Formulation of Anti-Plane Shear 28
2.4.2.2 Solution to the Problem 29
2.5 Mode I and Mode II Fields 31
2.5.1 Review of Plane Stress and Plane Strain Field Equations 31
2.5.1.1 Plane Strain 31
2.5.1.2 Plane Stress 32
2.5.1.3 Stress Function 32
2.5.2 Asymptotic Mode I Field 32
2.5.2.1 Stress Field 32
2.5.2.2 Displacement Field 34
2.5.3 Asymptotic Mode II Field 36
2.6 Complex Variables Method for Mode I and Mode II Cracks 36
2.6.1 Westergaard Approach for Mode-I 37
2.6.2 Westergaard Approach for Mode-II 37
2.6.3 General Solution for Internal Crack with Applied Tractions 37
2.6.4 Full Stress Field for Mode-I Crack in an In?nite Plate 38
2.6.5 Stress Intensity Factor Under Remote Shear Loading 40
2.6.6 Stress Intensity Factors for Cracks Loaded with Tractions 41
2.6.7 Asymptotic Mode I Field Derived from Full Field Solution 41
2.6.8 Asymptotic Mode II Field Derived from Full Field Solution 43
2.6.9 Stress Intensity Factors for Semi-in?nite Crack 43
2.7 Some Comments 43
2.7.1 Three-Dimensional Cracks 44
2.8 Exercises 46
References 47
Chapter 3: Energy Flows in Elastic Fracture 48
3.1 Generalized Force and Displacement 48
3.1.1 Prescribed Loads 48
3.1.2 Prescribed Displacements 49
3.2 Elastic Strain Energy 50
3.3 Energy Release Rate, G 51
3.3.1 Prescribed Displacement 51
3.3.2 Prescribed Loads 52
3.3.3 General Loading 53
3.4 Interpretation of G from Load-Displacement Records 53
3.4.1 Multiple Specimen Method for Nonlinear Materials 53
3.4.2 Compliance Method for Linearly Elastic Materials 56
3.4.3 Applications of the Compliance Method 57
3.4.3.1 Determination of G in DCB Sample 57
3.4.3.2 Use of Compliance to Determine Crack Length 58
3.5 Crack Closure Integral for G 58
3.6 G in Terms of KI, KII, KIII for 2D Cracks That Grow Straight Ahead 62
3.6.1 Mode-III Loading 62
3.6.2 Mode I Loading 63
3.6.3 Mode II Loading 63
3.6.4 General Loading (2D Crack) 63
3.7 Contour Integral for G (J-Integral) 64
3.7.1 Two Dimensional Problems 64
3.7.2 Three-Dimensional Problems 66
3.7.3 Example Application of J-Integral 66
3.8 Exercises 67
References 69
Chapter 4: Criteria for Elastic Fracture 70
4.1 Introduction 70
4.2 Initiation Under Mode-I Loading 70
4.3 Crack Growth Stability and Resistance Curve 73
4.3.1 Loading by Compliant System 75
4.3.2 Resistance Curve 76
4.4 Mixed-Mode Fracture Initiation and Growth 78
4.4.1 Maximum Hoop Stress Theory 78
4.4.2 Maximum Energy Release Rate Criterion 80
4.4.3 Crack Path Stability Under Pure Mode-I Loading 81
4.4.4 Second Order Theory for Crack Kinking and Turning 84
4.5 Criteria for Fracture in Anisotropic Materials 85
4.6 Crack Growth Under Fatigue Loading 86
4.7 Stress Corrosion Cracking 89
4.8 Exercises 89
References 91
Chapter 5: Determining K and G 92
5.1 Analytical Methods 92
5.1.1 Elasticity Theory 92
5.1.1.1 Finite Crack in an In?nite Body 92
5.1.1.2 Semi-in?nite Crack in an In?nite Body 93
5.1.1.3 Array of Cracks Under Remote Loading 93
5.1.2 Energy and Compliance Methods 94
5.1.2.1 4-Point Bending Debond Specimen: Energy Method 94
5.2 Stress Intensity Handbooks and Software 95
5.3 Boundary Collocation 95
5.4 Computational Methods: A Primer 99
5.4.1 Stress and Displacement Correlation 99
5.4.1.1 Stress Correlation 99
5.4.1.2 Displacement Correlation 100
5.4.2 Global Energy and Compliance 100
5.4.3 Crack Closure Integrals 101
5.4.3.1 Nodal Release 101
5.4.3.2 Modi?ed Crack Closure Integral 102
5.4.4 Domain Integral 104
5.4.5 Crack Tip Singular Elements 105
5.4.6 Example Calculations 109
5.4.6.1 Displacement Correlation and Domain Integral with 1/4 Point Elements 110
5.4.6.2 Global Energy 110
5.4.6.3 Modi?ed Crack Closure Integral 111
5.5 Experimental Methods 112
5.5.1 Strain Gauge Method 113
5.5.2 Photoelasticity 115
5.5.3 Digital Image Correlation 116
5.5.4 Thermoelastic Method 118
5.6 Exercises 120
References 121
Chapter 6: Fracture Toughness Tests 123
6.1 Introduction 123
6.2 ASTM Standard Fracture Test 124
6.2.1 Test Samples 124
6.2.2 Equipment 126
6.2.3 Test Procedure and Data Reduction 126
6.3 Interlaminar Fracture Toughness Tests 127
6.3.1 The Double Cantilever Beam Test 127
6.3.1.1 Geometry and Test Procedure 127
6.3.1.2 Data Reduction Methods 128
6.3.1.3 Example Results 130
6.3.2 The End Notch Flexure Test 131
6.3.3 Single Leg Bending Test 132
6.4 Indentation Method 134
6.5 Chevron-Notch Method 136
6.5.1 KIVM Measurement 137
6.5.2 KIV Measurement 138
6.5.3 Work of Fracture Approach 139
6.6 Wedge Splitting Method 141
6.7 K-R Curve Determination 144
6.7.1 Specimens 144
6.7.2 Equipment 145
6.7.2.1 Optical Measurement of Crack Length 145
6.7.2.2 Compliance Method for Crack Length 145
6.7.2.3 Other Methods for Crack Length 145
6.7.3 Test Procedure and Data Reduction 147
6.7.3.1 By Measurement of Load and Crack Length 147
6.7.3.2 By Measurement of Load and Compliance 147
6.7.3.3 Indirect Approach Using Monotonic Load-Displacement Data 148
6.7.4 Sample K-R curve 148
6.8 Exercises 148
References 149
Chapter 7: Elastic Plastic Fracture: Crack Tip Fields 151
7.1 Introduction 151
7.2 Strip Yield (Dugdale) Model 151
7.2.1 Effective Crack Length Model 157
7.3 A Model for Small Scale Yielding 158
7.4 Introduction to Plasticity Theory 160
7.5 Anti-plane Shear Cracks in Elastic-Plastic Materials in SSY 164
7.5.1 Stationary Crack in Elastic-Perfectly Plastic Material 164
7.5.2 Stationary Crack in Power-Law Hardening Material 168
7.5.3 Steady State Growth in Elastic-Perfectly Plastic Material 170
7.5.4 Transient Crack Growth in Elastic-Perfectly Plastic Material 174
7.6 Mode-I Crack in Elastic-Plastic Materials 176
7.6.1 Stationary Crack in a Power Law Hardening Material 176
7.6.1.1 Deformation Theory (HRR Field) 176
7.6.1.2 Incremental Theory 179
7.6.2 Slip Line Solutions for Rigid Plastic Material 179
7.6.2.1 Introduction to Plane Strain Slip Line Theory 179
7.6.2.2 Plane-Strain, Semi-in?nite Crack 181
7.6.2.3 Plane-Stress, Semi-in?nite Crack 183
7.6.3 Large Scale Yielding (LSY) Example 183
7.6.4 SSY Plastic Zone Size and Shape 184
7.6.5 CTOD-J Relationship 186
7.6.6 Growing Mode-I Crack 187
7.6.7 Three Dimensional Aspects 191
7.6.8 Effect of Finite Crack Tip Deformation on Stress Field 193
7.7 Exercises 195
References 196
Chapter 8: Elastic Plastic Fracture: Energy and Applications 198
8.1 Energy Flows 198
8.1.1 When Does G=J? 198
8.1.2 General Treatment of Crack Tip Contour Integrals 199
8.1.3 Crack Tip Energy Flux Integral 201
8.1.3.1 Global Path Independence for Steady State Crack Growth 201
8.1.3.2 Energy Flux as Gamma-> 0
8.1.3.3 Energy Flux for Gamma Outside Plastic Zone 202
8.1.3.4 Thermal Field Visualization of Energy Flow 204
8.2 Fracture Toughness Testing for Elastic-Plastic Materials 206
8.2.1 Samples and Equipment 206
8.2.2 Procedure and Data Reduction 207
8.2.2.1 Test Procedure 207
8.2.2.2 Data Reduction 208
8.2.2.3 Validation of Results 209
8.2.3 Examples of J-R Data 210
8.3 Calculating J and Other Ductile Fracture Parameters 210
8.3.1 Computational Methods 211
8.3.2 J Result Used in ASTM Standard JIC Test 213
8.3.2.1 Rigid Plastic Material 215
8.3.2.2 Elastic Material 215
8.3.2.3 Elastic-Plastic Material 215
8.3.3 Engineering Approach to Elastic-Plastic Fracture Analysis 215
8.3.3.1 Sample Calculation 217
8.4 Fracture Criteria and Prediction 218
8.4.1 J Controlled Crack Growth and Stability 218
8.4.2 J-Q Theory 220
8.4.3 Crack Tip Opening Displacement, Crack Tip Opening Angle 223
8.4.4 Cohesive Zone Model 226
8.4.4.1 Cohesive Zone Embedded in Elastic Material 228
8.4.4.2 Cohesive Zone Embedded in Elastic-Plastic Material 229
References 231
Index 233