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Micromechanics of Fracture and Damage

Micromechanics of Fracture and Damage

Luc Dormieux, Djimedo Kondo

ISBN: 978-1-119-29218-0

Mar 2016

332 pages

$116.99

Description

This book deals with the mechanics and physics of fractures at various scales.  Based on advanced continuum mechanics of heterogeneous media, it develops a rigorous mathematical framework for single macrocrack problems as well as for the effective properties of microcracked materials.  In both cases, two geometrical models of cracks are examined and discussed: the idealized representation of the crack as two parallel faces (the Griffith crack model), and the representation of a crack as a flat elliptic or ellipsoidal cavity (the Eshelby inhomogeneity problem).

The book is composed of two parts:

- The first part deals with solutions to 2D and 3D problems involving a single crack in linear elasticity. Elementary solutions of cracks problems in the different modes are fully worked. Various mathematical techniques are presented, including Neuber-Papkovitch displacement potentials, complex analysis with conformal mapping and Eshelby-based solutions.

- The second part is devoted to continuum micromechanics approaches of microcracked materials in relation to methods and results presented in the first part. Various estimates and bounds of the effective elastic properties are presented. They are considered for the formulation and application of continuum micromechanics-based damage models.

Notations  xiii

Preface xv

Part 1. Elastic Solutions to Single Crack Problems  1

Chapter 1. Fundamentals of Plane Elasticity 3

1.1. Complex representation of Airy’s biharmonic stress function 3

1.2. Force acting on a curve or an element of arc 7

1.3. Derivation of stresses  9

1.4. Derivation of displacements 11

1.5. General form of the potentials φ and ψ 12

1.6. Examples 15

1.6.1. Circular cavity under pressure 15

1.6.2. Circular cavity in a plane subjected to uniaxial traction at infinity 16

1.7. Conformal mapping 18

1.7.1. Application of conformal mapping to plane elasticity problems 18

1.7.2. The domain Σ is the unit disc |ζ| ≤ 1 20

1.7.3. The domain Σ is the complement Σ− of the unit disc 23

1.8. The anisotropic case 26

1.8.1. General features  26

1.8.2. Stresses, displacements and boundary conditions 28

1.9. Appendix: mathematical tools 29

1.9.1. Theorem 1  30

1.9.2. Theorem 2  31

1.9.3. Theorem 3  31

Chapter 2. Fundamentals of Elasticity in View of Homogenization Theory  33

2.1. Green's function concept  33

2.2. Green’s function in two-dimensional conditions 34

2.2.1. The general anisotropic case 34

2.2.2. The isotropic case 35

2.3. Green’s function in three-dimensional conditions 38

2.3.1. The general anisotropic case 38

2.3.2. The isotropic case 39

2.4. Eshelby’s problems in linear microelasticity 41

2.4.1. The (elastic) inclusion problem 41

2.4.2. The Green operator of the infinite space 44

2.4.3. The Green operator of a finite domain 48

2.4.4. The inhomogeneity problem 50

2.4.5. The inhomogeneity problem with stress boundary conditions 51

2.4.6. The infinite heterogeneous elastic medium 52

2.5. Hill tensor for the elliptic inclusion 54

2.5.1. Properties of the logarithmic potential 54

2.5.2. Integration of the r,ir,l term 57

2.5.3. Components of the Hill tensor 59

2.6. Hill’s tensor for the spheroidal inclusion 60

2.6.1. Components of the Hill tensor 63

2.6.2. Series expansions of the components of the Hill tensor for flat spheroids 64

2.7. Appendix 65

2.8. Appendix: derivation of the χij 67

Chapter 3. Two-dimensional Griffith Crack 71

3.1. Stress singularity at crack tip 72

3.1.1. Stress singularity in plane elasticity: modes I and II 73

3.1.2. Stress singularity in antiplane problems in elasticity: mode III 78

3.2. Solution to mode I problem 80

3.2.1. Solution of PI 82

3.2.2. Solution of PI 90

3.2.3. Displacement jump across the crack surfaces 91

3.3. Solution to mode II problem 92

3.3.1. Solution of PII 93

3.3.2. Solution of PII 96

3.3.3. Displacement jump across the crack surfaces 97

3.4. Appendix: Abel’s integral equation 98

3.5. Appendix: Neuber–Papkovitch displacement potentials 101

Chapter 4. The Elliptic Crack Model in Plane Strains 103

4.1. The infinite plane with elliptic hole 103

4.1.3. Elliptic cavity in a plane subjected to a remote stress state at infinity 107

4.1.4. Stress intensity factors 108

4.1.5. Some remarks on unilateral contact 111

4.2. Infinite plane with elliptic hole: the anisotropic case 112

4.2.1. General properties 112

4.2.2. Complex potentials for an elliptic cavity in the presence of traction at infinity 115

4.2.3. Complex potentials for an elliptic cavity in the case of shear at infinity 116

4.2.5. Displacement discontinuities 121

4.2.6. Closed cracks 123

4.3. Eshelby approach 130

4.3.1. Mode I 130

4.3.2. Mode II 133

Chapter 5. Griffith Crack in 3D 137

5.1. Griffith circular (penny-shaped) crack in mode I 138

5.1.1. Solution of PI 139

5.1.2. Solution of PI 143

5.2. Griffith circular (penny-shaped) crack under shear loading 144

5.2.1. Solution of PII 146

5.2.2. Solution of PII 151

Chapter 6. Ellipsoidal Crack Model: the Eshelby Approach 155

6.1. Mode I 156

6.2. Mode II 159

Chapter 7. Energy Release Rate and Conditions for Crack Propagation 163

7.1. Driving force of crack propagation 163

7.2. Stress intensity factor and energy release rate 167

Part 2. Homogenization of Microcracked Materials 173

Chapter 8. Fundamentals of Continuum Micromechanics 175

8.1. Scale separation 175

8.2. Inhomogeneity model for cracks 177

8.2.1. Uniform strain boundary conditions 177

8.2.2. Uniform stress boundary conditions 181

8.2.3. Linear elasticity with uniform strain boundary conditions 182

8.2.4. Linear elasticity with uniform stress boundary conditions 185

8.3. General results on homogenization with Griffith cracks 187

8.3.1. Hill’s lemma with Griffith cracks 187

8.3.2. Uniform strain boundary conditions 188

8.3.3. Uniform stress boundary conditions 190

8.3.4. Derivation of effective properties in linear elasticity: principle of the approach 190

8.3.5. Appendix 194

Chapter 9. Homogenization of Materials Containing Griffith Cracks 197

9.1. Dilute estimates in isotropic conditions 197

9.1.1. Stress-based dilute estimate of stiffness  199

9.1.2. Stress-based dilute estimate of stiffness with closed cracks 202

9.1.3. Strain-based dilute estimate of stiffness with opened cracks 204

9.1.4. Strain-based dilute estimate of stiffness with closed cracks 205

9.2. A refined strain-based scheme 206

9.3. Homogenization in plane strain conditions for anisotropic materials 208

9.3.1. Opened cracks 208

9.3.2. Closed cracks 211

Chapter 10. Eshelby-based Estimates of Strain Concentration and Stiffness  213

10.1. Dilute estimate of the strain concentration tensor: general features 213

10.1.1. The general case 213

10.2. The particular case of opened cracks 215

10.2.1. Spheroidal crack 215

10.2.2. Elliptic crack 216

10.2.3. Crack opening change 218

10.3. Dilute estimates of the effective stiffness for opened cracks 220

10.3.1. Opened parallel cracks 222

10.3.2. Opened randomly oriented cracks 224

10.4. Dilute estimates of the effective stiffness for closed cracks 226

10.4.1. Closed parallel cracks 228

10.4.2. Closed randomly oriented cracks 228

10.5. Mori–Tanaka estimate of the effective stiffness 229

10.5.1. Opened cracks 231

10.5.2. Closed cracks 233

Chapter 11. Stress-based Estimates of Stress Concentration and Compliance 235

11.1. Dilute estimate of the stress concentration tensor 235

11.2. Dilute estimates of the effective compliance for opened cracks 236

11.2.1. Opened parallel cracks 237

11.2.2. Opened randomly oriented cracks 239

11.2.3. Discussion 239

11.3. Dilute estimate of the effective compliance for closed cracks 240

11.3.1. 3D case 241

11.3.2. 2D case 242

11.3.3. Stress concentration tensor 243

11.3.4. Comparison with other estimates 244

11.4. Mori–Tanaka estimates of effective compliance 244

11.4.1. Opened cracks 246

11.4.2. Closed cracks 246

11.5. Appendix: algebra for transverse isotropy and applications 246

Chapter 12. Bounds 251

12.1. The energy definition of the homogenized stiffness 252

12.2. Hashin–Shtrikman’s bound 255

12.2.1. Hashin–Shtrikman variational principle 255

12.2.2. Piecewise constant polarization field 259

12.2.3. Random microstructures 261

12.2.4. Application of the Ponte-Castaneda and Willis (PCW) bound to microcracked media 270

Chapter 13. Micromechanics-based Damage Constitutive Law and Application 273

13.1. Formulation of damage constitutive law 273

13.1.1. Description of damage level by a single scalar variable 274

13.1.2. Extension to multiple cracks 276

13.2. Some remarks concerning the loss of uniqueness of the mechanical response in relation to damage 277

13.3. Mechanical fields and damage in a hollow sphere subjected to traction 280

13.3.1. General features 280

13.3.2. Case of damage model based on the dilute estimate 284

13.3.3. Complete solution in the case of the damage model based on PCW estimate  285

13.4. Stability of the solution to damage evolution in a hollow sphere 296

13.4.1. The MT damage model 298

13.4.2. The general damage model [13.44] 300

Bibliography 305

Index 309