Mechanics of Dislocation Fields

Accompanying the present trend of engineering systems aimed at size reduction and design at microscopic/nanoscopic length scales, Mechanics of Dislocation Fields describes the self-organization of dislocation ensembles at small length scales and its consequences on the overall mechanical behavior of crystalline bodies.

The account of the fundamental interactions between the dislocations and other microscopic crystal defects is based on the use of smooth field quantities and powerful tools from the mathematical theory of partial differential equations. The resulting theory is able to describe the emergence of dislocation microstructures and their evolution along complex loading paths. Scale transitions are performed between the properties of the dislocation ensembles and the mechanical behavior of the body.

Several variants of this overall scheme are examined which focus on dislocation cores, electromechanical interactions of dislocations with electric charges in dielectric materials, the intermittency and scale-invariance of dislocation activity, grain-to-grain interactions in polycrystals, size effects on mechanical behavior and path dependence of strain hardening.

Acknowledgements ix

Introduction xi

Chapter 1 Continuous Dislocation Modeling 1

1.1 Introduction 1

1.2 Lattice incompatibility 2

1.3 Burgers vector 5

1.4 Compatibility conditions 8

1.5 Dislocation fields 10

1.6 Tangential continuity at interfaces 13

1.7 Curvatures and rotational incompatibiliy 19

1.8 Incompatibility tensor 22

1.9 Conclusion 23

1.10 Problems 23

1.10.1 Discrete versus continuous modeling of crystal defects 23

1.10.2 Incompatibility in simple shear 25

1.10.3 Frank’s relation 26

1.11 Solutions 27

1.11.1 Discrete versus continuous modeling of crystal defects 27

1.11.2 Incompatibility in simple shear 28

1.11.3 Frank’s relation 29

Chapter 2 Elasto-static Field Equations 31

2.1 Introduction 31

2.2 Elasto-static solution to field equations 31

2.2.1 Stokes-Helmholtz decomposition and Poisson-type equations 32

2.2.2 Navier-type equations for compatible elastic distortion fields 34

2.3 Straight screw dislocation in a linear isotropic elastic medium 35

2.4 Straight edge dislocation in a linear isotropic elastic medium 37

2.5 Conclusion 38

2.6 Problems 39

2.6.1 Screw dislocation 39

2.6.2 Twist boundary 39

2.6.3 Tilt boundary 41

2.6.4 Zero-stress everywhere dislocation fields 41

2.7 Solutions 42

2.7.1 Screw dislocation 42

2.7.2 Twist boundary 43

2.7.3 Tilt boundary 45

2.7.4 Zero-stress everywhere dislocation fields 46

Chapter 3 Dislocation Transport 49

3.1 Introduction 49

3.2 Dislocation flux and plastic distortion rate 50

3.3 Coarse graining 52

3.4 Compatibility versus incompatibility of plasticity 54

3.5 Tangential continuity of plastic distortion rate 57

3.6 Transport equations 60

3.6.1 Small transformations 60

3.6.2 Finite transformations 62

3.7 Transport waves 64

3.7.1 Annihilation 66

3.7.2 Expansion of dislocation loops 68

3.7.3 Initiation of a Frank–Read source 69

3.8 Numerical algorithms for dislocation transport 71

3.9 Conclusion 76

3.10 Problems 76

3.10.1 Propagation of a discontinuous dislocation density 76

3.10.2 Dislocation loop expansion 78

3.10.3 Stability / instability of homogeneous dislocation distributions 79

3.10.4 Dislocation nucleation 80

3.11 Solutions 81

3.11.1 Propagation of a discontinuous dislocation density 81

3.11.2 Expansion of dislocation loops 84

3.11.3 Stability / instability of homogeneous dislocation distributions 85

3.11.4 Dislocation nucleation 86

Chapter 4 Constitutive Relations 89

4.1 Introduction 89

4.2 Dissipation 90

4.3 Pressure independence 92

4.4 Dislocation climb versus dislocation glide 93

4.5 Viscoplastic relationships 94

4.6 Coarse graining 96

4.7 Contact with conventional crystal plasticity 97

Chapter 5 Elasto-plastic Field Equations 99

5.1 Introduction 99

5.2 Fundamental field equations 99

5.3 Boundary conditions 101

5.4 Coarse graining 102

5.5 Resolution algorithm 104

5.6 Reduced field equations 105

5.6.1 Plane dislocations 107

5.7 Augmented crystal plasticity 109

5.8 Dynamics of a twist boundary 111

5.9 Conclusion 116

5.10 Problems 117

5.10.1 Helical dislocations 117

5.11 Solutions 118

5.11.1 Helical dislocations 118

Chapter 6 Case Studies 121

6.1 Introduction 121

6.2 Dislocation core structure 123

6.3 Piezoelectricity and dislocations 132

6.3.1 Coupling piezoelectricity, lattice incompatibility and transport 132

6.3.2 Piezoelectric polarization and dislocations in GaN layers 134

6.3.3 Dislocation transport and electric displacement in GaN layers 137

6.4 Intermittent plasticity 139

6.5 Effects of size on mechanical response 150

6.6 Complex loading paths 159

6.7 Strain localization 170

6.7.1 Experimental data in Al–Cu–Li alloys 171

6.7.2 Simulation results 174

Chapter 7 Review and Conclusions 181

7.1 Comparisons with conventional crystal plasticity 181

7.2 Alternative approaches 183

7.2.1 Peierls-Nabarro model 183

7.2.2 Atomistic simulations 184

7.2.3 Phase field methods 186

7.2.4 Discrete dislocation dynamics 187

7.3 Shortcomings and extensions 190

7.3.1 Fracture and disconnections 190

7.3.2 Rotational incompatibility and disclinations 191

7.3.3 Phase transformation and generalized disclinations 193

7.4 Final remarks 196

Appendix 197

Bibliography 203

Index 217

Claude Fressengeas, University of Lorraine, France.