Multiscale Modelling of Plasticity and Fracture by Means of Dislocation Mechanics (eBook)

Title Page 
3 
Copyright Page 
4 
Preface 
5 
Table of Contents 
7 
Atomistic Simulation Methods and their Application on Fracture 
8 
1 Introduction 
8 
2 Description of Interatomic Bonds 
10 
2.1 Quantum Mechanics Based Descriptions of the Atomic Bound- ing 
10 
2.2 Atomic Interaction Models, Potentials 
13 
3 The Molecular Dynamics Method 
17 
3.1 Force Calculation 
18 
3.2 Integrating the Equations of Motion 
18 
3.3 Relaxation Algorithms 
21 
3.4 Boundary and initial conditions 
23 
3.5 Stable Defects under Load 
27 
3.6 Visualisation and Analysis of Defects 
28 
4 Comcurrent Multiscale Methods 
31 
4.1 Introduction and Classification of Multiscale Methods 
31 
4.2 The Finite Element Atomistic (FEAt) Method 
33 
4.3 The Quasicontinuum-Method Based on the Cauchy Born Rule 
36 
4.4 The Fully Nonlocal Cluster-Based Quasicontinuum-Method 
42 
4.5 Other Concurrent Multiscale Methods 
48 
4.6 'Learn-On-The-Fly' - LOTF 
48 
5 Atomistic Aspects of Fracture 
50 
5.1 Lattice Trapping and the Directional Cleavage Anisotropy 
51 
5.2 Metastable Fracture Surfaces 
53 
6 Dynamics of Brittle Crack Propagation 
54 
7 Summary 
57 
Biblliography 
57 
Fundamental dislocation theory and 3D dislocation mechanics 
65 
1 Basic Dislocation Theory 
65 
1.1 Heuristic Dislocation Creation 
65 
1.2 Basic Dislocation Types 
68 
1.3 Moving Dislocations 
72 
1.4 Dislocations In Real Crystals 
82 
2 Curved Dislocations 
88 
2.1 Line Tension Model 
88 
2.2 Dislocation Self-Interaction 
97 
3 2-D Applications 
107 
3.1 Simulation Technique 
108 
3.2 Static Simulations Using the Line Tension Model 
115 
3.3 Simulation Using Dislocation Self Interaction: Particle Strengthening 
122 
3.4 Simulations of Thermally Activated Dislocation Glide 
128 
4 3-D aspects 
135 
4.1 Non-elastic 3-D effects 
136 
4.2 Computational Aspects for 3-D Simulations 
145 
Bibliography 
151 
Plasticity of moderately loaded cracks and the consequence of the discrete nature of plasticity to fatigue and fracture 155
1 Introduction 
155 
2 Stress field of a crack in a linear elastic material 
156 
3 Dislocation crack tip interaction 
163 
3.1 Linear elastic analysis of the stresses and deformations induced by an edge dislocation near a crack tip 
164 
3.2 Moderate cyclic loading of a crack 
169 
3.3 The involved length scales 
171 
4 Modelling of plasticity, crack propagation and fracture surface contact 
172 
4.1 The cyclic plastic deformation as a function of load amplitude and ber of cycles 
174 
4.2 The effect of boundaries 
183 
4.3 The threshold of cyclic plastic deformation and the effective threshold of stress intensity range 
186 
4.4 Other discrete discrete dislocation simulations of fatigue crack propagation 
187 
Bibliography 
188 
Discrete Dislocation Plasticity Analysis of Cracks and Fracture 
191 
1 Introducton 
191 
2 Elastic Models of Dislocations 
193 
2.1 General Idea 
193 
2.2 Edge Dislocations 
195 
3 Boundary Value Problems 
196 
4 Dislocation Dynamics 
198 
4.1 Annihilation 
199 
4.2 Frank-Read sources 
199 
5 Methodology for Crack Problems 
201 
6 Cracks in Single Crystals 
204 
6.1 Stationary Crack - Tip Fields 
204 
6.2 Crack Propagation under Monotonic Loading 
206 
7 Fatigue Crack Growth 
209 
8 Cracks in Polycrystals 
212 
Bibliography 
216 
Statistical physical approach to describe the collective properties of dislocations 
219 
1 Introduction 
219 
2 Kroner-Kosevich field theory of dislocations 
224 
2.1 Nye's dislocation density tensor 
224 
2.2 Internal stress generated by the dislocation system 
226 
2.3 Second order stress function tensor 
228 
2.4 2D problems 
229 
2.5 Time evolution of the dislocation density tensor 
230 
2.6 Time evolution of the displacement field 
231 
2.7 Problems related to coarse graining 
232 
3 Linking micro- to mesoscale for a 2D dislocation system 
234 
3.1 Discrete dislocation dynamics simulations in 2D 
234 
3.2 Continuum theories developed for other systems, analogies and differences 
238 
3.3 Hierarchy of the different oder density ffunctions 
239 
3.4 Evolution of the plastic shear 
244 
3.5 Self-consistent field approximation 
245 
3.6 Stability ananlysis 
246 
3.7 Numerical studies 
248 
3.8 The role of dislocation-dislocation correlation 
248 
3.9 Deformation of a constrained channel 
254 
3.10 Application to metal-matrix composite 
259 
3.11 Boltzmann thery of dislocation 
261 
3.12 Hydrodynamics approach proposed by Kratochvil and Sedlacek 
264 
4 Internal stress distribution generated by the dislocations 
266 
4.1 General considerations 
266 
4.2 Stress distribution at the dislocations 
269 
4.3 The mean values of distributions P(r) and Po(r) 
270 
4.4 Asymptomic properties of the stress distribution function 
271 
Bibliography 
273 
Basic ingredients, development of phenomenological models and practical use of crystal plasticity 
277 
1 Introduction 
277 
2 A thermodynamical approach to single crytal plasticity 
280 
2.1 General fframework 
280 
2.2 Derivation of single crystal models 
285 
2.3 Yield surfaces 
290 
2.4 Identification under tension and tension-shear loaadings 
293 
2.5 Slip system selection 
296 
2.6 Other crystal plasticity models 
297 
3 Finite element computations of single crystalline components 
299 
3.1 Algorithm for the numerical integration 
300 
3.2 Laboratory specimens 
301 
3.3 Turbine Blades 
301 
4 Finite Element Crystal Plasticity 
305 
5 Uniform field models 
317 
5.1 Yield surfaces 
317 
5.2 Scale transition rules 
319 
5.3 Complex paths 
323 
6 Conclusion and perspecives 
326 
Bibliography 
326 
Computational homogenization 
333 
1 Introduction 
333 
2 Underlying principles and assumptions 
336 
2.1 Scale separation 
336 
2.2 Local periodicity 
337 
2.3 Homogenization principles 338
2.4 Computational homogenization scheme 
338 
2.5 Kinematically driven multi-scale scheme 
339 
3 The micro-scale problem 
339 
3.1 The representative volume element 
339 
3.2 Micro-scale characterization & equilibrium
340 
3.3 The macro-micro scale transition 
341 
3.4 Micro-scale RVE boundary comditions 
344 
4 The macro-scale problem 
345 
4.1 The micro-macro scale transition 
345 
4.2 Macroscopec stress tensors 
349 
5 Two-scale numerical solution strategy 
350 
5.1 RYE Boundary value problem 
350 
5.2 Extraction of the macroscopic stress 
353 
5.3 Extraction of the macroscopic tangent operator 
355 
5.4 Nested solution strategy 
359 
6 Example: two-scale coupled analysis in bending 
361 
7 The RVE in first-order computational homogenization 
364 
7.1 General concept of an RVE 
364 
7.2 Unit cells versus RVEs 
365 
8 Second-order computational homogenization 
372 
8.1 Principles 
373 
8.2 Two-scale higher-order kinematics 
374 
8.3 Extracting stress tensors 
378 
8.4 Two-scale computational solution strategy 
380 
8.5 Parallel solution of the multi-scale nested boundary value problens 
383 
9 Higher-order issues 
384 
9.1 First-order versus second-order 
384 
9.2 Full gradient versus couple stress 
385 
9.3 Geometrical size effects 
387 
9.4 Large macroscopic gradients 
387 
9.5 Macroscopic localization 
388 
9.6 The higher-order RVE 
392 
10 Conclusions 
393 
Bibliography 
394