Continuum Scale Simulation of Engineering Materials: Fundamentals - Microstructures - Process Applications

Continuum Scale Simulation of Engineering Materials......Page 4
Contents......Page 8
Preface......Page 24
List of Contributors......Page 26
I Fundamentals and Basic Methods......Page 34
1.1 Introduction......Page 36
1.2.1 Diffusion......Page 38
1.2.2 Boundary Conditions......Page 41
1.2.3 Cell Size......Page 47
1.3.1 LE, LENP and PE in Fe-Mn-C......Page 48
1.3.2 LE, LENP and PE in Fe-Si-C......Page 50
1.3.3 PE in Fe-Ni-C......Page 53
1.3.4 Effect of Traces on the Growth of Grain Boundary Cementite......Page 54
1.3.5 Continuous Cooling......Page 55
1.3.6 Competitive Growth of Phases: Multi-Cell Calculations......Page 56
1.3.7 Gas-Metal-Reactions: Carburization......Page 59
1.4 Outlook......Page 66
References......Page 67
2.1 Introduction......Page 70
2.3.1 Representation of a Microstructure......Page 71
2.3.2 Thermodynamics of Microstructures......Page 73
2.3.3 The Evolution Equations......Page 79
2.5 Typical Fields of Applications and Examples......Page 80
2.6 Summary and Opportunities......Page 82
References......Page 84
3.1.1 Introduction......Page 90
3.1.2 Formal Description and Classes of Cellular Automata......Page 91
3.1.3 Cellular Automata in Materials Science......Page 93
3.1.4 Recrystallization Simulations with Cellular Automata......Page 94
3.2.2 The HPP and FHP Lattice Gas Cellular Automata......Page 100
3.2.3 The Lattice Boltzmann Automaton......Page 103
3.3 Conclusions and Outlook......Page 106
References......Page 107
4.2 History of the Monte Carlo Method......Page 110
4.2.1 Ising and Potts Models......Page 111
4.2.2 Metropolis Algorithm......Page 113
4.2.3 n-fold Way Algorithm......Page 114
4.3.1 Discretization of Microstructure......Page 118
4.3.2 Evolution of the Microstructure......Page 119
4.3.4 Lattices......Page 120
4.3.6 Parallelization of the Monte Carlo Algorithm......Page 122
4.4 Nucleation in Recrystallization......Page 125
4.5 Initialization of MC Simulations......Page 126
4.6 Verification of the Monte Carlo Model......Page 127
4.7 Scaling of Simulated Grain Size to Physical Grain Size......Page 130
4.8 Recrystallization Kinetics in the Monte Carlo model......Page 131
4.9.2 Static Recrystallization......Page 132
4.9.4 Recrystallization in the Presence of Particles......Page 134
4.9.5 Texture Development......Page 136
4.9.6 Texture......Page 138
4.9.7 Dynamic Recrystallization......Page 142
4.10 Summary......Page 143
References......Page 144
5.2.1 Mechanical Response of Single Crystals......Page 148
5.2.2 Lattice Orientation Distributions for Polycrystals......Page 153
5.2.3 Mechanical Response of Polycrystals......Page 155
5.3.1 Generalities......Page 157
5.3.2 Yield Surfaces Defined by Expansions......Page 159
5.3.3 Yield Surfaces Defined by Hyperplanes......Page 160
5.3.4 Isoparametric Flow Surface......Page 162
5.3.5 Direct Polycrystal Plasticity Implementation......Page 164
5.4.2 Finite Element Formulations......Page 165
5.5.1 Application to Explosive Forming......Page 167
5.5.2 Application to the Limiting Dome Height Test......Page 168
5.6 Summary......Page 172
References......Page 174
6.1 Introduction......Page 178
6.2.1 Isotropic Yield Conditions for Perfect Plasticity......Page 179
6.2.2 Flow Rules......Page 182
6.2.3 Subsequent Yield Surfaces during Plastic Hardening......Page 183
6.2.4 Anisotropic Plasticity......Page 185
6.2.5 Direct Generalizations of Isotropic Yield Conditions......Page 186
6.3.1 Texture......Page 187
6.3.2 Dislocations......Page 189
6.3.3 Porosity and Second Phases......Page 193
6.4.1 Quadratic Yield Functions......Page 194
6.4.2 Non-Quadratic Yield Functions......Page 195
6.4.5 BBC2000 Yield Criterion......Page 198
6.4.7 CB2001 Yield Criterion......Page 200
6.5.1 Mechanical testing......Page 202
6.5.2 Analysis and Treatment of the Test Results......Page 205
6.5.4 Plastic Flow Localization......Page 207
6.5.5 Cup Drawing Simulation......Page 208
6.6 Conclusions......Page 210
References......Page 211
7.1 Introduction......Page 218
7.3.2 Empirical Modeling......Page 219
7.4.1 Software......Page 220
7.5.1 Multilayer Perceptron......Page 221
7.5.2 Radial Basis Function Networks......Page 224
7.5.3 More Network Types......Page 226
7.6.3 Reinforcement Learning......Page 227
7.7.1 Network Type Selection and Configuration......Page 228
7.7.4 Prevention of Overfitting......Page 229
7.7.6 Diagnostics of the Internal State......Page 230
References......Page 231
8.1 Introduction......Page 234
8.2.2 Kinetics and Interaction Forces......Page 236
8.2.3 Dislocation Equation of Motion......Page 237
8.2.4 The Dislocation Stress and Force Fields......Page 241
8.2.5 The Stochastic Force and Cross-slip......Page 243
8.2.6 Modifications for Long-Range Interactions: The Super-Dislocation Principle......Page 245
8.2.8 The DD Numerical Solution: An Implicit-Explicit Integration Scheme......Page 246
8.3.1 Continuum Elasto-Viscoplasticity......Page 247
8.3.2 Modifications for Finite Domains......Page 248
8.4 Typical Fields of Applications and Examples......Page 250
8.4.1 Evolution of Dislocation Structure during Monotonic Loading......Page 251
8.4.2 Dislocation Crack Interaction: Heterogeneous Deformation......Page 253
8.4.3 Dislocations Interaction with Shock Waves......Page 256
8.5 Summary and Concluding Remarks......Page 258
References......Page 259
9.2 Recent Trends in Modelling Materials Behavior......Page 264
9.2.1 Analytical Models......Page 265
9.2.2 Computer Simulations......Page 266
9.2.3 Materials Modelling and Materials Design: Some Examples......Page 268
9.2.4 Sophisticated Statistical Analysis......Page 269
9.3.1 Recovery of Aluminum Alloys......Page 270
9.3.2 Competition Between Recrystallization and Precipitation......Page 273
9.3.3 Optimizing Casting Process in Precipitation Hardenable Alloys......Page 276
9.4 Perspectives......Page 278
References......Page 280
II Application to Engineering Microstructures......Page 282
10.1 Introduction......Page 284
10.2.1 Primary Phase Formation......Page 287
10.2.2 Secondary Phases Formation......Page 289
10.3.1 Primary Phase Formation......Page 290
10.4 Cellular Automaton – Finite Element Model......Page 291
10.4.2 Growth Law......Page 292
10.4.3 Coupling of CA and FE Methods......Page 293
10.5.1 PFT Model......Page 294
10.5.2 CAFE Model......Page 297
10.6 Conclusion......Page 299
References......Page 301
11.1 Introduction......Page 304
11.2 Phenomenological Description of Solid State Phase Transformations......Page 305
11.3 Phase-Field Model of Solid State Phase Transformations......Page 307
11.5 Bulk Microstructures with Periodic Boundary Conditions......Page 309
11.6 A Single Crystal Film with Surface and Substrate Constraint......Page 311
11.7 Elastic Coupling of Structural Defects and Phase Transformations......Page 312
11.9 Isostructural Phase Separation......Page 313
11.10 Precipitation of Cubic Intermetallic Precipitates in a Cubic Matrix......Page 315
11.11 Structural Transformations Resulting in a Point Group Symmetry Reduction......Page 317
11.12 Ferroelectric Phase Transformations......Page 319
11.13 Phase Transformation in a Reduced Dimensions: Thin Films and Surfaces......Page 321
11.14 Summary......Page 323
References......Page 325
12.2.1 The Concept......Page 330
12.3 Irregular Shapeless Cellular Automata for Grain Growth......Page 331
12.3.1 Curvature Driven Grain Growth......Page 332
12.3.2 In the Presence of Additional Driving Forces......Page 335
12.4.1 The Deformation Model......Page 337
12.4.2 The Annealing Model......Page 338
References......Page 340
13.1 Introduction......Page 342
13.2.1 Definition of Parameters......Page 345
13.2.2 The Grain Sizes and Shapes and their Distributions......Page 346
13.3.1 Grain Boundaries (GBs) and Triple Points (TPs)......Page 347
13.3.3 Size Correlations of Nearest Neighbor Grains (Function k(ij))......Page 348
13.3.4 Space Filling (Function q(ij))......Page 349
13.4.2 Weaire–Aboav Equation (WAE, Partial Randomness)......Page 350
13.5.1 Direct Simulations......Page 352
13.5.2 Simulations by the Statistical Theory......Page 353
13.6 Summarizing Remarks......Page 356
References......Page 358
14.1 Introduction......Page 360
14.2 The Diffuse Interface Model......Page 362
14.3 Free Energies......Page 364
14.4 Numerical Simulations......Page 366
14.4.1 Grain Growth and Coarsening......Page 367
14.4.2 Multicomponent Multiphase Solidification......Page 368
14.5 Outlook......Page 373
References......Page 374
15.1 Background......Page 376
15.1.1 Formation of Deformation Zones......Page 377
15.1.2 Formation and Growth of Particle Stimulated Nuclei......Page 378
15.2 Computational Approach......Page 381
15.3 Simulations......Page 382
15.4.1 Microstructure and Kinetics......Page 383
15.4.2 Texture......Page 388
15.5 Summary......Page 391
References......Page 392
16.1 Introduction......Page 394
16.2 Discretization......Page 398
16.3 Numerical Implementation......Page 399
16.4 Numerical Results......Page 402
16.5 Conclusion......Page 403
References......Page 404
17.1 Introduction......Page 408
17.2.1 Basis of the Method......Page 410
17.2.2 Dislocation Segmentation......Page 411
17.2.3 Dislocation Self-Interaction......Page 412
17.2.4 Simulation Procedure and Accuracy......Page 414
17.3 Particle Arrangement......Page 415
17.4.1 Dispersion Strengthening......Page 417
17.4.2 Order Strengthening......Page 419
17.4.3 Lattice Mismatch Strengthening......Page 421
References......Page 426
18.1 Thin Film Plasticity......Page 430
18.2 Simulation of Dislocations in Thin Films......Page 432
18.2.1 Boundary Conditions......Page 433
18.3.1 Mobility Controlled Deformation......Page 435
18.3.2 Source Controlled Deformation......Page 436
References......Page 442
19.1 Introduction......Page 446
19.2 Model......Page 447
19.3 Crack-Tip Plasticity......Page 452
19.4 Scaling Relations......Page 454
19.5 Discussion......Page 457
References......Page 458
20.1 Introduction......Page 462
20.2.2 Phase-Field Methods......Page 463
20.3 Static Coarse-Grained Properties......Page 464
20.3.1 Continuous Dislocation Theory......Page 465
20.3.2 Extensions to the Continuous Theory......Page 468
20.4 Dynamic Coarse-Grained Properties......Page 472
20.5 Conclusions......Page 474
References......Page 475
21.1 Introduction......Page 478
21.2.1 Phenomenological Model......Page 481
21.2.2 Materials Science Approach......Page 483
21.3.1 Various Approaches......Page 484
21.3.2 Composite Framework......Page 486
21.4 Conclusions......Page 488
References......Page 489
22.1 Introduction......Page 492
22.2 Local Constitutive Laws (Mesoscopic Scale)......Page 493
22.3 The Taylor Ambiguity......Page 495
22.4 Full Constraints (FC) Taylor Theory......Page 496
22.5 Classical Relaxed Constraints (RC) Models......Page 497
22.6.1 Introduction......Page 498
22.6.2 The Lamel Model......Page 499
22.6.3 The Advanced Lamel Model......Page 500
22.8 Conclusions......Page 502
References......Page 504
23.1 Introduction......Page 506
23.2.1 Local Constitutive Behavior and Homogenization......Page 508
23.2.2 Green Function Method and Fourier Transform Solution......Page 510
23.2.3 Viscoplastic Inclusion and Eshelby Tensors......Page 511
23.2.4 Interaction and Localization Equations......Page 513
23.2.5 Selfconsistent Equations......Page 514
23.2.7 Algorithm......Page 515
23.3.1 Kinematics......Page 516
23.3.2 Hardening......Page 518
23.3.3 Twinning Reorientation......Page 519
23.4.1 Tension and Compression of FCC......Page 520
23.4.2 Torsion (Shear) of FCC......Page 521
23.4.3 Twinning and Anisotropy of HCP Zr......Page 525
23.4.4 Compression of Olivine (MgSiO(4))......Page 526
23.5 Further Selfconsistent Models and Applications......Page 528
References......Page 530
24.1 Introduction......Page 534
24.2 Basic Considerations and Results......Page 535
24.3 The Case of Small Deformation......Page 539
24.4 Simple Shear of a Crystalline Strip......Page 540
References......Page 543
25.1.1 Scope of this Chapter......Page 546
25.1.2 Motivations for Generalized Continuum Crystal Plasticity......Page 547
25.2.1 Cosserat Single Crystal Plasticity......Page 548
25.2.2 Second Gradient Single Crystal Plasticity......Page 550
25.2.3 Gradient of Internal Variable Approach......Page 551
25.3.1 Introduction to Multiscale Asymptotic Method......Page 552
25.3.2 Extension of Classical Homogenization Schemes......Page 555
25.4.2 Plasticity at the Crack Tip in Single Crystals......Page 556
25.4.3 Grain Size Effects in Polycrystalline Aggregates......Page 557
References......Page 559
26.2 Theoretical Background......Page 562
26.2.2 Total Lagrangian versus Updated Lagrangian Schemes......Page 563
26.2.3 Fully Implicit Time Integration Procedures......Page 565
26.3 Micro-Mechanical Finite Element Models......Page 567
26.4.2 Predictions of Micro-Texture......Page 568
References......Page 575
27.1.1 Slip versus Deformation Twinning......Page 576
27.1.2 Major Consequences of Deformation Twinning......Page 577
27.2 Historical Perspective......Page 579
27.2.2 Volume Fraction Transfer Scheme......Page 580
27.3.1 Relaxed Configuration......Page 581
27.3.3 Plastic Flow Rule......Page 583
27.3.4 Evolution of Twin Rotations......Page 585
27.3.5 Slip-Twin Hardening Functions......Page 587
27.4 Examples......Page 589
References......Page 592
28.2 The Texture Component Method......Page 594
28.2.2 Representation of Texture Components in a Crystal Plasticity FEM......Page 595
28.3 The Crystal Plasticity Model......Page 598
28.4.2 Prediction of Earing Behavior......Page 599
28.5 Outlook......Page 604
References......Page 605
29.1 Introduction......Page 606
29.2 Program Overview......Page 608
29.3 Modeling of Piezoelectric Microstructures......Page 611
29.4 Modeling of Electrochemical Solids: Rechargeable Lithium Ion Batteries......Page 613
29.5 The OOFTWO Project: A Preview......Page 618
References......Page 620
30.1 Introduction......Page 622
30.2.1 Matricity Model......Page 623
30.2.3 Realisation of the Adjustability of Matricity by Weighting Factors......Page 625
30.2.4 Calculation of Stress-strain Curves......Page 626
30.2.5 Mechanical Constants......Page 627
30.3.1 Comparison to Cluster Parameter r......Page 628
30.4 Conclusion......Page 637
References......Page 638
31.1 Introduction......Page 640
31.2 Empirical Relations......Page 641
31.3.1 Homogeneous Glide Activity......Page 642
31.3.2 Heterogeneous Glide Activity......Page 643
31.4 Models......Page 644
31.4.1 Two-parameter Model for Homogeneous Glide......Page 645
31.4.2 Composite Model for Heterogeneous Glide......Page 647
31.5 Concluding Remarks......Page 649
References......Page 651
32.1 Introductory Remarks on Inelastic Material Behaviour......Page 654
32.2.1 General Aspects and Examples......Page 656
32.2.2 Singular Elements for Stationary Cracks......Page 657
32.2.3 Regular Element Arrangements for Extending Cracks......Page 658
32.3.1 Foundation......Page 659
32.3.2 The Domain Integral or VCE Method......Page 660
32.3.3 Path Dependence of the J-Integral in Incremental Plasticity......Page 661
32.4.1 Fundamentals......Page 662
32.4.2 Example: Simulation of Ductile Tearing in a Laser Weld......Page 665
32.5 Summary......Page 666
References......Page 667
33.1 Introduction......Page 672
33.2 Behaviour of Suspensions: The Generation of Clusters......Page 673
33.3 Conclusions......Page 676
References......Page 677
III Application to Engineering Materials Processes......Page 680
34.1 Introduction......Page 682
34.2 Dendritic Microstructures......Page 683
34.3 Inverse Problems and Optimal Design......Page 685
34.4 Conclusion......Page 687
References......Page 688
35.1 Introduction......Page 690
35.4 Powder Compaction......Page 691
35.4.1 The Drucker-Prager-Cap Model and Finite Element Implementations......Page 692
35.4.2 Experiments to Determine the Drucker-Prager-Cap Parameters......Page 694
35.4.3 Example......Page 696
35.5 Sintering......Page 697
35.5.1 Models for Solid-State Sintering......Page 698
35.5.2 Liquid-Phase Sintering......Page 700
35.5.3 Parameters of the Liquid-Phase Sintering Model for an Alumina Ceramic......Page 701
35.5.4 Finite-Element Implementations and Applications......Page 702
35.6 Sizing and Post-Sintering Mechanical Densification......Page 703
References......Page 704
36 Integration of Physically Based Materials Concepts......Page 708
36.1 Through-process Modeling of Aluminum Alloy AA2024 from Solidification through Homogenization and Hot Rolling......Page 710
36.2 Through-process Texture Modeling of Aluminum Alloy AA5182 during Industrial Multistep hot Rolling, Cold Rolling, and Annealing......Page 714
36.3 Through-thickness Texture Evolution during Hot Rolling of an IF-Steel......Page 716
References......Page 717
37.1 Introduction......Page 720
37.2 Features of the Al Production Chain for Rolled Products......Page 721
37.3 TP Modelling of the Al Process Chain for Rolled Products......Page 723
37.4 Application of Through Process Modelling......Page 724
37.4.1 Tracing of Dislocation Density......Page 726
37.4.2 Tracing of Texture......Page 732
37.4.3 Tracing of Microchemistry......Page 734
37.5 Conclusions......Page 736
References......Page 737
38.1 Introduction......Page 738
38.2 Optimization Strategies in Sheet Processing and Material Quality......Page 739
38.3 Processing and Microstructure Features of Aluminum Sheet......Page 740
38.4 Thermomechanical Simulation of Rolling Processes......Page 741
38.5 Microstructure Evolution During hot Rolling......Page 744
38.6 Material Properties of Industrially Processed Aluminum Sheet......Page 750
38.7 Simulation of Anisotropic Sheet Properties......Page 752
38.7.2 Tensile Test and r-Value Simulation......Page 753
38.7.3 Earing During Cup Deep Drawing......Page 754
38.8 Formability of Aluminum Sheets......Page 756
38.9 Summary and Outlook......Page 757
References......Page 758
39.1 Introduction......Page 760
39.2.1 Equations for Microstructure Evolutions......Page 761
39.2.2 Integration into a Finite Element Code......Page 763
39.2.3 2D Simulation Results......Page 764
39.3 Case II: Warm Forming of Two-Phase Steels......Page 766
39.4 Case III: Texture Evolution in an Hexagonal Alloy......Page 769
39.4.1 Calibrating the Polycrystalline Model with Simple Mechanical Tests......Page 771
39.4.3 Application to Hot Forming......Page 772
39.5 Conclusions......Page 773
References......Page 776
40.1 Introduction......Page 778
40.2.1 Solidification Problem with the Sharp Interface......Page 780
40.2.2 Numerical Solution......Page 781
40.3 Verification of the CA-FDM Solidification Model......Page 787
40.4.1 Macroscopic Modelling of Solidification Conditions......Page 788
40.4.2 Microscopic Simulation of Solidification Structures......Page 789
40.5 Conclusions......Page 791
References......Page 793
41.1 Introduction......Page 796
41.2 Ideal Forming Design Theory for Tube Hydroforming......Page 799
41.3 Strain-Rate Potential: Srp98......Page 802
41.4 Preform Design for Hydroforming Parts......Page 803
References......Page 805
42.1 Introduction......Page 810
42.2 Review of Simulation Literature......Page 811
42.3 Review of the Experimental Literature......Page 813
42.4 Draw-Bend Springback......Page 815
References......Page 821
43.1 Background......Page 828
43.2 The EWK Fracture Model......Page 830
43.3 Academic Validation......Page 831
43.4 Semi-Industrial Validation......Page 832
43.5 Conclusions......Page 835
References......Page 836
44.1 Introduction......Page 838
44.3 Material – Characterization of Second Phase Particle Fields......Page 840
44.4 GTN-based FE Model......Page 841
44.5 Coupled damage percolation model......Page 844
44.6 Results......Page 845
44.7 Discussion......Page 847
References......Page 849
45.1 Introduction......Page 850
45.2 Plastic Potentials and Porosity......Page 851
45.3 Model Parameter Identification......Page 854
45.4 Strain Softening, Damage and Lengthscale......Page 856
45.5 Hints for Application......Page 858
References......Page 860
46.1 Introduction......Page 862
46.2 Artificial Neural Networks in Process Simulation......Page 865
46.3.1 Physical Model......Page 869
46.3.2 Physical Model plus Neural Network......Page 871
46.3.3 Off-line System, on-line System and in-line System......Page 872
46.3.4 Results from Hot Strip Mills......Page 874
46.4 Conclusions......Page 875
References......Page 876
Index......Page 878