Preface XIII
List of Contributors XV
1 Materials Modeling - Challenges and Perspectives 1
Miguel Vaz Jr., Eduardo A. de Souza Neto, and Pablo Andre´s Munoz-Rojas
1.1 Introduction 1
1.2 Modeling Challenges and Perspectives 3
1.2.1 Mechanical Degradation and Failure of Ductile Materials 3
1.2.2 Modeling of Cellular Structures 8
1.2.3 Multiscale Constitutive Modeling 15
1.3 Concluding Remarks 18
Acknowledgments 19
References 19
2 Local and Nonlocal Modeling of Ductile Damage 23
Jose Manuel de Almeida Cesar de Sa, Francisco Manuel Andrade Pires, and Filipe Xavier Costa Andrade
2.1 Introduction 23
2.2 Continuum Damage Mechanics 25
2.2.1 Basic Concepts of CDM 25
2.2.2 Ductile Plastic Damage 26
2.3 Lemaitre's Ductile Damage Model 27
2.3.1 Original Model 27
2.3.2 Principle of Maximum Inelastic Dissipation 31
2.3.3 Assumptions Behind Lemaitre's Model 32
2.4 Modified Local Damage Models 33
2.4.1 Lemaitre's Simplified Damage Model 33
2.4.2 Damage Model with Crack Closure Effect 37
2.5 Nonlocal Formulations 42
2.5.1 Aspects of Nonlocal Averaging 44
2.5.2 Classical Nonlocal Models of Integral Type 45
2.5.3 Numerical Implementation of Nonlocal Integral Models 47
2.6 Numerical Analysis 57
2.6.1 Axisymmetric Analysis of a Notched Specimen 57
2.6.2 Flat Grooved Plate in Plane Strain 62
2.6.3 Upsetting of a Tapered Specimen 63
2.7 Concluding Remarks 68
Acknowledgments 69
References 69
3 Recent Advances in the Prediction of the Thermal Properties of Metallic Hollow Sphere Structures 73
Thomas Fiedler, Irina V. Belova, Graeme E. Murch, and Andreas Öchsner
3.1 Introduction 73
3.2 Methodology 74
3.2.1 Lattice Monte Carlo Method 75
3.2.2 Finite Element Method 77
3.2.3 Numerical Calculation Models 89
3.3 Finite Element Analysis on Regular Structures 91
3.4 Finite Element Analysis on Cubic-Symmetric Models 94
3.5 LMC Analysis of Models of Cross Sections 98
3.5.1 Modeling 98
3.5.2 Results 101
3.6 Computed Tomography Reconstructions 103
3.6.1 Computed Tomography 104
3.6.2 Numerical Analysis 104
3.6.3 Results 106
3.7 Conclusions 108
References 109
4 Computational Homogenization for Localization and Damage 111
Thierry J. Massart, Varvara Kouznetsova, Ron H. J. Peerlings, and Marc G. D. Geers
4.1 Introduction 111
4.1.1 Mechanics Across the Scales 111
4.1.2 Some Historical Notes on Homogenization 112
4.1.3 Separation of Scales 113
4.1.4 Computational Homogenization and Its Application to Damage and Fracture 114
4.2 Continuous-Continuous Scale Transitions 115
4.2.1 First-Order Computational Homogenization 115
4.2.2 Second-Order Computational Homogenization 119
4.2.3 Application of the Continuous-Continuous Homogenization Schemes to Ductile Damage 121
4.3 Continuous-Discontinuous Scale Transitions 125
4.3.1 Scale Transitions and RVE for Initially Periodic Materials 126
4.3.2 Localization of Damage at the Fine and Coarse Scales 129
4.3.3 Localization Band Enhanced Multiscale Solution Scheme 135
4.3.4 Scale Transition Procedure for Localized Behavior 139
4.3.5 Solution Strategy and Computational Aspects 142
4.3.6 Applications and Discussion 147
4.4 Closing Remarks 159
References 160
5 A Mixed Optimization Approach for Parameter Identification Applied to the Gurson Damage Model 165
Pablo Andre´s Munoz-Rojas, Luiz Antonio B. da Cunda, Eduardo L. Cardoso, Miguel Vaz Jr., and Guillermo Juan Creus
5.1 Introduction 165
5.2 Gurson Damage Model 166
5.2.1 Influence of the Parameter Values on Behavior of the Damage Model 171
5.2.2 Recent Developments and New Trends in the Gurson Model 175
5.3 Parameter Identification 177
5.4 Optimization Methods - Genetic Algorithms and Mathematical Programming 179
5.4.1 Genetic Algorithms 180
5.4.2 Gradient-Based Methods 184
5.5 Sensitivity Analysis 187
5.5.1 Modified Finite Differences and the Semianalytical Method 188
5.6 A Mixed Optimization Approach 192
5.7 Examples of Application 192
5.7.1 Low Carbon Steel at 25 ?C 192
5.7.2 Aluminum Alloy at 400 ?C 197
5.8 Concluding Remarks 200
Acknowledgments 200
References 201
6 Semisolid Metallic Alloys Constitutive Modeling for the Simulation of Thixoforming Processes 205
Roxane Koeune and Jean-Philippe Ponthot
6.1 Introduction 205
6.2 Semisolid Metallic Alloys Forming Processes 207
6.2.1 Thixotropic Semisolid Metallic Alloys 208
6.2.2 Different Types of Semisolid Processing 209
6.2.3 Advantages and Disadvantages of Semisolid Processing 215
6.3 Rheological Aspects 216
6.3.1 Microscopic Point of View 216
6.3.2 Macroscopic Point of View 222
6.4 Numerical Background in Large Deformations 223
6.4.1 Kinematics in Large Deformations 223
6.4.2 Finite Deformation Constitutive Theory 225
6.5 State-of-the-Art in FE-Modeling of Thixotropy 237
6.5.1 One-Phase Models 237
6.5.2 Two-Phase Models 244
6.6 A Detailed One-Phase Model 246
6.6.1 Cohesion Degree 247
6.6.2 Liquid Fraction 248
6.6.3 Viscosity Law 248
6.6.4 Yield Stress and Isotropic Hardening 250
6.7 Numerical Applications 250
6.7.1 Test Description 250
6.7.2 Results Analysis 251
6.8 Conclusion 254
References 255
7 Modeling of Powder Forming Processes; Application of a Three-invariant Cap Plasticity and an Enriched Arbitrary Lagrangian-Eulerian FE Method 257
Amir R. Khoei
7.1 Introduction 257
7.2 Three-Invariant Cap Plasticity 260
7.2.1 Isotropic and Kinematic Material Functions 262
7.2.2 Computation of Powder Property Matrix 264
7.2.3 Model Assessment and Parameter Determination 265
7.3 Arbitrary Lagrangian-Eulerian Formulation 269
7.3.1 ALE Governing Equations 270
7.3.2 Weak Form of ALE Equations 272
7.3.3 ALE Finite Element Discretization 273
7.3.4 Uncoupled ALE Solution 274
7.3.5 Numerical Modeling of an Automotive Component 279
7.4 Enriched ALE Finite Element Method 282
7.4.1 The Extended-FEM Formulation 283
7.4.2 An Enriched ALE Finite Element Method 286
7.4.3 Numerical Modeling of the Coining Test 291
7.5 Conclusion 295
Acknowledgments 295
References 296
8 Functionally Graded Piezoelectric Material Systems - A Multiphysics Perspective 301
Wilfredo Montealegre Rubio, Sandro Luis Vatanabe, Glaucio Hermogenes Paulino, and Emilio Carlos Nelli Silva
8.1 Introduction 301
8.2 Piezoelectricity 302
8.3 Functionally Graded Piezoelectric Materials 304
8.3.1 Functionally Graded Materials (FGMs) 304
8.3.2 FGM Concept Applied to Piezoelectric Materials 306
8.4 Finite Element Method for Piezoelectric Structures 309
8.4.1 The Variational Formulation for Piezoelectric Problems 309
8.4.2 The Finite Element Formulation for Piezoelectric Problems 310
8.4.3 Modeling Graded Piezoelectric Structures by Using the FEM 312
8.5 Influence of Property Scale in Piezotransducer Performance 314
8.5.1 Graded Piezotransducers in Ultrasonic Applications 314
8.5.2 Further Consideration of the Influence of Property Scale: Optimal Material Gradation Functions 319
8.6 Influence of Microscale 322
8.6.1 Performance Characteristics of Piezocomposite Materials 326
8.6.2 Homogenization Method 328
8.6.3 Examples 332
8.7 Conclusion 335
Acknowledgments 335
References 336
9 Variational Foundations of Large Strain Multiscale Solid Constitutive Models: Kinematical Formulation 341
Eduardo A. de Souza Neto and Raul A. Feijoo
9.1 Introduction 341
9.2 Large Strain Multiscale Constitutive Theory: Axiomatic Structure 343
9.2.1 Deformation Gradient Averaging and RVE Kinematics 346
9.2.2 Actual Constraints: Spaces of RVE Velocities and Virtual Displacements 348
9.2.3 Equilibrium of the RVE 349
9.2.4 Stress Averaging Relation 351
9.2.5 The Hill-Mandel Principle of Macrohomogeneity 352
9.3 The Multiscale Model Definition 353
9.3.1 The Microscopic Equilibrium Problem 354
9.3.2 The Multiscale Model: Well-Posed Equilibrium Problem 354
9.4 Specific Classes of Multiscale Models: The Choice of V? 356
9.4.1 Taylor Model 356
9.4.2 Linear RVE Boundary Displacement Model 359
9.4.3 Periodic Boundary Displacement Fluctuations Model 359
9.4.4 Minimum Kinematical Constraint: Uniform Boundary Traction 360
9.5 Models with Stress Averaging in the Deformed RVE Configuration 361
9.6 Problem Linearization: The Constitutive Tangent Operator 362
9.6.1 Homogenized Constitutive Functional 363
9.6.2 The Homogenized Tangent Constitutive Operator 364
9.7 Time-Discrete Multiscale Models 366
9.7.1 The Incremental Equilibrium Problem 367
9.7.2 The Homogenized Incremental Constitutive Function 367
9.7.3 Time-Discrete Homogenized Constitutive Tangent 368
9.8 The Infinitesimal Strain Theory 371
9.9 Concluding Remarks 372
Appendix 373
Acknowledgments 376
References 376
10 A Homogenization-Based Prediction Method of Macroscopic Yield Strength of Polycrystalline Metals Subjected to Cold-Working 379
Kenjiro Terada, Ikumu Watanabe, Masayoshi Akiyama, Shigemitsu Kimura, and Kouichi Kuroda
10.1 Introduction 379
10.2 Two-Scale Modeling and Analysis Based on Homogenization Theory 382
10.2.1 Two-Scale Boundary Value Problem 383
10.2.2 Micro-Macro Coupling and Decoupling Schemes for the Two-Scale BVP 385
10.2.3 Method of Evaluating Macroscopic Yield Strength after Cold-Working 386
10.3 Numerical Specimens: Unit Cell Models with Crystal Plasticity 387
10.4 Approximate Macroscopic Constitutive Models 390
10.4.1 Definition of Macroscopic Yield Strength 391
10.4.2 Macroscopic Yield Strength at the Initial State 391
10.4.3 Approximate Macroscopic Constitutive Model 393
10.4.4 Parameter Identification for Approximate Macroscopic Constitutive Model 393
10.5 Macroscopic Yield Strength after Three-Step Plastic Forming 395
10.5.1 Forming Condition 395
10.5.2 Two-Scale Analyses with Micro-Macro Coupling and Decoupling Schemes 396
10.5.3 Evaluation of Macroscopic Yield Strength after Three-Step Plastic Forming 398
10.6 Application for Pilger Rolling of Steel Pipe 401
10.6.1 Forming Condition 401
10.6.2 Decoupled Microscale Analysis 403
10.6.3 Evaluation of Macroscopic Yield Strength after Pilger Rolling Process 406
10.7 Conclusion 408
References 409
Index 413
