Preface ................................................... XIII
List of Contributors ........................................ XV
1 Materials Modeling - Challenges and Perspectives ............. 1
Miguel Vaz Jr., Eduardo A. de Souza Neto, and Pablo Andres
Munoz-Rojas
1.1 Introduction ............................................ 1
1.2 Modeling Challenges and Perspectives .................... 3
1.2.1 Mechanical Degradation and Failure of Ductile
Materials ........................................ 3
1.2.1.1 Remarks ................................. 7
1.2.2 Modeling of Cellular Structures .................. 8
1.2.2.1 Remarks ................................ 14
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
José Manuel de Almeida César de Sá, 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
2.3.1.1 The Elastic State Potential ............ 28
2.3.1.2 The Plastic State Potential ............ 29
2.3.1.3 The Dissipation Potential .............. 29
2.3.1.4 Evolution of Internal Variables ........ 30
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
2.4.1.1 Constitutive Model ..................... 33
2.4.1.2 Numerical Implementation ............... 34
2.4.2 Damage Model with Crack Closure Effect .......... 37
2.4.2.1 Constitutive Model ..................... 37
2.4.2.2 Numerical Implementation ............... 40
2.5 Nonlocal Formulations .................................. 42
2.5.1 Aspects of Nonlocal Averaging ................... 44
2.5.1.1 The Averaging Operator ................. 44
2.5.1.2 Weight Functions ....................... 45
2.5.2 Classical Nonlocal Models of Integral Type ...... 45
2.5.2.1 Nonlocal Formulations for Lemaitre's
Simplified Model ....................... 46
2.5.3 Numerical Implementation of Nonlocal Integral
Models .......................................... 47
2.5.3.1 Numerical Evaluation of the
Averaging Integral ..................... 48
2.5.3.2 Global Version of the Elastic
Predictor/Return Mapping Algorithm ..... 49
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
2.6.3.1 Damage Prediction Using the
Lemaitre's Simplified Model ............ 65
2.6.3.2 Damage Prediction Using the
Lemaitre's Model with Crack Closure
Effect ................................. 67
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 Óchsner
3.1 Introduction ........................................... 73
3.2 Methodology ............................................ 74
3.2.1 Lattice Monte Carlo Method ...................... 75
3.2.2 Finite Element Method ........................... 77
3.2.2.1 Basics of Heat Transfer ................ 77
3.2.2.2 Weighted Residual Method ............... 77
3.2.2.3 Discretization and Principal Finite
Element Equation ....................... 78
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
3.6.2.1 Microstructure ........................ 105
3.6.2.2 Mesostructure ......................... 206
3.6.3 Results ........................................ 206
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 ........................... 223
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
4.3.1.1 First-Order Scale Transitions ......... 126
4.3.1.2 Choice of the Mesoscopic
Representative Volume Element ......... 127
4.3.1.3 Boundary Conditions for the Unit
Cell .................................. 128
4.3.2 Localization of Damage at the Fine and
Coarse Scales .................................. 129
4.3.2.1 Fine-Scale Localization - Implicit
Gradient Damage ....................... 229
4.3.2.2 Detection of Coarse-Scale
Localization as a Bifurcation into
an Inhomogeneous Deformation
Pattern ............................... 130
4.3.2.3 Illustration of the Localization
Analysis .............................. 132
4.3.2.4 Identification and Selection of the
Localization Orientation .............. 235
4.3.3 Localization Band Enhanced Multiscale
Solution Scheme ................................ 135
4.3.3.1 Introduction of the Localization
Band .................................. 136
4.3.3.2 Coupled Multiscale Scheme for
Localization .......................... 137
4.3.4 Scale Transition Procedure for Localized
Behavior ....................................... 139
4.3.4.1 Multiscale Solution Procedure ......... 139
4.3.4.2 Causes of Snapback in the Averaged
Material Response ..................... 239
4.3.4.3 Strain Jump Control for Embedded
Band Snapback ......................... 240
4.3.4.4 Dissipation Control for Unit-Cell
Snapback .............................. 242
4.3.5 Solution Strategy and Computational Aspects .... 242
4.3.5.1 Governing Equations for the
Macroscopic and Mesoscopic Solution
Procedures ............................ 142
4.3.5.2 Extraction of Consistent Tangent
Stiffness for Unit-Cell Snapback
Control ............................... 244
4.3.5.3 Discretization and Linearization of
the Macroscopic Solution Procedure .... 244
4.3.5.4 Introduction of Localization Bands
upon Material Bifurcation ............. 146
4.3.6 Applications and Discussion .................... 147
4.3.6.1 Selection of Localized Solutions ...... 147
4.3.6.2 Mesostructural Snapback in
a Tension-Compression Test ............ 149
4.3.6.3 Size Effect in a Shear-Compression
Test .................................. 351
4.3.6.4 Masonry Shear Wall Test ............... 152
4.4 Closing Remarks ....................................... 159
References ................................................. 160
5 A Mixed Optimization Approach for Parameter
Identification Applied to the Curson Damage Model .......... 165
Pablo Andreś Muñoz-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 ............................. 15O
5.4.1.1 Formulation ........................... 181
5.4.1.2 Implementation ........................ 184
5.4.2 Gradient-Based Methods ......................... 184
5.4.2.1 General Procedure ..................... 184
5.4.2.2 Sequential Linear Programming
(SLP) ................................. 185
5.4.2.3 Globally Convergent Method of
Moving Asymptotes (GCMM A) ............ 185
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 ............................... 292
5.7.1 Low Carbon Steel at 25°C ....................... 292
5.7.2 Aluminum Alloy at 400°C ........................ 297
5.8 Concluding Remarks .................................... 200
Acknowledgments ............................................ 200
References ................................................. 202
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
6.2.2.1 Production of Spheroidal
Microstructure ........................ 210
6.2.2.2 Reheating ............................. 212
6.2.2.3 Forming ............................... 213
6.2.3 Advantages and Disadvantages of Semisolid
Processing ..................................... 215
6.3 Rheological Aspects ................................... 216
6.3.1 Microscopic Point of View ...................... 216
6.3.1.1 Origins of Thixotropy ................. 216
6.3.1.2 Transient Behavior .................... 217
6.3.1.3 Effective Liquid Fraction ............. 222
6.3.2 Macroscopic Point of View ...................... 222
6.3.2.1 Temperature Effects ................... 222
6.3.2.2 Yield Stress .......................... 222
6.3.2.3 Macrosegregation ...................... 223
6.4 Numerical Background in Large Deformations ............ 223
6.4.1 Kinematics in Large Deformations ............... 223
6.4.1.1 Lagrangian Versus Eulerian
Coordinate Systems .................... 223
6.4.1.2 Deformation Gradient and Strain
Rate Tensors .......................... 225
6.4.2 Finite Deformation Constitutive Theory ......... 225
6.4.2.1 Principle of Objectivity .............. 225
6.4.2.2 Different Classes of Materials ........ 226
6.4.2.3 A Corotational Formulation ............ 228
6.4.2.4 Linear Elastic Solid Material
Model ................................. 229
6.4.2.5 Linear Newtonian Liquid Material
Model ................................. 230
6.4.2.6 Hypoelastic Solid Material Models ..... 231
6.4.2.7 Liquid Material Models ................ 236
6.4.2.8 Comparison of Solid and Liquid
Approaches ............................ 236
6.5 State-of-the-Art in FE-Modeling of Thixotropy ......... 237
6.5.1 One-Phase Models ............................... 237
6.5.1.1 Apparent Viscosity Evolution .......... 238
6.5.1.2 Yield Stress Evolution ................ 243
6.5.2 Two-Phase Models ............................... 244
6.5.2.1 Two Coupled Fields .................... 244
6.5.2.2 Coupling Sources ...................... 245
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 ............................... 253
6.7.2.1 First Validation of the Model under
Isothermal Conditions ................. 251
6.7.2.2 Thermomechanical Analysis ............. 252
6.7.2.3 Residual Stresses Analysis ............ 253
6.7.2.4 Internal Variables Analysis ........... 253
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
7.2.3.1 Model Assessment ...................... 265
7.2.3.2 Parameter Determination ............... 267
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
7.3.4.1 Material (Lagrangian) Phase ........... 275
7.3.4.2 Smoothing Phase ....................... 276
7.3.4.3 Convection (Eulerian) Phase ........... 278
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
7.4.2.1 Level Set Update ...................... 287
7.4.2.2 Stress Update and Numerical
Integration ........................... 288
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,
Gláucio 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
8.6.1.1 Low-Frequency Applications ............ 326
8.6.1.2 High-Frequency Applications ........... 328
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 Raúl A. Feijóo
9.1 Introduction .......................................... 341
9.2 Large Strain Multiscale Constitutive Theory:
Axiomatic Structure ................................... 343
9.2.1 Deformation Gradient Averaging and RVE
Kinematics ..................................... 346
9.2.1.1 Consequence: Minimum RVE
Kinematical Constraints ............... 346
9.2.1.2 Minimum Constraint on Displacement
Fluctuations .......................... 347
9.2.2 Actual Constraints: Spaces of RVE Velocities
and Virtual Displacements ...................... 348
9.2.3 Equilibrium of the RVE ......................... 349
9.2.3.1 Strong Form of Equilibrium ............ 350
9.2.3.2 Solid-Void/Pore Interaction ........... 350
9.2.4 Stress Averaging Relation ...................... 351
9.2.4.1 Macroscopic Stress in Terms of RVE
Boundary Tractions and Body Forces .... 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 νμ ................................................. 356
9.4.1 Taylor Model ................................... 356
9.4.1.1 The Taylor-Based Constitutive
Functional: the Rule of Mixtures ...... 357
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
9.7.3.1 Taylor Model .......................... 369
9.7.3.2 The General Case ...................... 369
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
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