Multiscale Modelling in Sheet Metal Forming (eBook)

Preface 7
Contents 11
Contributors 12
1 Plastic Behaviour of Sheet Metals 13
1.1 Anisotropy of Sheet Metals 13
1.1.1 Uniaxial Characteristics of Plastic Anisotropy 13
1.1.2 Biaxial Characteristics of Plastic Anisotropy 17
1.2 Classical Yield Criteria for Anisotropic Sheet Metals 18
1.2.1 Hill (1948) Yield Criterion 18
1.2.2 Barlat (1989) Yield Criterion 19
1.3 BBC (2005) Yield Criterion 21
1.3.1 Equation of the Yield Surface 21
1.3.2 Flow Rule Associated to the Yield Surface 22
1.3.3 BBC (2005) Equivalent Stress 22
1.3.4 Identification Procedure 24
1.3.5 Theoretical Yield Stress in Pure Tension 25
1.3.6 Theoretical Coefficient of Uniaxial Plastic Anisotropy 25
1.3.7 Theoretical Yield Stress in Biaxial Tension Along RD and TD 28
1.3.8 Theoretical Coefficient of Biaxial Plastic Anisotropy 29
1.3.9 Identification Constraints 31
1.3.10 Particular Formulations of the BBC (2005) Yield Criterion 35
1.4 BBC (2008) Yield Criterion 36
1.4.1 BBC 2008 Equivalent Stress 36
1.4.2 Basic Identification Procedure 37
1.4.3 Enhanced Identification Procedure 41
1.5 3D Extensions of the BBC (2005, 2008) Yield Criteria 46
1.6 Advanced Anisotropic Yield Criteria 47
1.6.1 Barlat Yield Criteria 48
1.6.2 Cazacu-Barlat Yield Criteria 52
1.6.3 Vegter Yield Criterion 55
1.7 Recommendations on the Choice of the Yield Criteria 56
1.8 Perspectives 57
References 57
2 Crystallographic Texture and Plastic Anisotropy 59
2.1 The Structure of Polycrystalline Materials 59
2.2 Definition of Crystallographic Texture 60
2.2.1 Crystal Orientation 61
2.3 Experimental Determination of Textures 62
2.4 Texture and Properties of Materials 67
2.5 Plasticity of Polycrystalline Materials 69
2.5.1 The Taylor Model (Full-Constraints) 72
2.5.2 Special Plasticity Parameters 79
2.5.3 Plasticity of Cubic Metals 82
2.5.4 Deformation Hardening 85
2.5.5 Plasticity of Macroscopic Bodies 85
2.6 Parameterization of the Texture Function 86
2.7 Other Modes of Plasticity 87
References 88
3 Multiscale Modelling of Mechanical Anisotropy 91
3.1 Introduction 91
3.2 Multiscale Frameworks in Crystal Plasticity 95
3.2.1 Statistical Crystal Plasticity 96
3.2.1.1 Sachs-Type Models 96
3.2.1.2 Taylor-Type Models 97
3.2.1.3 Grain Interaction Models 97
3.2.1.4 Self-consistent Schemes 98
3.2.2 Full-Field Approaches 98
3.2.2.1 Crystal Plasticity Finite Element Method 99
3.2.2.2 Crystal Plasticity FFT 100
3.3 Multi-scale Modelling of Plastic Anisotropy 101
3.3.1 Direct Micro-Macro Coupling 102
3.3.1.1 Embedded Full-Field Models 102
3.3.1.2 Embedded Mean-Field Models 103
3.3.1.3 Embedded Reduced Texture Models 104
3.3.2 Hierarchical Coupling 105
3.3.2.1 Database and Sampling Techniques 105
3.3.2.2 Spectral Crystal Plasticity (SCP) 105
3.3.3 Yield Criteria Based on Crystal Plasticity 107
3.3.3.1 Yield Criteria Defined by Interpolation 107
3.3.3.2 Yield Criteria Defined by Approximation 108
3.3.3.3 Evolving BBC2008 Yield Criterion 110
3.3.4 Other Concepts in Multi-scale Modelling of Plastic Anisotropy 135
Acknowledgments 137
References 137
4 Modelling the Voids Growth in Ductile Fracture 147
4.1 Models for Ductile Fracture 147
4.1.1 Void Shape Effects 149
4.1.2 Anisotropic Plasticity 150
4.2 Anisotropic GTN Model for Sheet Metal Forming 150
4.2.1 GTN Models for Sheet Metal Forming 151
4.2.2 Anisotropic GTN Model with Hill 48 Yield Criterion 151
4.2.3 Determination of GTN Parameters from Uniaxial Tests 154
4.2.4 Simulation of a Deep Drawing Process 155
4.3 Development of a Gurson Type Model for Some Advanced Yield Criteria for Sheet Metals 158
4.3.1 Limit Analysis and Homogenization 159
4.3.2 An Introduction to Gurson Type Models 162
4.3.3 Dissipation Functions for Some Non-quadratic Anisotropic Yield Criteria 167
4.3.3.1 Yield Criteria Yld91 and Yld2004-18p 168
4.3.3.2 Dissipation Function for the Yld91 Criterion 170
4.3.3.3 Dissipation Function for the Yld2004-18p Criterion 173
4.3.3.4 BBC2005 Criterion and Dissipation Function 175
4.3.4 Gurson-Type Models for Some Anisotropic Yield Criteria Based on Linear Transformations 178
4.4 Mie Decomposition of Incompressible Vector Fields in Ellipsoidal Coordinates 188
4.4.1 Natural Ellipsoidal Coordinates 189
4.4.2 Laplace’s Equation in Natural Ellipsoidal Coordinates 192
4.4.3 Some Properties of Surface Ellipsoidal Harmonics 194
4.4.4 Incompressible Vector Fields by Piola Transforms 196
4.4.5 The Ellipsoidal Mie Decomposition for Incompressible Vector Fields 200
4.5 Calibration of Gurson Type Models via the Mie Decomposition 204
4.5.1 The Homogenization Limit Analysis Problem 204
4.5.2 Boundary Conditions 206
4.5.3 Calibration of Gurson Type Models 210
4.6 Conclusions 212
References 213
5 Advanced Models for the Prediction of Forming Limit Curves 216
5.1 Failure in Sheet Metal Forming Operations 216
5.1.1 Diffuse Necking—Localized Necking—Ductile Fracture 217
5.1.2 Diffuse Necking—Localized Necking—Shear Instability—Ductile Fracture 217
5.1.3 Diffuse Necking—Shear Instability—Ductile Fracture 217
5.2 Forming Limit Diagram: Introduction 219
5.3 Experimental Formability Tests 223
5.3.1 An Overview of Experimental Formability Tests 224
5.3.2 Experimental Formability Observations Concerning the Influence of Sheet Curvature 227
5.3.3 Experimental Formability Observations Concerning the Influence of Sheet Thickness 229
5.3.4 Experimental Formability Observations Concerning the Combined Influence of Sheet Curvature and Thickness 230
5.3.5 Experimental Formability Observations Concerning the Influence of Temperature 231
5.3.6 Experimental Formability Observations Concerning the Influence of Strain Rate 232
5.4 Forming Limit Models 234
5.4.1 Diffuse Necking Models 235
5.4.1.1 Swift’s Model 235
5.4.1.2 Modified Maximum Force Criterion (MMFC) 236
5.4.2 Localized Necking Model (Hill’s Model) 239
5.4.3 Assessing the Formability of Metallic Sheets by Means of Localized and Diffuse Necking Models 240
5.4.3.1 Constitutive Equations 240
5.4.3.2 Localized and Diffuse Necking Models 242
5.4.4 Marciniak-Kuckzynski (M-K) Model 248
5.4.4.1 Overview 248
5.4.4.2 Implicit Formulation of the M-K and H-N Models 251
5.4.4.3 Comparison of the FLC’s Predicted by Different Theoretical Models 261
5.4.4.4 Non-zero Thickness Stress 262
5.4.4.5 Non-zero Through-Thickness Shear Stress 265
5.4.5 Crystal Plasticity Based FLC Prediction 269
5.4.5.1 Crystal Plasticity in MK Analysis 271
5.4.5.2 Formability Modelling Through Crystal Plasticity in FEM (CPFEM) 273
5.4.6 Void Growth Based FLC Prediction 274
5.4.6.1 Modelling of FLC using the GTN Model 274
5.4.6.2 FLC Prediction by Numerical Simulation of Traditional Formability Tests 274
5.4.6.3 FLC Prediction by Combined M-K and GTN Models 283
5.4.6.4 Theoretical Model for Forming Limit Diagram Predictions Without Initial Inhomogeneity 288
Limit Analysis Interpretation of the MK Model 288
Coalescence Models for Ductile Porous Materials 290
5.4.6.5 Necking Model Based on Limit Analysis for Porous Sheets 291
Numerical Results 294
5.4.7 Other Models 295
5.4.7.1 Bifurcation Models 295
5.4.7.2 Perturbation Models 296
5.4.8 Semi-empirical Models 296
5.5 Commercial Programs for FLC Prediction 297
5.5.1 FORM-CERT Program 297
5.5.1.1 Calculation and Displaying the FLC 299
5.5.1.2 “Experimental Data” Module 299
5.5.2 Other Programs 300
5.6 Conclusions 301
References 301
6 Anisotropic Damage in Elasto-plastic Materials with Structural Defects 312
6.1 Introduction 312
6.1.1 List of Notation 316
6.2 Damage State 317
6.2.1 Isotropic Damage 318
6.2.2 Void Volume Fraction 319
6.2.3 Effect of Stress Triaxiality 320
6.2.4 Undamaged Configuration 321
6.3 Models with Damage State Variables 324
6.3.1 Model with Multiple Undamaged Configurations 324
6.3.2 Crystal Plasticity Model Coupled with Anisotropic Damage 329
6.3.3 Lemaitre and Chaboche Models 334
6.4 Model with Stress-Free Undamaged Configuration and Deformation-like Damage Tensor {/textbf{F}}^{d} 337
6.4.1 Elastic Type Response Dependent on Damage 339
6.4.2 Equations for Damage and Plasticity 341
6.4.3 Dissipative Nature of the Irreversible Behaviour 342
6.4.4 Constitutive Models 345
6.5 Models with Non-metric Property 347
6.5.1 Constitutive Hypotheses 348
6.5.2 Dissipation Postulate 351
6.5.3 Constitutive and Evolution Equations with Respect to the Reference Configuration 353
6.5.4 Model Proposed by Aslan et al. (2011) 357
6.6 Conclusion 359
References 360
7 Modelling the Portevin-Le Chatelier Effect—A Study on Plastic Instabilities and Pattern Formation 362
7.1 Introduction 362
7.1.1 Experimental and Physical Aspects 364
7.1.2 Main Ideas for the Constitutive Modelling of the PLC Effect 369
7.2 An Elastic-Viscoplastic Model with ‘‘Negative Strain-Rate Sensitivity’’ of McCormick Type 373
7.3 One-Dimensional Stress State 377
7.3.1 Constitutive Relations 377
7.3.2 Field Equations and Initial-Boundary Value Problems 379
7.3.3 A Numerical Investigation 381
7.3.3.1 Strain-Controlled Experiments 382
7.3.3.2 Stress-Controlled Experiments 389
7.4 A Methodology for Investigating Mechanical Parameters for Critical Conditions on PLC Effect 391
7.4.1 Temporal Stability Analysis of Serrated Curves 392
7.4.2 Calibration of Mechanical Parameters 397
7.5 Conclusions and Outlook 407
Acknowledgments 408
Appendix: Numerical Scheme 408
References 412
8 Erratum to: Multiscale Modelling in Sheet Metal Forming 415
Erratum to: D. Banabic (ed.), Multiscale Modelling in Sheet Metal Forming, ESAFORM Bookseries on Material Forming, DOI 10.1007/978-3-319-44070-5 415
Author Index 416