Multiscale Modelling in Sheet Metal Forming

Preface

Contents

Contributors

1 Plastic Behaviour of Sheet Metals

1.1 Anisotropy of Sheet Metals

1.1.1 Uniaxial Characteristics of Plastic Anisotropy

1.1.2 Biaxial Characteristics of Plastic Anisotropy

1.2 Classical Yield Criteria for Anisotropic Sheet Metals

1.2.1 Hill (1948) Yield Criterion

1.2.2 Barlat (1989) Yield Criterion

1.3 BBC (2005) Yield Criterion

1.3.1 Equation of the Yield Surface

1.3.2 Flow Rule Associated to the Yield Surface

1.3.3 BBC (2005) Equivalent Stress

1.3.4 Identification Procedure

1.3.5 Theoretical Yield Stress in Pure Tension

1.3.6 Theoretical Coefficient of Uniaxial Plastic Anisotropy

1.3.7 Theoretical Yield Stress in Biaxial Tension Along RD and TD

1.3.8 Theoretical Coefficient of Biaxial Plastic Anisotropy

1.3.9 Identification Constraints

1.3.10 Particular Formulations of the BBC (2005) Yield Criterion

1.4 BBC (2008) Yield Criterion

1.4.1 BBC 2008 Equivalent Stress

1.4.2 Basic Identification Procedure

1.4.3 Enhanced Identification Procedure

1.5 3D Extensions of the BBC (2005, 2008) Yield Criteria

1.6 Advanced Anisotropic Yield Criteria

1.6.1 Barlat Yield Criteria

1.6.2 Cazacu-Barlat Yield Criteria

1.6.3 Vegter Yield Criterion

1.7 Recommendations on the Choice of the Yield Criteria

1.8 Perspectives

References

2 Crystallographic Texture and Plastic Anisotropy

2.1 The Structure of Polycrystalline Materials

2.2 Definition of Crystallographic Texture

2.2.1 Crystal Orientation

2.3 Experimental Determination of Textures

2.4 Texture and Properties of Materials

2.5 Plasticity of Polycrystalline Materials

2.5.1 The Taylor Model (Full-Constraints)

2.5.2 Special Plasticity Parameters

2.5.3 Plasticity of Cubic Metals

2.5.4 Deformation Hardening

2.5.5 Plasticity of Macroscopic Bodies

2.6 Parameterization of the Texture Function

2.7 Other Modes of Plasticity

References

3 Multiscale Modelling of Mechanical Anisotropy

3.1 Introduction

3.2 Multiscale Frameworks in Crystal Plasticity

3.2.1 Statistical Crystal Plasticity

3.2.1.1 Sachs-Type Models

3.2.1.2 Taylor-Type Models

3.2.1.3 Grain Interaction Models

3.2.1.4 Self-consistent Schemes

3.2.2 Full-Field Approaches

3.2.2.1 Crystal Plasticity Finite Element Method

3.2.2.2 Crystal Plasticity FFT

3.3 Multi-scale Modelling of Plastic Anisotropy

3.3.1 Direct Micro-Macro Coupling

3.3.1.1 Embedded Full-Field Models

3.3.1.2 Embedded Mean-Field Models

3.3.1.3 Embedded Reduced Texture Models

3.3.2 Hierarchical Coupling

3.3.2.1 Database and Sampling Techniques

3.3.2.2 Spectral Crystal Plasticity (SCP)

3.3.3 Yield Criteria Based on Crystal Plasticity

3.3.3.1 Yield Criteria Defined by Interpolation

3.3.3.2 Yield Criteria Defined by Approximation

3.3.3.3 Evolving BBC2008 Yield Criterion

3.3.4 Other Concepts in Multi-scale Modelling of Plastic Anisotropy

Acknowledgments

References

4 Modelling the Voids Growth in Ductile Fracture

4.1 Models for Ductile Fracture

4.1.1 Void Shape Effects

4.1.2 Anisotropic Plasticity

4.2 Anisotropic GTN Model for Sheet Metal Forming

4.2.1 GTN Models for Sheet Metal Forming

4.2.2 Anisotropic GTN Model with Hill 48 Yield Criterion

4.2.3 Determination of GTN Parameters from Uniaxial Tests

4.2.4 Simulation of a Deep Drawing Process

4.3 Development of a Gurson Type Model for Some Advanced Yield Criteria for Sheet Metals

4.3.1 Limit Analysis and Homogenization

4.3.2 An Introduction to Gurson Type Models

4.3.3 Dissipation Functions for Some Non-quadratic Anisotropic Yield Criteria

4.3.3.1 Yield Criteria Yld91 and Yld2004-18p

4.3.3.2 Dissipation Function for the Yld91 Criterion

4.3.3.3 Dissipation Function for the Yld2004-18p Criterion

4.3.3.4 BBC2005 Criterion and Dissipation Function

4.3.4 Gurson-Type Models for Some Anisotropic Yield Criteria Based on Linear Transformations

4.4 Mie Decomposition of Incompressible Vector Fields in Ellipsoidal Coordinates

4.4.1 Natural Ellipsoidal Coordinates

4.4.2 Laplace’s Equation in Natural Ellipsoidal Coordinates

4.4.3 Some Properties of Surface Ellipsoidal Harmonics

4.4.4 Incompressible Vector Fields by Piola Transforms

4.4.5 The Ellipsoidal Mie Decomposition for Incompressible Vector Fields

4.5 Calibration of Gurson Type Models via the Mie Decomposition

4.5.1 The Homogenization Limit Analysis Problem

4.5.2 Boundary Conditions

4.5.3 Calibration of Gurson Type Models

4.6 Conclusions

References

5 Advanced Models for the Prediction of Forming Limit Curves

5.1 Failure in Sheet Metal Forming Operations

5.1.1 Diffuse Necking—Localized Necking—Ductile Fracture

5.1.2 Diffuse Necking—Localized Necking—Shear Instability—Ductile Fracture

5.1.3 Diffuse Necking—Shear Instability—Ductile Fracture

5.2 Forming Limit Diagram: Introduction

5.3 Experimental Formability Tests

5.3.1 An Overview of Experimental Formability Tests

5.3.2 Experimental Formability Observations Concerning the Influence of Sheet Curvature

5.3.3 Experimental Formability Observations Concerning the Influence of Sheet Thickness

5.3.4 Experimental Formability Observations Concerning the Combined Influence of Sheet Curvature and Thickness

5.3.5 Experimental Formability Observations Concerning the Influence of Temperature

5.3.6 Experimental Formability Observations Concerning the Influence of Strain Rate

5.4 Forming Limit Models

5.4.1 Diffuse Necking Models

5.4.1.1 Swift’s Model

5.4.1.2 Modified Maximum Force Criterion (MMFC)

5.4.2 Localized Necking Model (Hill’s Model)

5.4.3 Assessing the Formability of Metallic Sheets by Means of Localized and Diffuse Necking Models

5.4.3.1 Constitutive Equations

5.4.3.2 Localized and Diffuse Necking Models

5.4.4 Marciniak-Kuckzynski (M-K) Model

5.4.4.1 Overview

5.4.4.2 Implicit Formulation of the M-K and H-N Models

5.4.4.3 Comparison of the FLC’s Predicted by Different Theoretical Models

5.4.4.4 Non-zero Thickness Stress

5.4.4.5 Non-zero Through-Thickness Shear Stress

5.4.5 Crystal Plasticity Based FLC Prediction

5.4.5.1 Crystal Plasticity in MK Analysis

5.4.5.2 Formability Modelling Through Crystal Plasticity in FEM (CPFEM)

5.4.6 Void Growth Based FLC Prediction

5.4.6.1 Modelling of FLC using the GTN Model

5.4.6.2 FLC Prediction by Numerical Simulation of Traditional Formability Tests

5.4.6.3 FLC Prediction by Combined M-K and GTN Models

5.4.6.4 Theoretical Model for Forming Limit Diagram Predictions Without Initial Inhomogeneity

Limit Analysis Interpretation of the MK Model

Coalescence Models for Ductile Porous Materials

5.4.6.5 Necking Model Based on Limit Analysis for Porous Sheets

Numerical Results

5.4.7 Other Models

5.4.7.1 Bifurcation Models

5.4.7.2 Perturbation Models

5.4.8 Semi-empirical Models

5.5 Commercial Programs for FLC Prediction

5.5.1 FORM-CERT Program

5.5.1.1 Calculation and Displaying the FLC

5.5.1.2 “Experimental Data” Module

5.5.2 Other Programs

5.6 Conclusions

References

6 Anisotropic Damage in Elasto-plastic Materials with Structural Defects

6.1 Introduction

6.1.1 List of Notation

6.2 Damage State

6.2.1 Isotropic Damage

6.2.2 Void Volume Fraction

6.2.3 Effect of Stress Triaxiality

6.2.4 Undamaged Configuration

6.3 Models with Damage State Variables

6.3.1 Model with Multiple Undamaged Configurations

6.3.2 Crystal Plasticity Model Coupled with Anisotropic Damage

6.3.3 Lemaitre and Chaboche Models

6.4 Model with Stress-Free Undamaged Configuration and Deformation-like Damage Tensor {\textbf{F}}^{d}

6.4.1 Elastic Type Response Dependent on Damage

6.4.2 Equations for Damage and Plasticity

6.4.3 Dissipative Nature of the Irreversible Behaviour

6.4.4 Constitutive Models

6.5 Models with Non-metric Property

6.5.1 Constitutive Hypotheses

6.5.2 Dissipation Postulate

6.5.3 Constitutive and Evolution Equations with Respect to the Reference Configuration

6.5.4 Model Proposed by Aslan et al. (2011)

6.6 Conclusion

References

7 Modelling the Portevin-Le Chatelier Effect—A Study on Plastic Instabilities and Pattern Formation

7.1 Introduction

7.1.1 Experimental and Physical Aspects

7.1.2 Main Ideas for the Constitutive Modelling of the PLC Effect

7.2 An Elastic-Viscoplastic Model with ‘‘Negative Strain-Rate Sensitivity’’ of McCormick Type

7.3 One-Dimensional Stress State

7.3.1 Constitutive Relations

7.3.2 Field Equations and Initial-Boundary Value Problems

7.3.3 A Numerical Investigation

7.3.3.1 Strain-Controlled Experiments

7.3.3.2 Stress-Controlled Experiments

7.4 A Methodology for Investigating Mechanical Parameters for Critical Conditions on PLC Effect

7.4.1 Temporal Stability Analysis of Serrated Curves

7.4.2 Calibration of Mechanical Parameters

7.5 Conclusions and Outlook

Acknowledgments

Appendix: Numerical Scheme

References

8 Erratum to: Multiscale Modelling in Sheet Metal Forming

Erratum to: D. Banabic (ed.), Multiscale Modelling in Sheet Metal Forming, ESAFORM Bookseries on Material Forming, DOI 10.1007/978-3-319-44070-5

Author Index