Principles of the Draw-Bend Springback Test

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PRINCIPLES OF THE DRAW-BEND SPRINGBACK TEST

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Jianfeng Wang, B.S., M.S.

*****

The Ohio State University

2004

Dissertation Committee: Approved by

Robert H. Wagoner, Adviser Glenn S. Daehn Adviser Peter M. Anderson Department of Materials Science and Engineering © Copyright by

Jianfeng Wang

2004 ABSTRACT

This thesis focuses on the principles of springback for 6022-T4 aluminum sheets, using a special draw-bend test with a range of sheet tensions and tool radii. To model the anisotropic yielding of 6022-T4 sheet, Barlat’91, ’96 and 2000 yield functions were implemented into Abaqus/Standard through user material subroutines. A nonlinear kinematic hardening model with multiple back stresses was constructed to closely reproduce the reversed strain hardening behavior of sheet metals. The new material constitutive models are as accurate as the previous work based on two-surface plasticity, but they have simpler mathematical forms and require fewer model parameters.

The mechanics of the persistent anticlastic curvature were studied by draw-bend experiments, ﬁnite element analysis and an elastic plate-bending theory. The transverse cross-section shape was dictated by a dimensionless parameter, β, which depends on the specimen geometry, tool radius and sheet tension. The rapid decrease of springback angle as the front sheet tension approaches yielding is correlated to a critical values of β, above which the retained anticlastic curvature is small and hence has little impact on the springback angle. However, the anticlastic curvature built up during the forming step persisted after unloading when β is less than 10–15, and thus greatly reduced the ﬁnal springback.

ii In order to quantify the time-dependent springback phenomenon and infer its physical basis, several aluminum alloys were draw-bend tested under conditions promoting the time-dependent response. The time-dependent springback angles are approximately linear with log(time) for times up to a few months, after which the kinetics become slower and saturation is reached in approximately 15 months. Resid- ual stress-driven creep and anelasticity are discussed as the possible sources of the time-dependent springback. For 6022-T4, qualitative agreement was obtained using a crude ﬁnite element model, with creep laws derived from constant load creep tests. For the second possibility, novel anelasticity tests following a reverse loading path were performed for 6022-T4 and DQSK steel. Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys.

iii This is dedicated to my beloved parents.

iv ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Professor Robert H.

Wagoner, for providing encouragement and guidance throughout the course of my study. I am also grateful to Professor Peter M. Anderson and Professor Glenn S.

Daehn for serving as members of my dissertation committee. I also want to thank

Professor David K. Matlock from Colorado School of Mines, for kindly hosting my visit in 2001 and assisting the draw-bend experiment.

I gratefully appreciate the help of Dr. Lumin Geng from General Motors Com- pany, for his valuable advice on Abaqus/UMAT programming. I would like to thank

Mr. William D. Carden and Mr. Vijay Balakrishnan for providing some of their experimental results for this research. Ms. Christine Putnam is greatly appreciated for her administrative support and assistance on proof-reading various publications.

Richard Boger and Lloyd Barnhart are also acknowledged for their kind help on some of the mechanical testings.

Finally, I would like to thank my parents for their enormous support and understanding throughout my study.

v VITA

January 18, 1973 ...... Born — Zhejiang, P. R. China

1989-1993 ...... B.S. Materials Science and Engineer- ing, Shanghai Jiaotong University, P. R. China 1991-1993 ...... B.S. Computer Engineering and Appli- cation, Shanghai Jiaotong University, P. R. China 1993-1996 ...... M.S. Materials Science and Engineer- ing, Shanghai Jiaotong University, P. R. China 1999-present ...... Graduate Research Associate, The Ohio State University.

PUBLICATIONS

Research Publications

Wang, J.F., Wagoner, R.H. and Matlock, D.K., “Creep and Anelasticity in the Spring- back of Aluminum”. International Journal of Plasticity, submitted in Feburary 2004.

Wang, J.F., Wagoner, R.H. and Matlock, D.K., “Eﬀect of Anticlastic Curvature on Springback of Aluminum Sheets after the Draw-Bend Test”. The 8th International Conference on Numerical Methods in Industrial Forming Processes, to appear in June 2004.

Wang, J.F., Wagoner, R.H. and Matlock, D.K., “Anticlastic Curvature in Draw- Bend Springback”. International Journal of Solids and Structures, submitted in March 2004.

vi Wang, J.F., Wagoner, R.H. and Matlock, D.K., “Simulation of Springback for Small Bending Radius after the Draw-Bend Test”. International Journal of Plasticity, in preparation.

Wang, J.F., Wagoner, R.H. and Matlock, D.K., “Creep Following Springback”. Proceedings of the Plasticity 2003: Dislocations, Plasticity and Metal Forming, ed. A.S. Khan, p211–213. Quebec City, Canada.

Wagoner, R.H. and Wang, J.F., “Springback”. ASM Metals Handbook, Volume 14 — Forming and Forging, to appear in September 2004.

FIELDS OF STUDY

Major Field: Materials Science and Engineering

Studies in: Plasticity and Metal Forming Prof. Robert H. Wagoner Mechanical Metallurgy Prof. Peter M. Anderson Prof. Glenn S. Daehn Finite Element Method Prof. Somnath Ghosh

vii TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita...... vi

List of Tables ...... xi

List of Figures ...... xiii

Chapters:

1. Introduction ...... 1

2. Review of Yield Criteria for Sheet Metal ...... 4

2.1 Introduction to Plastic Anisotropy ...... 6 2.2 Thermodynamic Framework of Elasto-plastic Materials ...... 8 2.3 Isotropic Yield Criteria ...... 14 2.3.1 Von Mises and Tresca Yield Criteria ...... 14 2.3.2 Hosford’s Non-quadratic Yield Criterion ...... 15 2.4 Anisotropic Yield Criteria ...... 16 2.4.1 Hill’s family of Anisotropic Yield Criteria ...... 17 2.4.2 Karaﬁllis and Boyce’s Yield Criterion ...... 21 2.4.3 Barlat’s Family of Anisotropic Yield Criteria ...... 22 2.4.4 Anisotropic Yielding of 6022-T4 Aluminum ...... 26

viii 3. Integration of Plastic Rate Equation ...... 29

3.1 Introduction to Stress Update Schemes ...... 29 3.2 Forward Integration ...... 31 3.3 Backward Integration ...... 32 3.4 Semi-backward Integration ...... 38 3.5 Implementation to Abaqus via UMAT ...... 39 3.6 Anisotropic Hardening of 6022-T4 Aluminum Sheet ...... 41

4. Springback Simulation of Draw-Bend Test with Finite Element Method . 46

4.1 Introduction ...... 47 4.2 Draw-Bend Experiment ...... 50 4.2.1 Materials ...... 51 4.2.2 Draw-Bend Test ...... 51 4.3 Experimental Results ...... 54 4.3.1 Effect of Back Force and Tool Radius ...... 55 4.3.2 Effect of Specimen Width ...... 57 4.4 Finite Element Results and Discussion ...... 58 4.4.1 Effect of Back Force ...... 60 4.4.2 Effect of Strip Width ...... 62 4.4.3 Choice of Element — Shell vs. Solid ...... 63 4.5 Conclusions ...... 66

5. Anticlastic Curvature in Draw-Bend Test ...... 68

5.1 Introduction ...... 69 5.2 Draw-Bend Experiment ...... 71 5.3 Finite Element Models ...... 73 5.4 Elastic Theory for Plate Bending ...... 76 5.4.1 Problem Statement and Closed-Form Solution ...... 76 5.4.2 Results for Pure Bending of an Initially Flat Plate ( 1 = Rx0 1 =0) ...... 80 Rx0 5.4.3 Bending of Initially Curved Plate ...... 86 5.4.4 Error Analysis of Elastic Theory ...... 89 5.5 Anticlastic Curvature in Draw-Bend Test ...... 91 5.5.1 Eﬀect of Back Force ...... 91 5.5.2 Eﬀect of Specimen Width ...... 95 5.5.3 Application of the Elastic Bending Theory ...... 96 5.6 Discussion ...... 101 5.7 Conclusions ...... 102

ix 6. Time-Dependent Springback ...... 105

6.1 Introduction ...... 106 6.2 Experimental ...... 108 6.2.1 Materials ...... 108 6.2.2 Draw-Bend Experiments ...... 111 6.2.3 Anelastic Tests ...... 114 6.3 Results ...... 115 6.3.1 Static (Time-Independent) Draw-Bend Tests ...... 115 6.3.2 Time-Dependent Draw-Bend Springback ...... 118 6.3.3 Room Temperature Creep Test for 6022-T4 ...... 125 6.3.4 Creep-Based Springback Simulation ...... 127 6.3.5 Anelastic Deformation after Unloading ...... 131 6.4 Discussions ...... 135 6.5 Conclusions ...... 140

7. Conclusions ...... 142

Appendices:

A. Numerical Algorithm for Abaqus UMAT ...... 147

B. Draw-Bend Test Data ...... 154

Bibliography ...... 158

x LIST OF TABLES

Table Page

2.1 Parameters for Barlat’91, ’96 and 2000 yield functions...... 28

3.1 Parameters ci (MPa) and γi for the mNLK model...... 43

4.1 Chemical composition of 6022-T4 aluminum sheet (in weight percent). 51

5.1 Parameters used in ﬁnite element model for elastic bending...... 82

5.2 Error introduced by using parabolic function...... 91

5.3 The radii of primary and anticlastic curvature in draw-bend test. . . . 97

6.1 Chemical composition (in weight pct.) and thickness of aluminum sheets...... 108

6.2 Grain size of aluminum alloys...... 110

6.3 Kinetics of time-dependent springback of 6022-T4 aluminum from mea- 7 surements at τ1 = 60s and τ2 = 4 × 10 s (15 months)...... 121

6.4 Long-time kinetics of springback for 6022-T4 aluminum based on re- 8 measurement of samples at τ1 = 60s and τ2 = 2.2 × 10 s (7 years). . . 124

6.5 Calculated times to reach fractions (0.2, 0.5 and 0.8) of the saturation strains (δε∞) or springback angles (δθ∞)...... 139

B.1 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 3.5 and W = 50mm...... 154

xi B.2 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 14 and W = 50mm...... 155

B.3 Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (TD) with R/t = 3.5 and W = 50mm...... 156

B.4 Measured springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) using R/t = 14 tool...... 157

xii LIST OF FIGURES

Figure Page

2.1 Schematics of coordinate systems of a sheet (X-Y -Z) and a tensile sample (x-y-z) cut at an angle θ at the sheet rolling direction. . . . . 6

2.2 Schematics of the yield surface evolution in the stress space: (a) with kinematic and isotropic hardening, (b) with kinematic hardening only, and (c) with isotropic hardening only...... 10

2.3 Evolution of the back stress ...... 13

2.4 von Mises and Tresca yield loci compared with experimental data. . . 16

2.5 Plane-stress isotropic yield functions...... 17

2.6 Eﬀect of r-value on the yield surface shape ...... 19

2.7 Eﬀect of m on stress ratio σb ...... 21 σRD

2.8 Comparison of the calculated and experimental results for 6022-T4: (a) normalized yield stress, and (b) r-value...... 27

3.1 Schematics of return mapping algorithm in stress space...... 34

3.2 Schematics of the return mapping schemes: (a) the backward Euler method, and (b) the tangent cutting plane method...... 39

3.3 Input and output of a user material subroutine...... 40

3.4 Schematics of the Bauschinger test: (a) the fork device, and (b) ﬂow curves after load reversal...... 42

xiii 3.5 Simulated and experimental reverse ﬂow curves after: (a) compression/tension test, and (b) tension/compression test. Markers are experimental data from Balakrishnan (1999), and solid lines are the corresponding simulation results...... 44

3.6 Simulated reverse ﬂow curves using various hardening laws: (a) isotropic hardening and NLK model with single back stress, and (b) mNLK and Geng-Wagoner models...... 45

4.1 Draw-bend experiment: (a) equipment at Colorado School of Mines, and (b) schematics of test procedure and geometry of a deformed sheet after springback ...... 52

4.2 Dependence of the springback angle and the anticlastic curvature on the normalized back force: (a) R/t = 3.5 and (b) R/t = 14.0...... 55

4.3 Eﬀect of back force on springback for samples with diﬀerent orientation: (a) ∆θ and (b) 1 ...... 56 Ra

4.4 Dependence of springback angle and anticlastic curvature on specimen width: (a) Fb = 0.5 and (b) Fb = 0.9...... 58

4.5 Comparison of the simulation results with the experimental data: (a) the springback angle ∆θ, and (b) the unloaded anticlastic curvature 1 . 61 Ra

4.6 Comparison of the simulated springback angles using (a) the Barlat’96 and 2000 yield functions and (b) the Geng-Wagoner model and the mNLK law...... 62

4.7 Comparison of the simulation and the experimental data for strips with various widths using Fb = 0.5: (a) the springback angle, and (b) the anticlastic curvature...... 64

4.8 Comparison of the simulated springback angles using diﬀerent elements for (a) Fb = 0.5 and (b) Fb = 0.9...... 64

4.9 Simulated springback angles using shell and solid elements...... 66

5.1 (a) An anticlastic surface, and (b) a synclastic surface...... 70

5.2 Schematics of the draw-bend test and an unloaded specimen...... 72

xiv 5.3 Simulated springback angle using various assumptions...... 75

5.4 Rectangular coordinate system for plate bending problems...... 77

5.5 Closed-form solution for: (a) normalized anticlastic deﬂection, and (b) normalized transverse stress...... 81

5.6 Anticlastic deﬂection and stress ratio for an elastic material with β = 5. 83

5.7 Anticlastic deﬂection and stress ratio for elastic material with β = 100. 84

5.8 Variation of the anticlastic factor φ with β...... 86

5.9 Bending an initially curved plate...... 87

5.10 Maximum anticlastic deﬂection from ﬁnite element simulation. . . . . 88

5.11 Error analysis for using parabolic function...... 90

5.12 Eﬀect of the back force on: (a) springback angle and (b) unloaded anticlastic curvature. Lines are FEM simulation results and markers are experimental data...... 92

5.13 Variation of the maximum anticlastic deﬂection with the front pulling force in the draw-bend test: (a) loaded and (b) unloaded...... 93

5.14 Moment and normalized section moment of inertia by ﬁnite element simulation...... 94

5.15 Springback angle and anticlastic curvature from draw-bend test for (a) Fb = 0.5 and (b) Fb = 0.9...... 95

5.16 Compare simulation and measurement for Fb = 0.5 case: (a) springback angle and (b) anticlastic curvature...... 96

5.17 Unbending and springback analysis for draw-bend test...... 98

5.18 Comparison of simulated and analytically predicted maximum anticlastic deﬂections for draw-bend tested samples after springback: (a) R/t = 3.5 and (b) R/t = 14.0...... 100

xv 5.19 Comparison of measured, simulated and analytically predicted anticlastic proﬁles for draw-bend tested samples: (a) Fb = 0.4 and (b) Fb = 0.8...... 100

5.20 Simulated anticlastic proﬁles of draw-bend tested samples: (a) for various back forces and (b) various widths at Fb = 0.5...... 101

5.21 Variation of Searle’s parameter β with (a) back force and (b) specimen width...... 102

6.1 Optical micrographs revealing grain structures...... 109

6.2 Uniaxial stress-strain curves for the tested aluminum alloys...... 110

6.3 Draw-bend specimens: (a) schematic geometry before and after springback; (b) tracings at various times following forming for 6022-T4 aluminum...... 112

6.4 Variation of measured time-independent springback angle on: (a) back force; (b) tool radius...... 113

6.5 Schematic of the tension/unloading test: (a) general load path, (b) detail of unloading region...... 114

6.6 Schematic of compression/tension/unloading test: (a) specimen geometry and stabilization ﬁxture; (b) general load path...... 115

6.7 Variation of simulated front force with back force for various R/t ratios.117

6.8 Eﬀect of bending radius and back force on residual stress: (a) through- thickness residual stress for various R/t at Fb = 0.5; (b) variation of maximum tensile and compressive residual stresses with back force for three R/t ratios...... 118

6.9 Change of springback angle with time after forming: (a) 2008-T4, (b) 5182-O, (c) 6022-T4, (d) 6111-T4. Multiple tests are diﬀerentiated by open and closed markers. Slopes shown are in degree/log(s)...... 120

6.10 Measured time-dependent springback angles for6022-T4 from two studies (Carden, 1996, and this work)...... 123

xvi 6.11 Room temperature creep curves for 6022-T4: (a) steady state creep law; (b) primary creep law...... 125

6.12 Room temperature creep behavior for 6022-T4 and DQSK: (a) creep strain; (b) creep rate...... 126

6.13 Abaqus simulation results at three stages: (a) through-thickness stress; (b) Comparison of measured and simulated time-dependent springback. 129

6.14 Eﬀect of creep law parameters on simulated time-dependent springback: (a) stress exponent N; (b) strength parameter K...... 130

6.15 Anelastic strain after uniaxial tension...... 132

6.16 Anelastic strain after unloading from compression–tension test: (a) 6022-T4; (b) DQSK steel...... 134

6.17 Strain paths in draw-bend test: (a) locations of 5 through-thickness integration points (IP); (b) change of strain paths...... 135

A.1 A ﬁrst guess for backward Euler algorithm...... 148

xvii CHAPTER 1

INTRODUCTION

Special draw-bend tests are utilized to study both the static and time-dependent springback of high-strength 6022-T4 aluminum alloy sheet. The main results are reported in Chapters 4, 5 and 6, with conclusions following in Chapter 7.

In Chapter 2, various yield functions are brieﬂy reviewed, with focus on the

Barlat’s family of yield criteria that are used for static springback simulation after the draw-bend test for 6022-T4 aluminum sheets. The Barlat’96 and Barlat 2000 yield functions can accurately describe the planar anisotropy, i.e., the angular variation of yield strength and r-value within the sheet plane.

Chapter 3 is mainly concerned with the numerical implementation of material constitutive models into ﬁnite element code, Abaqus/Standard, through user subroutines. After reviewing the return mapping algorithm with forward, backward and semi-backward integration methods, a nonlinear kinematic hardening model with multiple back stresses (mNLK ) is discussed. The combination of Barlat’s anisotropic yield functions and mNLK model ensures accuracy when modeling the anisotropic yielding and non-isotropic hardening behavior of 6022-T4 aluminum sheets.

In Chapter 4, the draw-bend device and test procedure are ﬁrst presented. Alu- minum sheets of 50mm wide are draw-bend tested, with the normalized sheet tension

1 varying from 0.1 to 1.1 and two tool radii: 3.2mm and 12.7mm. Finite element simulations are also conducted using Barlat’s yield functions and anisotropic hardening models (mNLK and a modiﬁed hardening law by Geng and Wagoner). The mNLK model parameters are calibrated from reverse ﬂow curves of in-plane tension/compression tests. Draw-bend test results for specimens with widths ranging from 12mm to 50mm are also reported, with 12.7mm tool radius and two selected back forces of 0.5 and 0.9. The role of persistent anticlastic curvature on springback is discussed. A manuscript to be submitted to the International Journal of Plasticity is in preparation.

Chapter 5 is dedicated to the study of anticlastic curvature that appeared during the forming step in the draw-bend test, but persisted after unloading for certain test conditions. A closed-form solution, derived from the classical large deformation theory of bending elastic plate, is employed to explain the correlation of the persistent anticlastic curvature and forming conditions. It is realized that the anticlastic surface after springback deviates from a circular shape for small tensions. This transition is determined by the interplay between specimen geometry (W/t) and the curvature of the curled region of unloaded strip (R ), via a dimensionless parameter β = W 2 . The x Rxt occurrence of the persistent anticlastic curvature corresponds to a critical β-value of

10–15. A manuscript of this work has been submitted to the International Journal of

Solids and Structures.

The time-dependent springback in draw-bend tested aluminum sheets is investigated in Chapter 6. The short-time response is linear on log(time) scale for times up to a few months; after that the kinetics are slower and the time-dependent springback

2 angle eventually saturates. Simulation based on residual stress driven creep under- estimates the experimental results by a factor of 2, albeit the qualitative agreement.

Anelastic strains after unloading from uniaxial tension and compression/tension tests are measured for both 6022-T4 aluminum and two forming grade steels. The fast kinetics of anelasticity suggest that it is not likely the dominant mechanism for long- term response, but it may contribute to the short-time behavior. A manuscript of this work has been submitted for publication in the International Journal of Plasticity.

Chapter 7 summarizes the conclusions from Chapters 3 to 6.

3 CHAPTER 2

REVIEW OF YIELD CRITERIA FOR SHEET METAL

The accuracy of sheet metal forming and springback simulation depends not only on the forming conditions (friction, tool and binder geometry), but also on the choice of the material constitutive models and their numerical implementations into ﬁnite element programs. Among these factors, the material constitutive law plays an important role in describing the mechanical behavior of sheet metals, because it is essential to obtain the accurate stress distribution in a formed part in order to correctly predict springback.

Two diﬀerent approaches have been used in sheet metal forming simulations, which are based on either crystal plasticity or phenomenological models. In the polycrystalline plasticity model, a sheet sample is treated as an aggregate of single crystals with preferred orientation distribution (i.e., crystallographic texture). Plastic deformation is then described by the discrete dislocation slips that take place on speciﬁc lattice planes and along particular crystallography directions. This method is very powerful, because texture evolution during plastic deformation can be considered.

However, it requires a huge amount of computation time to simulate any practical sheet metal forming process [1].

4 In this study, the second approach is employed. Within the framework of continuum plasticity theory, plastic ﬂow occurs when a yield criterion is satisﬁed. In stress space, the initial yielding can be described by a smooth, continuous surface,

F (σij) = 0, which deﬁnes the boundary between the elastic and plastic domains. All the points inside the yield surface represent elastic deformation, whereas stress points belonging to the surface are related to plastic states. Compared to the crystal plasticity model, this approach has a simpler mathematical form, and thus it is more feasible for ﬁnite element programming. In addition, only a small number of parameters are needed, and they can be easily determined from simple experiments, such as uniaxial tensile tests.

Theoretical considerations and experimental evidence have shown that the mathematical form of the yield surface is subject to some restrictions. For example, Bridg- man’s work [2] demonstrated that the hydrostatic pressure did not cause plasticity for metals, i.e., yielding is pressure independent. As a result, the yield function can be expressed as F (I2,I3) = 0, where I2 and I3 are the second and third invariants of the stress tensor. On the other hand, Drucker’s postulate [3] stated that a yield surface must be convex to ensure the uniqueness of plastic strain rate for a given stress state. An associated ﬂow rule was also implied from this postulate for stable

(non-softening) materials, so that the direction of the plastic strain rate is normal to the yield surface: ∂F ²˙ p = λ˙ (2.1) ∂σ where λ˙ is a plastic multiplier parameter. In other words, a yield surface also acts as a plastic potential from which the plastic ﬂow direction is derived.

5 2.1 Introduction to Plastic Anisotropy

Sheet metals naturally exhibit mechanical anisotropy because of the preferred grain orientation. The detailed crystallographic texture is determined by the thermo- mechanical manufacturing history (e.g., hot/cold rolling and annealing). Typically, sheet metals are orthotropic, with mirror symmetry axes aligned with the sheet rolling

(RD), transverse (TD) and normal (ND) directions, as shown in Figure 2.1 [4].

z y σ x Y (TD) Z ND σ ( ) x

X(RD)

Figure 2.1: Schematics of coordinate systems of a sheet (X-Y -Z) and a tensile sample (x-y-z) cut at an angle θ at the sheet rolling direction.

To characterize the plastic anisotropy of the sheet, the Lankford parameter or the plastic strain ratio is generally adopted [5]. The plastic strain ratio, also called r-value, is deﬁned as follows: p ε2 rψ = p , (2.2) ε3 or more precisely by p dε2 rψ = p (2.3) dε3

6 p p where ε2 and ε3 are true plastic strains in the sample width and thickness directions, respectively. The r-values can be measured by uniaxial tension tests for samples at diﬀerent angles to the sheet rolling direction, Figure 2.1.

In practice, it is quite diﬃcult to precisely measure small changes of the sheet

p p p thickness. Since plastic deformation conserves volume, i.e., ε1 + ε2 + ε3 = 0, it is convenient and more accurate to calculate r-value alternatively:

p ε2 rψ = − p p (2.4) ε1 + ε2

p while ε1 is the plastic strain in the longitudinal direction.

The plastic strain ratio has a profound eﬀect on sheet formability. It is a measure of the resistance to thinning or localized deformation, which usually precedes failure in sheet metal forming. Therefore, high r-value is desirable to achieve good formability.

Usually, r-value varies with orientation in the sheet plane, and it is also a function of plastic strain. An averaged r-value is deﬁned as

r + 2r + r r¯ = 0 45 90 (2.5) 4

which closely correlates to the deep drawability of sheet metals [4]. Here, r0, r45 and

r90 designate the r-value measured from samples cut at 0°, 45°and 90° from the sheet rolling direction. On the other hand, the in-plane variation of the r-value can be evaluated by r − 2r + r ∆r = 0 45 90 (2.6) 2

∆r is a measure of the planar anisotropy, and it relates to the earing proﬁle in deep- drawn products [4].

7 2.2 Thermodynamic Framework of Elasto-plastic Materials

In small deformation theory, the total strain is additively decomposed into elastic and plastic parts:

² = ²e + ²p (2.7)

For isotropic elasticity, the generalized Hooke’s law states that the Cauchy stress tensor σ is proportional to the elastic strain:

σ = Cel : ²e (2.8) where Cel is a fourth order isotropic tensor. It can be expressed as:

Cel = 2GIdev + K1 ⊗ 1 (2.9)

The superscript dev denotes the deviatoric part of a tensor. For example,

1 Idev = I − 1 ⊗ 1 (2.10) 3 where I and 1 are the fourth and second identity tensors respectively:

(1)ij = δij (2.11) 1 (I) = (δ δ + δ δ ) (2.12) ijkl 2 ik jl il jk where δij is the Kronecker-Delta symbol: ½ 1 if i = j δ , (2.13) ij 0 otherwise

The material constants, G and K, are the shear and bulk moduli respectively.

E They are related to the Young’s modulus E and Poisson’s ratio ν through G = 1+ν

E and K = 3(1−2ν) .

8 When the elastic limit stress is exceeded, plastic deformation occurs. The onset and continuation of plastic ﬂow is governed by a yield function. In principal stress space, the yield function deﬁnes the boundary of an elastic region in which plastic deformation is absent. The stress stays on the yield surface after plasticity occurs.

All the possible plastic stress states constitute a hypersurface in the stress space. If the associated ﬂow rule is adopted, this surface also acts as a plastic potential, from which the plastic strain rate is derived. Mathematically, a yield surface can be deﬁned as follows

f(σ; α,R) = σ (σ − α) − [σ + R(q)] = 0 (2.14) |{z} eq | 0 {z } V σy The motion of the yield surface (kinematic hardening) is described by the translation of its center, represented by the back stress tensor α. The evolutionary law of the back stress will be discussed in the last section of this chapter. The change of yield surface size (isotropic hardening) is indicated by the change of the yield strength,

σy. The initial yield strength (when q = 0) is σ0, while R represents the isotropic expansion of the yield surface, as graphically shown in Figure 2.2.

There are three commonly used isotropic hardening laws in the literature [6]:   hisoq linear hardening R(q) = Aqn power law hardening  −mq R∞(1 − e ) Voce’s law with saturation

The scalar quantity, q, describes material hardening due to the microstructural change in the course of plastic deformation, such as the increase of dislocation density. Usu- ally, q is the accumulated equivalent plastic strain, and it is deﬁned as Z Z r t t 2 q(t) = k²˙ pkdτ = ²˙ p : ²˙ p dτ (2.15) 0 0 3

Generally q(t) 6= k²p(t)k, except for a monotonic loading path.

9 σ3 Subsequent Yield Surface (a) σy (b) (c) α α

σ0 Initial Yield Surface

σ σ2 1

Figure 2.2: Schematics of the yield surface evolution in the stress space: (a) with kinematic and isotropic hardening, (b) with kinematic hardening only, and (c) with isotropic hardening only.

For an isothermal deformation process, where no heat exchange is involved, the

Helmholtz’s free energy per unit mass is expressed as [7]: Z 1 a q ρψ(²e, θ, q) = ²e : Cel : ²e + θ : θ + R(l)dl (2.16) 2 2 0 where, ²e, θ and q are strain-like internal variables which are macroscopical measures of an irreversible process, such as plastic deformation; σ, α and R are their associated thermodynamic forces (stress-like variables). ρ is the material’s density. The thermodynamic quantities are related through the following equations of state:

∂ψ σ = ρ = Cel :(² − ²p) (2.17) ∂²e ∂ψ α = ρ = aθ (2.18) ∂θ ∂ψ R = ρ = R(q) (2.19) ∂q

When the temperature ﬁeld is static during deformation, the Clausius-Duhem inequality reduces to

σ : ²˙ − ρψ˙ ≥ 0 (2.20)

10 where the equality sign only holds for reversible processes. Substituting Equations 2.17,

2.18 and 2.19 into the above inequality, a mechanical dissipation function can be de-

ﬁned as:

p ˙ Dmech(σ; α,R) = σ : ²˙ − ρ(Rq˙ + α : θ) (2.21)

Hill’s principal of maximum dissipation [8] states that among all the admissible stresses and internal variables, S = {(σ∗; α∗,R∗)|f(σ∗; α∗,R∗) ≤ 0}, the exact solution (σ; α,R) maximizes the mechanical dissipation for any given (²˙ p, q˙):

∗ ∗ ∗ Dmech(σ; α,R) = max {Dmech(σ ; α ,R )} (2.22) (σ∗;α∗,R∗) ∈ S

Equation 2.22 describes a constrained optimization problem. Using the standard procedure [9], Equation 2.22 is ﬁrst transformed into a minimization problem by changing the sign of the mechanical dissipation function. Then, a Lagrangian functional is constructed by adding function Dmech and the constraint equation f together:

˙ ˙ L(σ, λ) = −Dmech + λf(σ; α,R) (2.23) where λ˙ is a Lagrangian multiplier.

The exact solution will make the functional L stationary with respect to any variations of σ and λ˙ . Therefore, the associated ﬂow rule and the so-called Kuhn-

Tucker conditions are arrived at after optimization:

∂L ∂f = −²˙ p + λ˙ = 0 (2.24) ∂σ ∂σ f(σ; α,R) ≤ 0 (2.25)

λ˙ ≥ 0 (2.26)

λf˙ = 0 (2.27)

11 The ﬁrst equation states that the direction of plastic ﬂow is normal to the yield surface

(i.e., the normality rule), while the last two equations establish the loading-unloading criterion. The scalar quantity, λ˙ , is also called the plastic multiplier, and its sign is determined by the plastic yielding and loading-unloading conditions:

λ˙ > 0 if f = 0 and f˙ = 0 (2.28)

λ˙ = 0 if f = 0 and f˙ < 0, or f < 0 (2.29)

One additional conclusion can also be derived from the principle of maximum dissipation, that is, the yield surface must be convex [8].

The evolution laws for the internal variables can be derived from a thermodynamic potential b F (σ; α,R) = f(σ; α,R) + α : α (2.30) 2a

Therefore,

∂F ∂f ²˙ p = λ˙ = λ˙ = λ˙ a (2.31) ∂σ ∂σ ∂F b θ˙ = −λ˙ = ²˙ p − λ˙ α (2.32) ∂α a ∂F q˙ = −λ˙ = λ˙ (2.33) ∂R

From the second equation above, and assuming that α = aθ, the evolution of the back stress becomes:

α˙ = a²˙ p − bλ˙ α (2.34)

Equation 2.34 is the Armstrong-Frederic type of evolution law [10]. A generalization was made by Chaboche to include multiple back stress tensors in order to better describe cyclic plastic deformations [11,12], such as ratcheting and cyclic creep [13].

12 Some earlier works did not have a recall term in the evolution equation for the back stress. For example, the following equations were used to describe the change of the back stress [14,15]: ½ c d²p (Prager) α˙ = p (2.35) dµ (σ − α) (Ziegler) where cp is a material constant, and dµ is a parameter that can be determined from the yield condition. The diﬀerence between these two models lies in the direction of the back stress. In Prager’s model, i.e, the linear kinematic hardening, the yield surface moves in the direction of the plastic strain rate. As pointed out by Ziegler [15],

Prager’s model does not produce consistent results for 3D and 2D problems. A modiﬁed model was proposed to overcome this drawback, so that the yield surface center translates in the radial direction of the yield surface [15].

nβ

σβ 1 2 3 n dα (1). Prager σ (2). Ziegler (2). Ziegler σ − α α (3). Mroz

O fn =0

fn+1 =0

Figure 2.3: Evolution of the back stress

Figure 2.2 compares the diﬀerent evolution laws for the back stress tensor α. Case

1 and 2 correspond to Prager’s and Ziegler’s rule of back stress evolution respectively.

13 For the two-surface plasticity theory [16] and multi-layer theory [17], the evolution of α is determined by the relative position of a bounding surface with respect to the yield surface. As demonstrated by Case 3 in Figure 2.2, the direction of the yield surface motion is parallel to a vector that connects the current stress σ (on the yield

surface) and a mapping stress σβ (on the bounding surface). At σβ , the bounding surface has the same outward normal as the yield surface does at σ, Figure 2.2. In this case, fn = 0 and fn+1 = 0 are the yield (loading) and the bounding surfaces, respectively.

2.3 Isotropic Yield Criteria

This section brieﬂy summarizes the commonly used isotropic yield functions. Al- though they cannot describe the plastic anisotropy of sheet metals, they are founda- tions upon which anisotropic yield criteria are constructed.

2.3.1 Von Mises and Tresca Yield Criteria

The most commonly used yield criterion was proposed by von Mises in 1913 [18].

It stated that plastic deformation began when the elastic distortion energy reached a critical value. Usually, a von Miese equivalent stress is deﬁned as

2 2 2 2 2 2 2 2σeq = (σ11 − σ22) + (σ22 − σ33) + (σ33 − σ11) + 6(σ12 + σ23 + σ31) (2.36)

Plastic deformation occurs when σeq exceeds the elastic limit stress. For an isotropic material, σeq is also the yield stress in a uniaxial tensile test. When expressed in the principal stress space, Equation 2.36 becomes:

2 2 2 2 2σeq = (σ1 − σ2) + (σ2 − σ3) + (σ3 − σ1) (2.37)

14 In the case of plane-stress (σ3 = 0), it further reduces to

2 2 2 σeq = σ1 + σ2 − σ1σ2 (2.38)

An earlier yield criterion was based on the Tresca’s maximum shear principle [19].

The Tresca’s equivalent stress is

σeq = max {|σ1 − σ2|, |σ2 − σ3|, |σ3 − σ1|} (2.39)

For plane-stress state, Equation 2.39 becomes

2 2 2 σeq = (σ11 − σ22) + 4σ12 (2.40)

For many metallic materials, the experimentally measured yield loci fall between the predictions by von Miese and Tresca yield criteria. As shown in Figure 2.4, von

Miese yield function can ﬁt experimental data better than Tresca’s criterion for many ductile metals, such as aluminum, copper and mild steel [8].

2.3.2 Hosford’s Non-quadratic Yield Criterion

A more general, isotropic yield criterion with a non-quadratic form was proposed by Hosford [20]:

m m m m 2σeq = |σ1 − σ2| + |σ2 − σ3| + |σ3 − σ1| (2.41) where 1 ≤ m < ∞ to ensure convexity of the yield surface. With proper choice of the exponent m, Equation 2.41 can better ﬁt experimentally measured yield loci of sheet metals [21, 22]. It was found that m = 6 and m = 8 are suitable for BCC and

FCC polycrystal aggregates [4].

A 2D (plane-stress) version of the Hosford’s yield criterion is given by

m m m m 2σeq = |σ1| + |σ2| + |σ1 − σ2| (2.42)

15 1 Tresca von Mises 0.8 Mild steel Aluminum 0.6 Copper eq σ / 12

σ 0.4

0.2

Hill, 1950 00 0.2 0.4 0.6 0.8 1 σ /σ 11 eq

Figure 2.4: von Mises and Tresca yield loci compared with experimental data.

The yield locus describe by Equation 2.42 is plotted in the ﬁrst quadrant of the 2D principal stress space for various m, Figure 2.5. It is clear that the Hosford yield criterion reduces to the Tresca and von Mises yield criteria, when m = ∞ and m = 2 respectively. The von Miese yield locus is an ellipse that is circumscribed to the

Tresca polygon. When m increases from 2 to ∞, the resulted yield loci are bounded by the von Miese and Tresca yield locus. Larger m tends to increase the curvature of the yield surface at the uniaxial and biaxial stress states, but reduce the curvature near the plain-strain states, Figure 2.5.

2.4 Anisotropic Yield Criteria

In order to describe orthotropic plasticity of sheet metals, Hill ﬁrst proposed a quadratic yield function [23]. It has been widely used for sheet forming simulations,

16 m =2 (von Mises)

1 m = ∞ (Tresca) 0.8 eq

y /σ 0.6 2

σ m =8 0.4

0.2 m = 20

0 0.2 0.4 0.6x 0.8 1 σ1/σeq

Figure 2.5: Plane-stress isotropic yield functions.

because it can be applied to general stress states, and can be easily implemented into ﬁnite element codes. In fact, it is readily available from most commercial ﬁnite element codes, such as Abaqus [24].

2.4.1 Hill’s family of Anisotropic Yield Criteria

Hill’s anisotropic yield criteria is a generalization of the von Mises criterion [23,25].

The equivalent stress is deﬁned as

2 2 2 2 2 2 2 2σeq = F (σ22 −σ33) +G(σ33 −σ11) +H(σ11 −σ22) +2Lσ23 +2Mσ31 +2Nσ12 (2.43) where F , G, H, L, M and N are material constants. Obviously, Equation 2.43 reduces to the von Mises yield function when all coeﬃcients are set to unity. In case of plane-stress (σ23 = σ31 = σ33 = 0), Equation 2.43 reduces to

2 2 2 (G + H)σ11 − 2Hσ11σ22 + (H + F )σ22 + 2Nσ12 = 1 (2.44)

17 The coeﬃcients G, H and F can be determined from the r-values of the sheet metals measured at diﬀerent angles to the sheet rolling direction:

H H H 1 r = , r = , r = − (2.45) 0 G 90 F 45 F + G 2 while the determination of N requires shear test.

When the principal directions of the stress tensor are coincident with the anisotropic axes, i.e., σ12 = 0, Equation 2.44 becomes

2 2r0 r0(1 + r90) 2 2 σ1 − σ1σ2 + σ2 = σRD (2.46) 1 + r0 r90(1 + r0) where σ1 and σ2 are the principal stresses, and σRD is the yield stress in uniaxial tension for a sample parallel to the sheet rolling direction. Now, only three material

properties, i.e., the r-values r0 and r90, and the uniaxial tensile stress σRD, are needed to determine the yield surface. This makes the Hill’48 yield criterion easy and friendly to use in practice.

As a special case, normal anisotropy (or, planar isotropy) leads to r0 = r90 = r45 = r, and Equation 2.46 further reduces to

2r σ2 − σ σ + σ2 = σ2 (2.47) 1 1 + r 1 2 2 RD

At the balanced biaxial stress state (i.e., σ1 = σ2 = σb), the above equation leads to r 1 + r σ = σ =⇒ r > 1, σ > σ (2.48) b RD 2 b RD

Therefore, Equation 2.47 fails to describe the so-called anomalous behavior, which was commonly observed in aluminum alloys [26]. Many aluminum alloys have r- values less than unity, but their biaxial yield strengths are higher than the uniaxial tensile strength. Furthermore, it can be shown that s σ r (1 + r ) 0 = 0 90 (2.49) σ90 r90(1 + r0)

18 which always predicts that r0 > r90 if σ0 > σ90. Therefore, the Hill’48 cannot represent the so-called secondary order anomalous behavior either [27].

Hill’48 yield locus is always elliptic for the plane-stress case, because of its quadratic form. However, r-value can signiﬁcantly change the shape of the yield locus, Fig- ure 2.4.1. The eﬀect of increasing r-value gives larger biaxial tension yield stress, i.e., stronger thinning resistance. Meanwhile, the plane-strain state moves along the dashed line as indicated in Figure 2.4.1. It approaches the biaxial stress state as r-value increases.

r = 5 ε 1.5 d 2 = 0 r = 2 r = 1

eq r = 0.5 σ

/ 1 2

σ r = 0

0.5

0 0 0.5 1 1.5 2 σ σ 1/ eq

Figure 2.6: Eﬀect of r-value on the yield surface shape

To remedy the aforementioned drawbacks of the Hill’s 48 yield criterion, a more general general yield criterion with a non-quadratic form (Hill’79) was proposed [28]:

m m m m m σeq = f|σ2 − σ3| + g|σ3 − σ1| + h|σ1 − σ2| + a|2σ1 − σ2 − σ3| (2.50) m m + b|2σ2 − σ1 − σ3| + c|2σ3 − σ1 − σ2|

19 where m is a material constant. For planar anisotropy, four special cases can accom- modate the yield anomaly, but only the case IV assures convexity of the yield surface, with a = b = g = f = 0:

m m m σeq = c|σ1 + σ2| + h|σ1 − σ2| (2.51)

The constants c and h can be determined from uniaxial tension test along the sheet

h rolling direction: c = 2r + 1 and c + h = 1. Then, Equation 2.51 can be rewritten as

m m m 2(1 + r)σRD = |σ1 + σ2| + (1 + 2r)|σ1 − σ2| (2.52)

Now the ratio of the biaxial stress to the uniaxial stress can be computed: r σb m 1 + r = m−1 (2.53) σRD 2

As shown in Figure 2.4.1, this new yield criterion is capable of describing the anomalous yield phenomenon, if m < 2.

Hosford’s non-quadratic isotropic yield function can be extended to incorporate the plastic anisotropy [29]:

m m m m σeq = F |σ1 − σ2| + G|σ2 − σ3| + H|σ3 − σ1| (2.54)

The exponent m is related to the crystallographic texture: m = 6 and m = 8 are found to be feasible for BCC and FCC materials respectively. In Hill’s model, however, the exponent m is determined from experiments, and thus it is not necessarily an integer.

Since Hill’79 criterion can only be used when the principal axes of the stress tensor are aligned with the orthotropic symmetry axes, its application is somewhat limited.

A generalization of Hill 1979 yield criterion (case IV) was proposed to include shear terms [30]. Other extensions were also made [25, 31, 32], but they are not discussed

20 3 m=1.5 m=2 2.5 m=2.5

σ =σ 1.5 b RD

Stress ratio 1

0.5

0 0 0.5 1 1.5 2 2.5 3 r-value

Figure 2.7: Eﬀect of m on stress ratio σb . σRD

here. There are other yield criteria appeared in the literature, such as a polynomial yield function with fourth order [33, 34], an empirically constructed yield function based on the Bezier interpolation [35], and a yield criterion that was expressed in polar coordinates [36–38]. A more detailed review of the various yield functions can be found elsewhere [39].

2.4.2 Karaﬁllis and Boyce’s Yield Criterion

A generic anisotropic yield function was proposed by superposing two stress potentials with diﬀerent weights [40]:

m m m φ1 = |S1 − S2| + |S2 − S3| + |S3 − S1| (2.55) 3m φ = (|S |m + |S |m + |S |m) (2.56) 2 1 + 2m−1 1 2 3

φ = cφ1 + (1 − c)φ2 (2.57)

21 where c is a weight factor, and Si (i = 1-3) are the principal values of a transformed stress tensor S = L : σ. The fourth order tensor L is determined by the material symmetry and mechanical anisotropy. The tensor L is symmetric:

Lijkl = Ljikl = Ljilk = Lklij (2.58)

Unlike other works, this yield criterion (K&B’93) is not limited to describe plastic orthotropy. It can be used for materials with lower symmetries, such as monoclinic and trigonal, by properly choosing the the components of L. For orthotropic materials

(such as metal sheets), there are only 6 non-zero components in L:   (c2 + c3)/3 −c3/3 −c2/3 0 0 0    −c3/3 (c3 + c1)/3 −c1/3 0 0 0     −c2/3 −c1/3 (c1 + c2)/3 0 0 0  [L] =    0 0 0 c4 0 0   0 0 0 0 c5 0  0 0 0 0 0 c6

If c = 0, the K&B’93 yield function reduces to the Barlat’91 yield function, as will be discussed next. The yield locus predicted by the K&B’93 is in very good agreement with experimental data, as well as theoretic calculations based on the Bishop-Hill theory [41].

2.4.3 Barlat’s Family of Anisotropic Yield Criteria

In 3D stress space, the Barlat’91 yield surface is given by [42]:

f = σeq − σy = 0, (2.59)

The equivalent stress, σeq, is calculated from a stress potential:

m m m m φ = 2σeq = |S1 − S2| + |S2 − S3| + |S3 − S1| . (2.60)

22 where m is a material constant. Usually m = 6 and m = 8 are used for BCC and FCC metals [42]. Si (i=1,2,3) are the eigenvalues of the so-called transformed deviatoric stress tensor, S:   S11 S12 S13 [S ] =  S21 S22 S23  (2.61) S31 S32 S33 The components of S are c(σ − σ ) − b(σ − σ ) S = 11 22 33 11 11 3 a(σ − σ ) − c(σ − σ ) S = 22 33 11 22 22 3 b(σ − σ ) − a(σ − σ ) S = 33 11 22 33 33 3

S12 = hσ12

S23 = gσ13

S31 = fσ23

Six material parameters (a, b, c, f, g and h) are needed for the Barlat’91 yield function. Constants a, b and c can be determined from uniaxial tensile tests along the RD, TD and ND directions; while shear tests along these directions are required

for f, g and h. For plane-stress case (σ3 = 0), only a, b, c and h are relevant, and tension in 45° can be used instead of in-plane shear test to determine h. In practice, a biaxial tensile test is utilized instead of compression in the thickness direction for sheet metals, although a few sheet samples can be stacked together for compression test [43].

The Barlat’91 yield function works better than Hill’48 criterion for aluminum sheets.

However, some discrepancy was found between experimental data and the model predictions for some aluminum-magnesium sheets [43].

An improved version, the Barlat’94 yield criterion, was proposed [43]:

m m m φ = α1|S2 − S3| + α2|S3 − S1| + α3|S1 − S2| (2.62)

23 where Si are the principal values of the transformed stress tensor S. The coeﬃcients,

αk (k = 1–3), are calculated as follows:

2 2 2 αk = αxp1k + αyp2k + αzp3k (2.63)

where pij (i, j=1–3) are the direction cosines between the principal axes of anisotropy and the principal axes of S.

A further improvement was achieved by making αk depend on the orientation of the principal axes [43]:

2 2 αx = αx0 cos (2β1) + αx1 sin (2β1) (2.64)

2 2 αy = αy0 cos (2β2) + αy1 sin (2β2) (2.65)

2 2 αz = αz0 cos (2β3) + αz1 sin (2β3) (2.66)

where βi (i = 1–3) are the angles between the material symmetry axes and the principal axes of S: ½ y · 1 if |S1| ≥ |S3| cos β1 = y · 3 if |S1| < |S3| ½ z · 1 if |S1| ≥ |S3| cos β2 = z · 3 if |S1| < |S3| ½ x · 1 if |S1| ≥ |S3| cos β3 = x · 3 if |S1| < |S3|

Here, {x, y, z} indicate the orthogonal coordinate frame attached to the material, i.e., the symmetry axes of the sheet; while {1, 2, 3} are the principal directions of the transformed stress tensor S. For plane-stress, the non-zero components of the

24 transformed stress tensor S are:

c (σ − σ ) + c σ S = 1 11 22 2 11 11 3 c σ + c (σ − σ ) S = 1 22 3 11 22 22 3 −c σ − c σ S = 2 11 1 22 33 3

S12 = c6σ12

The coeﬃcients αk now become

2 2 αx = αx cos (2ψ) + αy sin (2ψ)

2 2 αy = αx sin (2ψ) + αy cos (2ψ)

2 2 αz = αz0 cos (2ψ) + αz1 sin (2ψ)

where αz0 = 1, and ψ is the angle between the 1-axis and the sheet rolling direction

x. In total, 7 parameters are needed in the Barlat’96 yield function: c1, c2, c3, c6, αx,

αy and αz1. These material constants can be determined from three uniaxial tensile tests in the 0°, 45°and 90° directions, plus a hydraulic bulging test.

One shortcoming of the Barlat’96 yield function is that there is no guarantee of convexity for this yield surface [43]. Therefore, numerical diﬃculties were encountered in ﬁnite element analysis. To remedy this drawback, a new plane-stress yield criterion has been recently formulated (i.e, the Barlat 2000 yield function), using two stress potentials φ1 and φ2 [44]:

φ + φ σm = 1 2 (2.67) eq 2 m φ1 = |X1 − X2| (2.68)

m m φ2 = |2Y2 + Y1| + |2Y1 + X2| (2.69)

25 where Xi and Yi (i=1,2) are the principal values of the following transformed stresses:       0 0 X11 C11 C12 0 s11    0 0    X22 = C21 C22 0 s22 0 X12 0 0 C33 s12       00 00 Y11 C11 C12 0 s11    00 00    Y22 = C21 C22 0 s22 00 Y12 0 0 C33 s12 where sij (i, j = 1, 2) are the deviatoric stress components in plane-stress. Since φ1

0 only depends on |X1 − X2|, matrix [C ] has only 3 independent components. Usually,

0 0 C12 = C21 = 0 is imposed [45]. It is proved that this yield function is convex [45]. In addition, its numerical implementation into ﬁnite element code is easier than the

Barlat’96 function. The 8 material parameters can be determined from the uniaxial tensile test and the balanced biaxial tension test.

2.4.4 Anisotropic Yielding of 6022-T4 Aluminum

Some of the previously discussed yield functions are used to describe the initial plastic anisotropy for 6022-T4 aluminum sheet. Figure 2.8 compares the predicted yield strengths and r-values with the experimental data for samples at diﬀerent orientations with respect to the sheet rolling direction.

The coeﬃcients of Hill’48 yield function are calibrated from the measured r-values in unaxial tensile tests. As shown in Figure 2.8, the predicted yield stresses cannot match the experimental results. On the other hand, the parameters of the Barlat’91

(2D version) yield function are calculated from the yield stresses, but the predicted r- values do not agree with the measured results. Calculated yield strengths and r-values using the Barlat’96 and 2000 yield functions are in very good agreement with the experimental data, because more parameters are ﬁtted from both the measured yield

26 1.08 1 6022-T4

0.8 von Mises 1.04 von Mises 0.6 1 r-value 0.4 Barlat'91 Barlat'91 Barlat'96 Barlat'96 0.96 Barlat 2000 Barlat 2000 0.2 Normalizedyield stress Hill'48 Hill'48 Exp. Exp. (Barlat, 1997) 0.92 0 0 15 30 45 60 75 90 0 15 30 45 60 75 90 Angle from RD (degree) Angle from RD (degree) (a) (b)

Figure 2.8: Comparison of the calculated and experimental results for 6022-T4: (a) normalized yield stress, and (b) r-value.

strengths and r-value. For the 6022-T4 aluminum sheet, the coeﬃcients of Hill’48,

Barlat’91, Barlat’96 and Barlat 2000 yield functions are listed in Table 2.1 [42–44].

27 Hill’48 FGHLMN —— — 0.531 0.775 0.910 — — — — — Barlat’91 a b c f g h —— — 1.0765 0.9187 1.0671 1.0 1.0 1.0772 — — Barlat’96 c1 c2 c3 c4 αx αy αz0 αz1 — 0.918 0.825 1.042 1.031 2.863 3.139 1.0 0.565 0 0 0 00 00 00 00 00 †Barlat 2000 C11 C22 C33 C11 C12 C21 C22 C33 — 0.6237 0.690 0.9594 0.7172 -0.3987 -0.3987 0.7463 1.1828 † These coeﬃcients are computed from the matrices [L0] and [L00] given in [45].

Table 2.1: Parameters for Barlat’91, ’96 and 2000 yield functions.

28 CHAPTER 3

INTEGRATION OF PLASTIC RATE EQUATION

In this chapter, the numerical implementation of Barlat’s anisotropic yield functions and a nonlinear kinematic hardening model with multiple back stress components (denoted as mNLK hereafter) are discussed. The general procedure to inte- grate the plastic rate equation is ﬁrst reviewed. Three algorithms, i.e., the forward, backward and semi-backward integration methods are explained. The procedure of implementing material constitutive models into a commercial ﬁnite element package

ABAQUS/Standard is discussed. To verify the implementation, tension/compression tests are simulated and the simulated ﬂow curves after a strain path reversal are compared with the experimental results for 6022-T4. The proposed algorithm is mathematically simpler than the previously developed Geng-Wagoner hardening law that was based on a two-surface plasticity model [46]. The mNLK model only requires six ﬁtting parameters, which can be derived from the uniaxial compression/tension

(C-T) or tension/compression (T-C) tests.

3.1 Introduction to Stress Update Schemes

For small strain deformation, rate-independent elasto-plasticity with an associated

ﬂow rule, the task of stress update scheme is to solve the the following diﬀerential

29 equations, using an appropriate integration method:

²˙ = ²˙ e + ²˙ p (3.1)

σ˙ = Cel : ²˙ e (3.2) ∂f ²˙ p = λ˙ a = λ˙ (3.3) ∂σ V˙ = λ˙ H(σ; V ) (3.4)

The above governing diﬀerential equations are subject to two constrains:

f(σ; V ) = σeq(σ − α) − σ0 − R(q) = 0 (3.5)

λ˙ ≥ 0, f ≤ 0 and λf˙ ≤ 0 (3.6) where V = {α,R} denotes the internal variables collectively, and H specifies the evolution laws of the internal variables. The first constraint states that the yield condition is always satisfied, while the second one is the loading-unloading criterion.

The evolution of the internal variables, given by Equation 3.4, can be rewritten in component form,

˙ α˙ = λhkin(σ, α) (3.7)

R = R(q) (3.8) where the ﬁrst equation speciﬁes a general kinematic hardening, and the second one is the isotropic hardening. In the stress space, Equations 3.7 and 3.8 describe the translation and expansion of the yield surface. The commonly used isotropic harden-

n ing laws are the linear hardening (R = hiso q), the power law hardening (R = R0 q )

−mq and the Voce’s hardening law with a saturation value (R = R∞ − R∞e ). When the associative ﬂow rule is used, such as in this study, the yield surface normal, a, is also the direction of the plastic ﬂow.

30 Due to the nonlinear nature of the governing diﬀerential equations, an incremental solution procedure is usually required. Let ∆(•) , (•)n+1 −(•)n denotes an increment over a time interval [tn, tn+1]. Then, the above equations can be incrementally solved © ª p p to obtain σn+1, ²n+1, ²n+1, V n+1 , if {σn, ²n, ²n, V n} and the total strain increment ∆² are known. For implicit ﬁnite element method, the so-called material Jacobian

alg δσ (algorithmic or consistent stiﬀness matrix, i.e., C = δ² |n+1) is also required in order to formulate the element stiﬀness matrix: Z XNIP e T alg e ¡ T ep ¢ K = B C BdV ' V wi B C B i (3.9) e V i=1 where B is the strain-displacement matrix (² = Bu), wi is the weight at the Gaus- sian integration point, and V e is the element volume. As pointed out by Simo and

Taylor [47], the use of consistent stiﬀness matrix ensured a quadratic rate of convergence, when the Newton-Ralphson method was used at structural level to solve the nodal displacement u from the global equilibrium equation Ku = f ext.

Based on how the elasto-plastic rate equations are discretized and integrated, there are three major integration methods, namely the forward, backward and semi- backward Euler integration methods, which are summarized in the following sections.

3.2 Forward Integration

The rate equations are ﬁrst discretized in the time domain, and the current incremental changes in the plastic strain and plastic multiplier are approximated as

p p ˙ ∆² ≈ ²˙ n∆t and ∆λ ≈ λn∆t, within the current time increment ∆t = tn+1 − tn. Notice that the rates at the beginning of the increment are used.

The plastic consistency condition can be derived from Equation 3.5 and 3.6:

˙ ˙ f = fσ : σ˙ + fV · V = 0 (3.10)

31 where the subscripts σ and V denote the partial derivatives with respect to the stress and the internal variables respectively. Following a standard derivation [48], the plastic multiplier λ˙ is calculated as

el ˙ a : C : ²˙ λ = el (3.11) a : C : a − fV · H

Then, the strain and stress are updated using the forward Euler integration method:

²n+1 = ²n + ∆² (3.12)

p p ²n+1 = ²n + ∆λan (3.13)

V n+1 = V n + ∆λH (σn; V n) (3.14)

el p σn+1 = σn + C : (∆² − ∆² ) (3.15)

Finally, the continuum tangent stiﬀness matrix Cep is obtained by substituting Equa- tion 3.11 back into the above equations:

el el ep el (C : a) ⊗ (C : a) C = C − el (3.16) a : C : a − fV · H

Since the yield condition is not enforced at the end of the current increment, i.e., fn+1 6= 0, the updated stress and internal variables deviate from the exact solutions.

Therefore, the forward Euler solution drifts away from the exact one, and an additional step is needed to correct this drift. However, the widely used return mapping algorithm, which is based on the backward Euler integration method, does not suﬀer from this shortcoming.

3.3 Backward Integration

In contrast to the forward method, the incremental quantities are calculated from rates at the end of the current increment in the backward method. That is, ∆²p ≈

32 p ˙ ²˙ n+1∆t and ∆λ ≈ λn+1∆t. In addition, the yield condition is always enforced at the end of the current increment, so that solution drifting is avoided:

²n+1 = ²n + ∆² (3.17)

p p ²n+1 = ²n + ∆λan+1 (3.18)

V n+1 = V n + ∆λHn+1 (σn+1; V n+1) (3.19)

el p σn+1 = σn + C : (∆² − ∆² ) (3.20)

fn+1 = σeq(σn+1 − αn+1) − σ0 − R(qn+1) = 0 (3.21)

p ˙ Since ²˙ n+1, αn+1 and λn+1 are unknown during the current time increment, an iterative solution procedure is required. According to the return mapping algorithm, which was firstly recognized by Simo and Taylor [47], the current stress is iteratively modified from an elastic trial value: el p σn+1 = σn + C : (∆² − ∆² ) (3.22) trial el = σn+1 − ∆λC : a Notice that the mapping direction is given by the flow vector a, and usually it is not constant during the iteration.

The return mapping algorithm consists of two steps, based on the methodology of operator splitting [47]:

1. Elastic Predictor

In this step, a trial stress is ﬁrst calculated, assuming that only elastic deforma-

p tion happens and all internal variables are frozen, i.e., ∆²n+1 = 0 and ∆q = 0:

trial el σn+1 = σn + C : ∆² (3.23)

2. Plastic Corrector

If the elastic trial stress lies inside the previous yield surface, then the elastic

33 trial stress is readily accepted as the ﬁnal solution. Otherwise, plasticity occurs

during the current increment, because the elastic trial stress exceeds the yield

trial trial stress, i.e., f = σeq(σn+1 − αn) − σ0 − R(qn) > 0. A plastic correction step is applied to restore the yield condition:

trial trial el σn+1 = σn+1 − ∆σn+1 = σn+1 − ∆λC : an+1 (3.24)

An iterative solution procedure is required to obtain the stress and internal

variables, since the ﬂow direction an+1 is also to be solved.

The return mapping algorithm is graphically shown in Figure 3.1. In the stress space, it can be envisioned as a geometric mapping by which the elastic trial stress is projected onto an updated yield surface. The projecting direction is the current yield surface normal, which is unknown in priori.

σtrial n+1 f trial > 0 el −∆λC : an+1 σ2

σn+1

σn

Elastic Region σ1

fn+1 =0

Figure 3.1: Schematics of return mapping algorithm in stress space.

34 In the plastic correction step, the equations are nonlinear with respect to the incremental plastic multiplier, ∆λ, while the total strain increment is kept constant. These nonlinear equations can be solved numerically by the Newton-Ralphson method. Let

χ(∆λ) = 0 denotes a nonlinear equation of ∆λ. An initial guess is ∆λ(0) = 0. After linearization using the ﬁrst order Taylor’s series expansion, the kth iterative change of ∆λ can be calculated as follows:

µ ¶(k) (k) (k) dχ (k) (k) χ χ + δλ = 0 =⇒ δλ = −¡ ¢ (3.25) d∆λ dχ (k) d∆λ Then, an update is made for the next iteration, ∆λ(k+1) = ∆λ(k) +δλ(k). The iteration ¯ ¯ ¯ δλ(k+1) ¯ procedure continues until convergence is achieved, i.e., δλ(k) ≤ TOLn where TOLn is a prescribed tolerance.

In the following context, the subscript n+1 will be omitted for quantities that are evaluated at the end of the current increment. After rearrangement, Equations 3.18,

3.19 and 3.21 become

p p Ψ = −² + ²n + ∆λa = 0 (3.26)

Υ = −V + V n + ∆λH = 0 (3.27)

f = 0 (3.28)

Both Ψ an Υ are functions of the iterative change of stress, ∆σ(k) and change of the internal variable, ∆V (k). Apply the Newton-Ralphson method: [48]: h¡ ¢ ¡ ¢ i (k) (k) (k) el−1 (k) (k) ∂a (k) (k) ∂a (k) (k) Ψ + δλ a + C : ∆σ + ∆λ ∂σ : ∆σ + ∂V · ∆V = 0 (3.29) h¡ ¢ ¡ ¢ i (k) (k) (k) (k) (k) ∂H (k) (k) ∂H (k) (k) Υ + δλ H − ∆V + ∆λ ∂σ : ∆σ + ∂V · ∆V = 0 (3.30) (k) (k) (k) (k) (k) f + fσ : ∆σ + fV · ∆V = 0 (3.31)

35 To solve for the unknowns ∆σ(k), ∆V (k) and δλ(k), the ﬁrst two equations are written in matrix form after some rearrangements: ½ ¾ h i−1 ∆σ(k) M(k) = −re(k) − δλ(k)ne(k) (3.32) ∆V (k) where · ¸ el−1 ∂a ∂a −1 C + ∆λ ∆λ [M] = ∂σ ∂V ∆λ ∂H −I + ∆λ ∂H ½ ¾ ∂σ ½ ¾ ∂V Ψ a re = , ne = Υ H

The stress and internal variable increments are then obtained: ½ ¾ ∆σ(k) = −M(k) : re(k) − δλ(k)M(k) : ne (k) (3.33) ∆V (k)

Substitute this into Equation (3.32), the kth iterative change of the plastic multiplier is obtained: (k) (k) e (k) (k) (k) f − ∂f : M : re δλ = (k) (3.34) ∂fe : M(k) : re(k) e where ∂f = {fσ, fV }. The plastic strain and internal variables are then updated:

−1 ²p(k+1) = ²p(k) − Cel : ∆σ(k) (3.35)

V (k+1) = V (k) + ∆V (k) (3.36)

∆λ(k+1) = ∆λ(k) + δλ(k) (3.37)

The complete algorithm is summarized in the following box [48].

36 Return Mapping Algorithm with Backward Euler Integration

1. Initialization: set initial values of plastic strain and internal variables to the last converged values: k = 0: p(0) (0) (0) (0) ² = 0, V = V n, ∆λ = 0 and σ = σtrial. 2. Check the yielding condition, evaluate the residuals and check convergence at the kth iteration:

(k) (k) (k) (k) f = σeq(σ − α ) − σ0 − R(q ) ½ ¾ Ψ (k) re(k) = Υ

(k) (k) If |f | < T OLf and kre k < T OLr, convergence is achieved after k- iterations. Otherwise, go to the next step.

3. Calculate the incremental plastic multiplier: · ¸ el−1 ∂a ∂a −1 C + ∆λ ∂σ ∆λ ∂V [M] = ∂H ∂H ∆λ ∂σ −I + ∆λ ∂V (k) (k) e (k) (k) (k) f − ∂f : M : re δλ = (k) ∂fe : M(k) : re(k)

4. Compute the incremental stress and internal variables: ½ ¾ ∆σ(k) = −M(k) : re(k) − δλ(k)M(k) : ne (k) ∆V (k)

5. Update the stress and internal variables:

−1 ²p(k+1) = ²p(k) − Cel : ∆σ(k) V (k+1) = V (k) + ∆V (k) ∆λ(k+1) = ∆λ(k) + δλ(k) σ(k+1) = σ(k) + ∆σ(k+1) k + 1 → k, go to 2.

37 3.4 Semi-backward Integration

The backward Euler method is robust and accurate. However, it involves the calculation of the yield surface curvature, i.e., the second derivatives of the yield

∂2f function with respect to the stress tensor, ∂σ2 . When a complicated yield criterion is used, these calculations can be quite painful and are possible sources of programming errors. An alternative to the backward Euler method is the tangent cutting plane algorithm [49]. It intends to solve the following diﬀerential equations:

²n+1 = ²n + ∆² (3.38)

p p ²n+1 = ²n + ∆λan (3.39)

V n+1 = V n + ∆λHn (σn; V n) (3.40)

el p σn+1 = σn + C : (∆² − ∆² ) (3.41)

fn+1 = 0 (3.42)

Notice that this algorithm is diﬀerent from the backward integration method, because it is implicit only in terms of λ. The ﬂow vector a at the beginning of the increment is used. Therefore, this method is often called the semi-backward integration.

Figure 3.2 illustrates the diﬀerence between the backward and the semi-backward integration methods. An iterative solution procedure can be geometrically envisioned as a projection that brings the elastic trial stress onto an updated yield surface in the stress space. In the backward Euler method, the projection always operates on the initial elastic trial stress, but the direction of return mapping is successively updated. However, in the semi-backward algorithm, each projection uses diﬀerent starting stresses. For some cases, the tangent cutting plane algorithm may cause stress drift away from the exact solution [50].

38 σtrial n+1 σtrial n+1 σ(1) σ(1) n+1 n+1 Cuts σ(k) n+1 σ(k) n+1 σ2 σ2 σ n+1 σn+1

σn σn

Elastic Region σ1 Elastic Region σ1

fn+1 =0 fn+1 =0

(a) (b)

Figure 3.2: Schematics of the return mapping schemes: (a) the backward Euler method, and (b) the tangent cutting plane method.

3.5 Implementation to Abaqus via UMAT

The commercial FE software Abaqus provides a powerful tool by allowing users to implement their own constitutive models through a user subroutine interface

(UMAT). This subroutine is called at every integration point, hence it needs to be accurate, robust and yet computationally eﬃcient [24].

The data passed into a UMAT include the stress, strain and internal valuables from the last converged increment, and the current total strain increment that is computed from the nodal displacement after solving the global equilibrium equation. The outputs of a UMAT are the updated stress and internal variables, and the consistent tangent stiﬀness matrix for implicit ﬁnite element program, Figure 3.3.

39 p p {σn, ǫ , αn, Rn} {σn+1, ǫ , αn+1, Rn+1} n UMAT n+1 ∆ǫ Calg = δσ δǫ |n+1

Figure 3.3: Input and output of a user material subroutine.

Springback prediction by FEM relies on the accurate description of the plastic anisotropy of the sheet material. In this study, the anisotropic yielding of a sheet metal is modeled by the Barlat’91, ’96 and 2000 yield functions [42–44]. In addition, the anisotropic hardening behavior, following a reversed strain path, is also included in the numerical implementation using the mNLK hardening model. The coeﬃcients in the yield functions are calibrated from the uniaxial yield strengths and the r-values of sheet specimens at 0, 45 and 90 degrees with respect to the sheet rolling direction.

The Barlat’96 yield function also requires the balanced biaxial ﬂow stress that can be measured form the bulge test [43].

The anisotropic hardening is modeled by a nonlinear kinematic hardening (NLK ) with an Armstrong-Frederic type of rule for the back stress evolution [10]. The basic

NLK hardening model (with one back stress) is available from Abaqus, but it can only be used with simple yield criteria, such as the von Mises and the Hill’48 [24].

Moreover, it cannot fully represent the stress-strain curves observed after a reversed strain path [46]. Therefore, an extended version of the nonlinear kinematic hardening with three back stress components [11] is adopted in this study.

40 The evolution of the back stresses is described by the commonly used Armstrong-

Frederick [10] law:

ci dαi = (σ − α)dq − γiαidq (3.43) σeq where σeq and α represent the size and center of the yield surface respectively, ci and γi are model parameters. The total back stress is the sum of N component back stresses, as suggested by Chaboche to better simulate cyclic plasticity [12]: XN α = αi, i = 1, 2,...N. (3.44) i=1

In this study, N = 3 and the model parameters ci and γi can be ﬁtted from in-plane

C-T or T-C tests. The method to obtain these parameters will be discussed in next section. Detailed numerical algorithm of Abaqus UMAT is included in Appendix A.

3.6 Anisotropic Hardening of 6022-T4 Aluminum Sheet

The 6 parameters in the mNLK model, i.e., ci and γi (i = 1-3), can be fitted from the Bauschinger test results. A special device was used to prevent buckling when a sheet sample was subject to compression [51], Figure 3.4(a). The sample was sandwiched between two pairs of sliding forks, on which stabilizing pressure was applied to avoid buckling. Teflon was used between the sheet sample and the forks to reduce friction. For isotropic hardening material, the reverse flow curve is schematically indicated by Curve a in Figure 3.4. When hardening becomes anisotropic, the yield strength in compression is lower than the monotonic flow stress just prior to unloading, and a region of fast strain hardening follows, as demonstrated by Curve b in Figure 3.4. The third characteristic of the Bauschinger effect is the permanent softening. At large strains after the load reversal the reverse flow curve becomes parallel to the monotonic one, with an offset in flow stress, ∆σps.

41 σ Pressure 1 Monotonic flow curve ∆σps Stress a b c (a). isotropic hardening (b) anisotropic hardening without permanent softening (c). with permannet softening σ1 Pressure Effective strain (a) (b)

Figure 3.4: Schematics of the Bauschinger test: (a) the fork device, and (b) ﬂow curves after load reversal.

The reversed stress-strain curves from uniaxial tension/compression tests were available for 6022-T4 aluminum with three or four prestrains ranging from 0.01 to

0.08 [52]. A simplex algorithm is used to obtain the optimum values of the six parameters in the mNLK model. The object function is the error deﬁned as follows v u K u X ¯ (k) (k) ¯2 1 ¯σexp − σ ¯ Error = t ¯ FEM ¯ (3.45) K (k) k=1 σexp where K is the number of experiment data points, σexp and σFEM are the measured and calculated ﬂow stresses after a load reversal respectively. A Fortran program was written to search for the optimal parameters that minimized the object function. The results after optimization are shown in Table 3.1, for both compression/tension and tension/compression tests.

To validate the numerical implementation of the mNLK model, the reverse ﬂow curves generated from the Bauschinger tests of 6022-T4 sheet were simulated by ﬁnite

42 c1 c2 c3 γ1 γ2 γ3 C-T ﬁt 520 5000 11250 10 250 750 T-C ﬁt 216 7040 4800 5.4 320 320

Table 3.1: Parameters ci (MPa) and γi for the mNLK model.

element method. One 4-node shell element (Abaqus type S4R) was used in the FE model to simulate the C-T and T-C tests. Figures 3.5(a) and 3.5(b) compare the simulated and measured reverse ﬂow curves after diﬀerent prestrains, for T-C and

C-T tests respectively. As can be seen, the Bauschinger eﬀect is closely reproduced by the mNLK model. The main features of the Bauschinger eﬀect, i.e, the reduced yielding and the fast transient hardening, are clearly captured. For small prestrains

(∼0.01), there is little permanent softening, i.e., the reverse ﬂow curve eventually joins the monotonic one, Figure 3.5. The permanent softening is also captured by the mNLK model.

In Figure 3.6, simulated reverse ﬂow curves from C-T tests, using diﬀerent hardening laws, are compared with the experimental data for one test case. Clearly, the isotropic law is incapable of describing neither the fast transient hardening nor the permanent softening behavior after a reversed strain path. The original NLK model, with single back stress, can either reproduce the transient hardening region or the permanent softening, Figure 3.6(a). It cannot capture all three features of the

Bauschinger eﬀect. On the other hand, both the Geng-Wagoner model based on two

43 350 350 6022-T4, C-T 6022-T4, T-C 300 300 Monotonic Monotonic 250 250 Exp. Exp. 200 200

ε =0.0466 150 com 150 ε =0.079 ten ε =0.0275 com ε 100 100 =0.047

True stress (MPa) ten ε =0.0111 True stress (MPa) com ε =0.029 50 50 ten ε =0.014 ten 0 0 0 0.04 0.08 0.12 0.16 0 0.04 0.08 0.12 0.16 Effective strain Effective strain (a) (b)

Figure 3.5: Simulated and experimental reverse ﬂow curves after: (a) compression/tension test, and (b) tension/compression test. Markers are experimental data from Balakrishnan (1999), and solid lines are the corresponding simulation results.

surface plasticity [46] and the mNLK model can closely simulate the Bauschinger effect. However, the mNLK model is simpler than the Geng-Wagoner model, in terms of the mathematical derivation and numerical implementation.

Taking the Bauschinger eﬀect into consideration gives more realistic prediction for draw-bend springback simulation, because the through-thickness stress distribution in forming step can be better represented [46]. If material softens after a load reversal, such as the 6022-T4 used in this study, the bending moment will be overestimated if an isotropic hardening model is used. As a result, the amount of springback is overpredicted [46].

44 350 350 6022-T4 Monotonic Isotropic 300 300 C-T

250 250 Exp. 200 200 Exp. (Balakrishnan, 1999) NLK (one α ) 150 150 ( γ mNLK ( α ) c=375, =5 ) 1,2,3 100 100 True stress (MPa) Truestress (MPa) NLK (one α ) Two surface ( γ (Geng 2001) 50 c=38000, =400 ) 50 ε =0.0466 ε =0.0466 com com 0 0 0.04 0.06 0.08 0.1 0.12 0.04 0.06 0.08 0.1 0.12 Effective strain Effective strain (a) (b)

Figure 3.6: Simulated reverse ﬂow curves using various hardening laws: (a) isotropic hardening and NLK model with single back stress, and (b) mNLK and Geng-Wagoner models.

45 CHAPTER 4

SPRINGBACK SIMULATION OF DRAW-BEND TEST WITH FINITE ELEMENT METHOD

Note: Some of the draw-bend experiment data were provided by W.D. Carden.

This chapter also utilized the tension-compression test results from V. Balakrishnan.

A manuscript to be submitted for publication in the International Journal of Plasticity is in preparation.

Abstract

Accurate springback prediction is essential for tool design and quality control in sheet forming processes. To understand the springback phenomenon, a series of draw-bend tests were carried out, under carefully controlled laboratory conditions. Aluminum alloy 6022-T4, which has been considered as a potential replacement for forming grade steel to reduce vehicle weight, was tested using a range of tool radii (3.2mm–12.7mm) and normalized sheet tension (0.2–1.1). Springback angles were measured from the unloaded strips that experienced sequential bending and unbending deformation. It was found that springback decreased with the tool radius and the applied stretching force, with the later dominating. A dramatic drop in springback angle was attributed to the persistent anticlastic curvature in the sheet width direction, when the sheet

46 tension approached the yielding force of the material. This secondary curvature significantly increased the section moment of inertia for bending, and thus greatly reduced springback. To understand the mechanics of the anticlastic curvature and its effect on springback, additional tests were conducted for specimens with various width (12mm–50mm). The effect of the anticlastic curvature on springback was clari-

fied. Finite element simulations were carried out, using Barlat’s family of anisotropic yield functions (Barlat’91, ’96 and 2000) and a non-isotropic hardening model that was developed to reproduce the Bauschinger effect after a load reversal. The effect of the tool radius on the springback was discussed. It is suggested that solid element should be used for small tool radius (R = 3.2mm) where shell assumptions are invalid.

4.1 Introduction

Springback denotes the undesired shape change of a formed part after the removal of the forming load. It often causes diﬃculty in part assembly, and extra work in tool design is needed to compensate for the shape deviation. Therefore, accurate prediction of springback in sheet metal forming operations is essential to reduce the try-out time in tool design, and to improve quality of the stamping parts. Many researchers have established empirical methods [53] for springback prediction and compensation during the past few decades, others have proposed analytical models based on engineering bending theory of beams [54–56].

There are many examples of predicting springback after simple forming operations, such as cylindrical tool bend [57], V-die bend [58–60], U-channel forming [61] and ﬂanging [62]. Although these experiments are not close to the real industrial sheet metal forming processes, scientiﬁc understanding was well developed on the

47 relationship between springback and material properties and process parameters. It was concluded that springback increased with the increase of material yield strength and the tool radius, but decreased with increasing elastic modulus for an elastic per- fectly plastic material after pure bending [55]. When considered strain hardening and plastic anisotropy, it was shown that strain hardening reduced springback [63] while the plastic strain ratio (i.e., r-value) promoted springback [63].

Among all the process variables that affect springback, sheet tension has the most prominent effect in reducing springback [64]. Hence, stretching became an efficient way to minimize springback in many real forming operations. Several shape control methods based on this idea have also been developed, and successfully applied to compensate for the springback in the flange operations. Friction also reduces springback, but in most studied cases it functioned because the sheet tension was increased by friction through a blank holder or a drawbead [65,66].

Numerous experiments have been conducted on springback behavior of sheet metals under simple test conditions, but there is little information for complicated sheet metal forming operations under controlled test conditions. For automotive industry, this desire becomes even more imperative because of the growing application of high-strength aluminum alloy sheet for structural components, driven by the more stringent requirements of improving fuel consumption and reducing green-house gas emission. The research challenge of springback prediction comes from the fact that aluminum is inferior to steel in formability, and it is prone to springback, primarily because of its lower elastic modulus. In addition, springback prediction becomes more obscured because a formed aluminum sheet part can continue to change shape over time [67–70].

48 While ordinary stretch-bending test is capable of providing direct control of sheet tension, it is unable to represent complicated deformations that are encountered in the practical forming processes in which a sheet metal slides over a rigid tool surface as it is drawn into a die cavity. However, the draw-bend test remedied this shortcoming, and it oﬀered a better opportunity to study the springback behavior [69].

The current draw-bend test was initially designed for friction measurement of coated steel sheets [71,72]. Although the basic concept is similar to the devices used by others [64, 73, 74], the most important improvement of the current test device is accurate measurement of the deformed shape because a longer drawing distance was adopted [67, 69]. This also makes it possible to observe and examine the side-wall curl [75] in the drawn region of the specimen with an improved accuracy. This sidewall curl presents another form of shape deviation, and it often appeared in U-channel forming and ﬂange operations [62].

A secondary curvature, which is orthogonal to the primary one, was observed in the curled region in the sheet width direction. This curvature has been known as the anticlastic curvature [76], and it has been reported by a few authors in bending ﬂat sheet and composite plate [77]. Although analytical methods are available for pure bending problems, using elastic theory of plate bending, there is a lack of knowledge of the anticlastic curvature in more complex forming processes, such as in the draw-bend test. Some conclusions were attained on how the specimen aspect ratio and bending radius are related to the magnitude of the anticlastic curvatures in four-point bending test [78], but they may not apply to the draw-bend test.

In addition to the forming parameters, springback simulation is also sensitive to the choice of the material constitutive models. To accurately calculate the stress/moment

49 distribution in the formed part, the mechanical anisotropy of the sheet metals has to be considered. An anisotropic yield function is required to describe the orientation- dependent initial yield strength and the plastic strain ratios (r-Value) [29]. The modeling of the anisotropic hardening is more complicated, and a few models are available in the literature [10,16,79,80]. In this study, a nonlinear kinematic hardening model was implemented into ﬁnite element program Abaqus/Standard via the user subroutine option. This model closely reproduces the characteristics of the Bauschinger eﬀect following a load reversal, yet it has a simpler form compared with the previous work based on a two-surface plasticity theory [46].

The objective of this chapter is to study the springback behavior of 6022-T4 alloy sheets in the draw-bend test under controllable conditions. In particular, the eﬀects of sheet tension, tool radius and anticlastic curvature on springback are clariﬁed.

The choice of elements (shell and solid) for springback simulation using ﬁnite element method is also discussed for small tool radius (R/t = 3.5).

4.2 Draw-Bend Experiment

Series of draw-bend tests were conducted using a special apparatus at Colorado

School of Mines that was initially designed to evaluate the friction behavior of coated steel sheets [71, 72]. Two groups of 6022-T4 sheet specimens were prepared for this study. For the ﬁrst group, rectangular blanks (500mm×50mm×1mm) were sheared with their lengths aligned with either the rolling or transverse direction of the sheet metal. Tooling radii of 3.2 and 12.7mm were used for these tests. Samples from the second group have the same length but various widths ranging from 12mm to 50mm.

Only the 12.7mm tool was employed for these strips, with two selected sheet tensions:

50 Fb = 0.5 and Fb = 0.9 (Fb is expressed as fractions of the yielding force of the strip in uniaxial tension). After the forming load was released, specimens were removed from the test device, and they were traced onto paper for geometry measurement.

In this chapter, only static springback results, measured approximately 60 seconds after unloading, were reported. Time-dependent springback results are presented and analyzed in Chapter 6.

4.2.1 Materials

To compare with the previous investigation [69], the same aluminum alloy 6022-T4 sheet, provided by the Partnership of Next Generation Vehicles (PNGV), was used in this study. The chemical composition of this alloy is given in Table 4.1 [69].

Si Fe Cu Mn Mg Ti Al 1.24 0.13 0.09 0.07 0.58 0.02 Bal.

Table 4.1: Chemical composition of 6022-T4 aluminum sheet (in weight percent).

4.2.2 Draw-Bend Test

The draw-bend test equipment consists of a standard servo-hydraulic mechanical test machine and an attached 90-degree bending frame, Figure 4.1(b). It has two independently controlled hydraulic actuators that are perpendicularly oriented to each other. During the draw-bend test, the upper right actuator was programmed to provide a constant restraining force (i.e, the back force), while the lower one was

51 set to move downward at a constant speed. The actual front and back tensions were measured by two load cells that were attached to each hydraulic cylinder.

start finish

Fb R 1 Sample 2 R′ r′ Upper grip start Roller 3

4 finish ∆θ 127mm Lower grip Unloaded Loaded springback angle: ∆θ 40mm/s

(a) (b)

Figure 4.1: Draw-bend experiment: (a) equipment at Colorado School of Mines, and (b) schematics of test procedure and geometry of a deformed sheet after springback

Figure 4.1(b) schematically shows the draw-bend test procedure. A pre-bend was

ﬁrst performed by hand to obtain an approximately 90° bent after unloading. Then, a prescribed back force was applied at the left side of the sheet, while the right side was held immobile. During the test, the strip was drawn over a ﬁxed or rotating cylindrical roller for a total travel of 127mm, while the back force was maintained constant. The drawing speed was 40mm/s, with a few exceptions where 10mm/s was used when a small back force (Fb < 0.1) was applied. The draw-bend test can closely simulate the realistic sheet metal forming operations, where sequential bending and

52 unbending deformation occurs under superposed tension as the strip slides over the tool surface [69].

After the forming operation was completed, the lower grip was opened to allow the sheet to springback freely. The unloaded sample was then traced onto paper for geometry measurement. A typical unloaded strip was schematically shown in Fig- ure 4.1(b), from which four distinguished regions of deformation were identiﬁed. In

Region 1 and 4, material underwent pure stretching only, and thus they were not interested for springback study. Region 2 is characterized by a relaxed radius, R0

(originally equals the tool radius R), or the corresponding angle change ∆θ1. Mate- rial in Region 3 has undergone sequential bending and unbending with superposed stretching. The radius of the Region 3 after springback is r0, with its corresponding

0 angle change ∆θ2. r and ∆θ2 are measures of the so-called side-wall curl which has been reported in channel forming operations [62]. Following an earlier analysis [69], a single parameter, ∆θ, was used to characterize the amount of springback for draw-bend tested specimens, Figure 4.1(b).

Since the sheet tension played a dominant role in reducing springback [64, 74], multiple tests were conducted with the normalized back forces ranging from 0.1 to

1.1, at an increment of 0.05–0.1 to improve resolution. Compared to the previous research [69], this study focuses more on the springback behavior in the small R/t range with refined back force intervals. Although an fixed or rotating tool can be used to provide different friction conditions, the effect of friction on springback was not the goal of this study. An industrial drawing lubricant [81] was carefully brushed on both the roller surface and the inner sheet surface to provide medium friction coefficient

(approximately 0.15) [69].

53 Previous study discovered a secondary curvature in the sheet width direction [69], which is perpendicular to the primary bending curvature in the curled region of an unloaded strip. This secondary curvature, often called as the anticlastic curvature in the literature [76], was originated from the differential contractions between the top and bottom fibers of a strip in elastic bending, caused by the Poisson’s effect [76,

78]. However, the anticlastic curvature persisted after unloading for certain test conditions in the draw-bend test [69,82]. As a result, the springback angle was greatly reduced, because the persistent anticlastic curvature substantially increases of the section moment of inertia for bending. To understand this phenomenon, specimens with various widths (from 12mm to 50mm) were tested at two selected back forces of

0.5 and 0.9 with R/t = 14. Since the anticlastic curvature does not vary signiﬁcantly within the sidewall curl region, measurements were made at the central position of the curl length, i.e., about 64mm away from the straight leg (Region 4 ).

The radius of anticlastic curvature, Ra, can be calculated from the following equation, assuming that the transverse cross-section is circular:

∆h W 2 R = + (4.1) a 2 8∆h where W is the strip width, and ∆h is the height of the transverse arc. A digital caliper with 0.01mm resolution was used to measure ∆h.

4.3 Experimental Results

The static springback results are presented in this section. The eﬀect of back force, tool radius and specimen geometry (width-to-thickness ratio, W/t) are discussed.

54 4.3.1 Eﬀect of Back Force and Tool Radius

Experiment results were tabulated in Appendix B, see Tables B.1 and B.2 for tool radius of 3.2 and 12.7mm, respectively. The measured springback angle, ∆θ, and the anticlastic curvature, 1 , are plotted in Figure 4.2. From the results, it is clear that Ra the stretching force plays a signiﬁcant role in reducing the springback. In addition to this commonly observed trend, a dramatic drop in the springback angle occurs near

Fb = 0.7–0.8. This is associated with an abrupt increase of the anticlastic curvature,

Figure 4.2(a) and 4.2(b). It is noted that the critical back force, at which the fast decline of springback angle occurs, weakly depends on tool radius.

100 5 80 5 ∆θ ∆θ Curvature 70 Curvature 80 4 4 1/R 60 1/R a a (10x 60 3 50 3 (10x 40 -3 -3 mm mm (degree) 2 (degree) 2 40 30 ∆θ 6022-T4 (RD) ∆θ -1 -1 ) R/t=3.5 20 6022-T4 (RD) ) 20 1 R/t=14 1 10

0 0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back force F Normalized back force F b b

(a) (b)

Figure 4.2: Dependence of the springback angle and the anticlastic curvature on the normalized back force: (a) R/t = 3.5 and (b) R/t = 14.0.

For the draw-bend test, the bending radius has less signiﬁcant impact on the springback angle than the sheet tension. Figure 4.2 shows that the springback angle

55 decreases as the R/t ratio increases. These results are in contrast with simple bending results, but they agree with other investigations [74].

The orientation of strips (RD or TD) also made a noticeable diﬀerence, in terms of the variation of the springback angle and the anticlastic curvature with back force. As shown in Figure 4.3(a), the springback angle of the transversely sheared specimens gradually decreases with increasing back force, while the RD samples show a fast decline of springback angle at Fb = 0.7–0.8. The anticlastic curvatures are smaller for TD samples, and the variation with Fb is less abrupt, Figure 4.3(b).

100 4 6022-T4 R/t=3.5

80 ) -1 3 mm

60 -3 2

(degree) 40 ∆θ

1 20 RD Curvature (10 TD RD TD 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back force F Normalized back force F b b

(a) (b)

Figure 4.3: Eﬀect of back force on springback for samples with diﬀerent orientation: (a) ∆θ and (b) 1 . Ra

56 4.3.2 Eﬀect of Specimen Width

In the literature, little work has been done systematically on the eﬀect of the anticlastic curvature on springback in sheet metal forming operations. According to the elementary bending theory of beam, the anticlastic curvature only appears in narrow beams, for which plane-stress state is a good approximation in the width direction [76]. For wide strips, the anticlastic deﬂection was assumed to nearly dis- appear or localize in the strip edges [83], so that the deformation is characterized as plane-strain. According to an elastic theory of plate bending [84], the transition from plane-stress state to plane-strain was gradual [85], and the shape of the anticlastic surface was determined by a dimensionless parameter β, which is often called the

Searle’s parameter [86]: W 2 (W/t)2 β = = (4.2) Rxt Rx/t where W and t are the strip width and thickness respectively, Rx is the radius of the primary (longitudinal) curvature. For draw-bend tested samples, r0 (Figure 4.1(b)) is used instead of Rx in Equation 4.2 when applying the elastic theory to analyze the draw-bend springback.

To explore the eﬀect of the anticlastic curvature on springback, additional draw- bend tests were conducted for 6022-T4 aluminum sheets with various widths (12–

50mm), using R = 12.7mm tool and two back forces: Fb = 0.5 and Fb = 0.9.

Measured springback angle and radius of the anticlastic curvature are reported in

Appendix B, Table B.4. As shown in Figure 4.4(a), both the springback angle and the anticlastic curvature decrease as the sheet width increases for large back force

(Fb = 0.9). At smaller back force (Fb = 0.5), the springback angle slightly decreases with the sheet width, followed by an increase for strips wider than 25mm. The

57 anticlastic curvature continuously decreases with the specimen width. The maximum change in springback angle from the narrowest (12mm) to the widest strip (50mm) is about 13° and 8° for Fb = 0.9 and Fb = 0.5, respectively. The maximum standard deviation of ∆θ is about 1.8°. The sources of scattering are probably caused by the non-uniform width, hardened strip edges due to shearing and poor alignment of narrow strips during the draw-bend test.

50 10 20 10 6022-T4 (RD) ∆θ ∆θ R/t=14 Curvature 8 Curvature 8

F =0.5 1/R 1/R 45 b 15 a a 6 (x10 6 (x10 40 10 -3 -3 (degree) (degree) 4 mm 4 mm ∆θ ∆θ -1 -1

35 ) 5 ) 2 2 R/t=14 F =0.9 30 0 0 b 0 10 20 30 40 50 60 10 20 30 40 50 60 Width (mm) Width (mm) (a) (b)

Figure 4.4: Dependence of springback angle and anticlastic curvature on specimen width: (a) Fb = 0.5 and (b) Fb = 0.9.

4.4 Finite Element Results and Discussion

The static draw-bend tests were simulated using ﬁnite element method, for a range of normalized back forces (0.1 ≤ Fb ≤ 1.2) and two R/t-ratios: 3.5 and 14.

Diﬀerent elements were chosen in the 3D analysis: 4-node shell element with reduced integration (S4R), 8-node brick element (C3D8R with reduced integration and

58 hybrid C3D8H) and 20-node brick element (C3D20R with reduced integration and

C3D20) [24]. For simulations using the shell elements, the sheet strip was discretized into 300 elements of non-uniform size in the longitudinal direction, and 8 element in the width direction. To ensure numerical accuracy, element size is smaller in the contact areas (Regions 2 and 3, Figure 4.1(b)), with one contact node per 4.5 degrees of turn angle [82]. Fifty one through-thickness integration points were used to minimize numerical error [82]. For FE models using the ﬁrst order solid elements (C3D8H and

C3D8R), the ﬁnite element mesh consists of 12,000 elements (with 300, 4, and 10 elements in length, width and thickness directions respectively). When higher oder elements (C3D20R and C3D20) were used, only 6 elements were used through the sheet thickness to avoid overwhelming computation cost. For all FE models, mirror symmetry was utilized, and only half of the physical sample was modeled.

The Coulomb friction coeﬃcient of 0.15 was used for the lubricated test conditions, while zero friction was assumed if the tool was free to rotate during the draw-bend test [69].

Since springback is determined by the stress/moment distribution in the forming step, accurate springback prediction relies on the proper choice of the material constitutive models. Previous work demonstrated that erroneous simulations were resulted if anisotropic yielding and non-isotropic hardening were neglected or poorly represented [46]. To closely represent the anisotropic yielding of 6022-T4 sheet, Barlat’91

(for solid elements), ’96 and 2000 (for shell elements) yield functions [42,43,45] were used. A modiﬁed anisotropic hardening model [46] and a nonlinear kinematic hardening law [11] were utilized to closely represent the features of the Bauschinger eﬀect after a load reversal. The constitutive models were implemented into a commercial

59 ﬁnite element package Abaqus/Standard through user material subroutines (UMAT).

Details of the Barlat’s yield functions and the nonlinear kinematic hardening model can be found in Chapters 2 and 3, while the UMAT algorithms are discussed in

Appendix A.

4.4.1 Eﬀect of Back Force

Previously implemented UMAT [46], which utilized the Barlat’96 yield function and a modiﬁed anisotropic hardening (the G-W model), was used ﬁrst to simulate the static springback after the draw-bend test. Compared to the previous study, more back forces are used in the current investigation, in order to reveal the details of the rapid decrease of the springback angle as the sheet tension approaches the yielding, a phenomenon that was reported before [67,69,82].

The simulated springback angles and anticlastic curvatures are compared with the experimental results in Figure 4.5(a) and Figure 4.5(b), respectively. The general trend confirms the earlier finding [69], that the sheet tension significantly reduces springback, while the tool radius has less profound effect. For small tensions (Fb <

0.7 − −0.8), an increase in the back force by 0.1 causes about 7 degrees of reduction in the springback angle. However, when R/t is increased from 3.5 to 14, ∆θ is decreased by 13 degrees for small back forces, Figure 4.5(a). The correlation between the sudden decrease of ∆θ near Fb = 0.7–0.8 and the substantial increase in the anticlastic curvature is in accord with the previous results [69].

The convexity of the Barlat’96 yield function is not proven [43]. To avoid numerical diﬃculties, the recently proposed Barlat 2000 yield function is implemented in this study [45]. In addition, a nonlinear kinematic hardening model with three back

60 100 6 Barlat'96 yield R/t=3.5 (Exp.) G-W hardening )

-1 R/t=3.5 (FEM) 80 5 R/t=14 (Exp.) R/t=14 (FEM) mm -3 4 60 (x10 a 3 (degree) 40

∆θ 2 R/t=3.5 (Exp.) 20 R.t=3.5 (FEM) R/t=14 (Exp.) 1 R/t=14 (FEM) Curvature1/R 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back forcre F Normalized back forcre F b b

(a) (b)

Figure 4.5: Comparison of the simulation results with the experimental data: (a) the springback angle ∆θ, and (b) the unloaded anticlastic curvature 1 . Ra

stresses (the mNLK model) is employed [11], in which the evolution of the back stress takes the Armstrong-Frederic type of law [10]. This model provides a simpler mathematical form, therefore the numerical implementation is easier than the G-W model [87]. Only 6 parameters are required in the mNLK model, as reported in

Table 3.1 of Chapter 3. The detailed procedure used to attain the mNLK model parameters from the Bauschinger test can be found in Chapter 3.

As shown in Figure 4.6(a), the use of the Barlat 2000 yield function gives nearly identical simulation results as the Barlat’96, when the G-W hardening model was used for both cases. This is expected because the variation of the yield strength and r-value with orientation are closely reproduced by both the Barlat’96 and 2000 yield functions, as can be seen from Figure 2.8 in Chapter 3. The computation cost by

61 80 80 R/t=14, 8x300 S4R mNLK parameters G-W hardening 70 γ =5.4, γ =γ =320 70 1 2 3 c /γ =40, c /γ =22, c /γ =15 60 60 1 1 2 2 3 3

50 50

40 40 (degree) (degree) 30 30 ∆θ ∆θ R/t=14 20 20 8x300 S4R Exp. Exp. Barlat'96 10 Barlat 2000 10 mNLK Barlat'96 G-W 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back force F Normalized back force F b b (a) (b)

Figure 4.6: Comparison of the simulated springback angles using (a) the Barlat’96 and 2000 yield functions and (b) the Geng-Wagoner model and the mNLK law.

using the Barlat 2000 yield function, however, is increased by approximately 50%, presumably because of two stress potentials are used in formulating the Barlat 2000 yield function (Equation 2.67).

The simulated springback angles by the mNLK and G-W models are compared in

Figure 4.6(b). The overall agreement is good, although the mNLK model gives less closer match with the experimental data for small back forces. The possible source of error may be that the hardening parameters (ci and γi, i=1–3) are not optimized, because a crude optimization algorithm was used, see Chapter 3.

4.4.2 Eﬀect of Strip Width

Finite element simulations were also carried out for draw-bend tests with specimens of various widths (12mm to 50mm), using the Barlat’96 yield function [43] and

62 the G-W hardening model [87]. As an ideal plane-stress case, beam element (Abaqus type B21) was also used for one case with R/t = 14 and Fb = 0.5. The simulation agrees with the experimental results, that is, ∆θ initially decrease with the strip width, Figure 4.7(a). For narrower samples (W < 25mm), however, an opposite trend was observed and confirmed by finite element simulations. In the limit case of W = 0, the problem can be treated as plane-stress and the springback is maximized. This conclusion is validated by FE simulation, since the 3D results appear to approach the plane-stress result (44.3° by beam element) as the width decreases. The experimental data also supported this conclusion with the smallest width of 12mm. The anticlastic curvatures (measured and simulated) monotonically decrease with strip width, Figure 4.7(b). This is inconsistent with the variation of the springback angle for small back forces, see Figure 4.7(a). Notice that Equation 4.1 was used to calculate the anticlastic curvature, assuming that the cross-section was circular. As will be discussed later in Chapter 5, this assumption is invalid for strips tested at low back forces. Experiments and simulations demonstrate that the anticlastic deflection is concentrated on the sheet edges, while the center area of the specimen is nearly flat.

Therefore, Equation 4.1 overestimates the magnitude of the anticlastic curvature for the same arc height ∆h.

4.4.3 Choice of Element — Shell vs. Solid

As mentioned in Section 4.3, small tool radii moderately promote the draw-bend springback. Previous experimental results [88] were analyzed by ﬁnite element simulations in this chapter. These experiments used a range of R/t (1.8–28) and two back forces (Fb = 0.5 and 0.9), with free rotating tool that provided nearly zero friction.

63 50 10 Exp. Exp. FEM (S4R) FEM (S4R) )

FEM (B21) -1 8 45 mm

-3 6 40

(degree) 4 ∆θ

35 R/t=14, F =0.5 b 2 Barlat'96 Yld Curvature (x10 G-W hardening 30 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Width (mm) Width (mm) (a) (b)

Figure 4.7: Comparison of the simulation and the experimental data for strips with various widths using Fb = 0.5: (a) the springback angle, and (b) the anticlastic curvature.

80 60 Exp. (Carden, 1996) F =0.9 S4R (Barlat'96) b 70 C3D8H (Barlat'91) C3D8R (Barlat'91) 40 C3D20R (Barlat'91)

60 20 (degree) 50 (degree) 0 ∆θ ∆θ Exp. (Carden, 1996) S4R (Barlat'96) 40 6022-T4 F =0.5 -20 C3D8H (Barlat'91) b C3D8R (Barlat'91) Friction=0 C3D20 (Barlat'91) 30 -40 0 5 10 15 20 25 30 0 5 10 15 20 25 30 R/t ratio R/t ratio (a) (b)

Figure 4.8: Comparison of the simulated springback angles using diﬀerent elements for (a) Fb = 0.5 and (b) Fb = 0.9.

64 As shown in Figure 4.8, ∆θ initially increases with the tool radius, but decreases when R/t < 3.5 (for Fb = 0.5) and R/t < 10.5 (for Fb=0.9). Shell element generally works well when the R/t ratio is larger than 5–6 [23, 82], but it fails to predict the decline of the springback angle for small bending radii. This is probably because the assumptions used in the shell formulation, namely the zero through-thickness stress and plane section remaining planar after deformation, are no longer valid. On the other hand, solid elements (linear or quadratic) can qualitatively reproduce the experimental trend. The quadratic elements (C3D20R and C3D20) are in better agreement with the experimental data than the linear elements (C3D8H and C3D8R), presumably because the second order elements are more accurate for bending dominant problems. However, the computation cost is overwhelming for the fully integrated,

20-node solid element (C3D20): about 2 weeks for a typical draw-bend simulation case with 7200 elements (300 in length, 4 in width and 6 through thickness). The use of the C3D20R element (with reduced integration) cuts down the computation time to about 5 days for the same ﬁnite element model, because only 8 integration points was used by the C3D20R element instead of 27 by C3D20. In view of the computation eﬃciency and bending performance, a recently developed, locking-free

8-node brick element appears to be suitable for the draw-bend simulations [89].

Figure 4.9 compares the measured springback angles with the simulation results using both the shell and solid elements for R/t = 3.5. The overall agreement with experiment is satisfactory, even though simulations with the shell element deviate from experiment noticeably near Fb = 0.7–0.8. For solid element, improvement is possible if ﬁnite element mesh is further reﬁned, or a better 3D yield function [90] is employed to replace the Barlat’91.

65 90 C3D20 (Barlat'91) 80 S4R (Barlat'96) 70 Exp. 60 50

40 (degree)

∆θ 30

20 R/t=3.5 10 Friction=0.15 G-W hardening 0 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back force F b

Figure 4.9: Simulated springback angles using shell and solid elements.

4.5 Conclusions

The following conclusions are reached from the draw-bend experiments and ﬁnite element simulation for 6022-T4 aluminum sheets:

1. Sheet tension has a dominant role in reducing the static springback of aluminum

alloy 6022-T4 after the draw-bend test, while the tool radius has moderate eﬀect

in reducing springback.

2. The sudden increase of the springback angle as the normalized back force ap-

proaches a critical value is attributed to the occurrence of the persistent anti-

clastic curvature in the sheet width direction. The critical back force is approx-

imately 0.7–0.8, and it slightly depends on the tool radius.

3. Sheet orientations (RD or TD) aﬀect the measured springback angle and an-

ticlastic curvature. For transverse strips, the decline of the springback angle

66 with sheet tension is gradual, and the corresponding increase of the anticlastic

curvature when the normalized back force is near 0.7–0.8 appears to be less

abrupt.

4. The Barlat’s 2000 yield function and a nonlinear kinematic hardening with mul-

tiple back stress components are as accurate as the Barlat’96 yield function and

the G-W modiﬁed anisotropic hardening that were used in a previous study [46],

but they have simpler mathematical forms and therefore are easier for numerical

implementation into ﬁnite element program.

5. Shell element fails to match the decrease of springback when R/t-ratio is less

than 4. Solid element with quadratic interpolation functions is advocated for

bending with small radius, but large computational cost is required. Linear

solid element, however, is not recommended because of its poor performance in

bending applications.

67 CHAPTER 5

ANTICLASTIC CURVATURE IN DRAW-BEND TEST

Note: A manuscript of this work has been submitted to the International Journal of Solids and Structures for publication.

Abstract

Draw-bend springback shows a sudden decline as the applied sheet tension approaches the force to yield the strip. This phenomenon coincides with the appearance of persistent anticlastic curvature, which develops during the forming operation and is maintained during unloading under certain test conditions. In order to understand the mechanics of the persistent anticlastic curvature and its dependence on the forming conditions, aluminum sheet strips of widths ranging from 12mm to 50mm were draw-bend tested with various sheet tensions and tool radii. Finite element simulations were also carried out, and the simulated and measured springback angle and anticlastic curvature were compared. Analytical methods based on a large deformation bending theory for elastic plates were employed to understand the occurrence and persistence of the anticlastic curvature. The results showed that the ﬁnal cross- section shape of a specimen is determined by a dimensionless parameter, which is a function of the sheet width, thickness and radius of the primary curvature in the

68 curled region of an unloaded sample. When the normalized sheet tension approaches

1, this parameter rapidly decreases, and signiﬁcant anticlastic deﬂection is retained after unloading. The retained anticlastic curvature greatly increases the moment of inertia for bending, and thus reduces springback angle.

5.1 Introduction

When a long, flat rectangular sheet of uniform thickness is bent about an axis parallel to one of its edges, say in the x-direction, a transverse curvature is developed in the direction parallel to the bending axis [91]. For elastic deformation, this happens by the differential lateral contraction caused by Poisson’s effect. Consequently, the initially flat surface becomes an anticlastic surface, with two orthogonal curvatures in opposite sign, Figure 5.1(a). If the centers of these two curvatures appear on the same side of the surface, the surface is synclastic, Figure 5.1(b). For narrow, initially

ﬂat sheets, the ratio between the longitudinal (i.e., primary, x-direction) and the transverse (i.e., secondary, y-direction) curvatures is given by the Poisson’s ratio ν, i.e., Ry = νRx, according to the fundamental bending theory of beams [76].

The shape of the cross-section of a bent beam or plate depends on a dimensionless parameter, β = W 2 , with W , t and R being the sheet width, thickness and radius Rxt x of the primary bending curvature respectively [86]. In the literature, β is also called the Searle’s parameter [93]. The anticlastic surface has a constant curvature of − ν , Rx when β is less than one [92]. In this case, the sheet behaves like a plane-stress beam.

However, if β is larger than 20, the anticlastic deflection is mainly confined to the sheet edges, while the sheet central area stays relatively flat [93–95]. Consequently, the deformation state can be characterized as more plate-like, with plane-strain the

69 Ry z(x, y) z(x, y)

Rx R Ry longitudinal, xx longitudinal, x transverse, y (a) transverse, y (b)

Figure 5.1: (a) An anticlastic surface, and (b) a synclastic surface.

limiting approximation. The deformation modes maybe interpreted in terms of “body forces” which tend to suppress the formation of a large circular cross-section [78].

When bending wide sheets to a small radius, the constrained anticlastic curvature causes a biaxial stress state on the tension side of the sheet [96].

The principles of simple elastic plate bending have been extended to bodies of varying thickness [97–100], to the measurement of elastic constants [101, 102], and to specimens plastically deformed in four-point bending tests [78]. In the last case, pertinent to the current work, it was concluded that plasticity aﬀected the magnitude of the anticlastic curvature, but had little eﬀect on springback.

There is little literature on anticlastic deﬂection for more complicated forming processes. Anticlastic displacements up to 1.5 times the sheet thickness have been measured after draw-bending and unloading [69,82], in marked contrast to the simple bending results where theory predicts a maximum deﬂection of about 10 percent of the sheet thickness. For small sheet tensions, the anticlastic curvature developed in

70 the forming step nearly disappeared during unloading, thus having little eﬀect on the ﬁnal specimen shape (consistent with observations for springback in simple bending). However, as sheet tension was increased to the yield stress of the material, the anticlastic distortion persisted after unloading. This persistent anticlastic distortion increased the moment of inertia of the specimen greatly, and thus reduced springback commensurately [69].

This investigation focuses on the role of anticlastic curvature in springback following draw-bend deformation. The mechanics of the persistent anticlastic curvature is sought, especially its dependence on the forming parameters and the specimen geometry. In order to proceed, the classic theory of bending elastic plate is reviewed.

Draw-bend test results are then presented, and they are considered with the aid of the theory and ﬁnite element simulations. Discussions and conclusions are then drawn.

5.2 Draw-Bend Experiment

The draw-bend test can closely mimic industrial forming processes, where sequential bending and unbending takes place under superposed tension as sheet material is drawn over a rigid tool surface [69]. Unlike other laboratory forming tests, where stretching is usually provided through various locking mechanisms (draw-bead or blank holder), sheet tension can be directly and precisely controlled in the draw-bend test, using a secondary hydraulic cylinder which is programmed to provide constant stretching force during test [71]. In this study, 6022-T4 aluminum sheet were tested using a special draw-bend machine at Colorado School of Mines. Details of this equipment can be found elsewhere [69,72].

71 Fb Initial Fb Loaded R 1 2 Unloaded R′ r′ 3

127 mm 4

∆θ X˙ = 40 mm/s 127 mm

Figure 5.2: Schematics of the draw-bend test and an unloaded specimen.

Rectangular specimens were sheared with their lengths parallel to the sheet rolling direction. As shown in Figure 5.2, the draw-bend test procedure is divided into three steps, after the strip was hand-formed around a cylindrical tool to get a 90 degree bent.

A prescribed stretching force was ﬁrst applied to the left end of the strip, while its right end was held immobile. Here, Fb (the back force) is the actual sheet tension divided by the yielding force of the specimen in uniaxial tension. Then, the strip was drawn over an unrotating tool by imposing a constant speed of 40mm/s to the right end of the strip, while the back force was kept constant. Standard industrial lubricant [81] was brushed on both the strip and tool surfaces to provide medium friction [69]. After the drawing distance reached 127mm, specimens were unloaded and removed from the test device. Their shapes were traced on paper and then recorded digitally. Traces were ﬁrst taken one minute after forming and unloading, then repeated at intervals

72 up to 15 months for time-dependent springback measurement [70]. The current work only studies the static springback, which is measured approximately one minute after unloading.

A typical unloaded specimen is depicted in Figure 5.2, with four deformation regions delineated. Regions 1 and 4 remain straight throughout the test, and thus are not interested for springback study. Regions 2 was in contact with the forming tool before unloading, and its radius of curvature changes from R to R0 after springback.

The important specimen geometry is deﬁned by 3 , which has a radius of curvature r0 after unloading. It is a measure of the so-called “sidewall curl” that was observed in many sheet-formed parts [103]. Springback is characterized by the angle ∆θ,

Figure 5.2.

A transverse curvature in the sheet width direction was discovered in draw-bend tested strips [69]. If the cross-section is assumed to be circular, the radius of this anticlastic curvature, Ra, is calculated by

∆h W 2 R = + , (5.1) a 2 8∆h where ∆h is the arc height of the cross-section [69]. ∆h was measured at the center point of the Region 3 (Figure 5.2) of an unloaded specimen, using a digital caliper with

0.01mm resolution. The cross-section proﬁle was measured by a 0.03mm resolution dial gauge for two draw-bend tested samples.

5.3 Finite Element Models

Simple bending of initially ﬂat and curved elastic plates were simulated by ﬁnite element to validate the closed-form solution (as will be discussed next). 4-node shell elements with reduced integration (type S4R) were used [24], with 15 integration

73 points through sheet thickness (t = 1). The elastic modulus and Poisson’s ratio

1 adopted in simulations were 65 GPa and 3 respectively. Mirror symmetry was utilized and only one quarter of the plate was modeled, with symmetric boundary conditions applied at the plate edges. Pure bending was attained by applying prescribed rotation to edge nodes.

The static springback in draw-bend test was simulated, for a range of back force

(0.1 ≤ Fb ≤ 1.2), specimen width (W = 12mm–50mm) and tool radius (3.2mm–

12.7mm). Both 2D and 3D analysis were carried out, using plane-stress beam element (Abaqus type B21), and 4-node shell element with reduced integration (S4R), respectively [24]. The sheet strip was modeled by 300 elements of non-uniform size in the longitudinal direction. Smaller elements were used in the contact areas (Regions

2 and 3, Figure 5.2) to ensure numerical accuracy, with one contact node per 4.5 degrees of turn angle [82]. In 3D FE models, only half of the physical strip was modeled because of the mirror symmetry, with 8 elements in the sheet-width direction.

Fifty-one integration points were used through the sheet thickness, for both B21 and

S4R elements, to minimize numerical error [82]. In order to closely represent the plastic anisotropy of 6022-T4 sheet, the Barlat’96 yield function [43] and a modiﬁed anisotropic hardening model [46] were adopted. A friction coeﬃcient of 0.15 was used for lubricated test conditions [69].

Simple bending theory suggests that the draw-bend process is closer to plane- strain deformation [83], considering the large width-to-thickness ratio (W/t=55).

However, 2D ﬁnite element simulations demonstrated that plane-stress (with B21 beam element) results were consistently better than plane-strain (with S4R shell element) [82]. In FE models, plane-strain assumption was enforced by prescribing zero

74 80 6022-T4, R/t=10.5 70 von Mises yield Isotropic hardening 60

40 (degree) 30 ∆θ 20 Exp. Plane-stress 10 Plane-strain 3D 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Normalized back force F b

Figure 5.3: Simulated springback angle using various assumptions.

lateral displacement to all nodes in the width direction [24]. For the sake of simplicity, only von Mises yield function and isotropic hardening law were used in these simulations. As shown in Figure 5.3, simulations using 2D beam elements are in better agreement with the experiment data than that of plane-strain, Figure 5.3. It is also noted that 2D simulations causes signiﬁcant error for larger back forces, because neither plane-stress nor plane-strain is able to represent the anticlastic curvature.

Full 3D simulations correctly reproduce the fast decline of springback angle when

Fb > 0.7. This phenomenon is closely related to the persistent anticlastic curvature, which will be explored in detail in the following text.

75 5.4 Elastic Theory for Plate Bending

Before interpreting the anticlastic curvature during and after draw-bending, the simpler case of pure bending is considered. The classic bending theory of elastic plate is summarized here, while details can be found elsewhere [84, 92]. The results are based on the work of Ashwell [84, 92], which makes use of Marguerre’s large deformation theory of plate bending [104] and extends von K´arm´an’sanalysis [105].

Unlike the small deformation theory, the membrane stress at the plate middle surface is considered.

The general theory is ﬁrst introduced, with approximations suitable for closed- form solution. Then, these results are applied to two cases of interest for the draw- bend application: 1) bending of an initially ﬂat sheet, and 2) unbending (straightening) of an initially curved sheet.

5.4.1 Problem Statement and Closed-Form Solution

The problem of bending an initially curved plate is shown schematically in Fig- ure 5.4. A rectangular plate (dimension L, W and t), with initial middle surface shape described by z0(x, y) and radii of curvature Rx0 and Ry0, is subjected to a uniform

L bending moment mx (per unit width) applied to the plate edges, i.e, at x = ± 2 . Following Ashwell’s work [84], the governing diﬀerential equation (GDE) for this problem is given as follows: · ¸ t q ∂2F ∂2z ∂2F ∂2z ∂2F ∂2z ∇4(z − z ) = + + − 2 (5.2) 0 D t ∂x2 ∂y2 ∂y2 ∂x2 ∂x∂y ∂x∂y "µ ¶ µ ¶ µ ¶# ∂2z 2 ∂2z 2 ∂2z ∂2z ∂2z ∂2z ∇4F = E − 0 − − 0 0 (5.3) ∂x∂y ∂x∂y ∂x2 ∂y2 ∂x2 ∂y2

76 mx z y Ry ,W n x x width q(x,y) nx length t , L mx

Figure 5.4: Rectangular coordinate system for plate bending problems.

4 ∂4 ∂4 ∂4 where ∇ = ∂x4 + 2 ∂x2∂y2 + ∂y4 is the bi-harmonic diﬀerential operator, q(x, y) is

Et3 the surface pressure acting along the z-axis and D = 12(1−ν2) is the bending ﬂexural rigidity of an elastic plate. F (x, y) is a stress function, from which the tensions per unit length in the plate can be derived:

∂2F ∂2F ∂2F n = t , n = t , n = −t (5.4) x ∂y2 y ∂x2 xy ∂x∂y

It is assumed that the plate middle plane has the following shapes before and after deformation [92], respectively:

x2 x2 z0 = ze0(y) + , z = ze(y) + (5.5) 2Rx0 2Rx where Rx0 and Rx are the initial and deformed radii of the primary curvature, respectively. It was further assumed that the undeformed transverse shape at any position of ﬁxed x can be similarly written as

y2 W 2 ze0(y) = − (5.6) 2Ry0 12Ry0 77 where Ry0 is the initial radius of anticlastic curvature. The unknown to be solved from Equations (5.2) and (5.3) is the anticlastic deﬂection, ze(y).

When there is no lateral pressure applied on the plate surface, i.e., q(x, y) =

0, the proﬁles given by Equations (5.5) and (5.6) satisfy the governing diﬀerential

Equations (5.2) and (5.3) if µ ¶ d4(ze − ze ) d2ze Et ze ze D 0 − T + − 0 = 0 (5.7) dy4 dy2 R R R ZZ µ x¶ x x0 ze ze T x2 −E − 0 dydy + = F (5.8) Rx Rx0 2t

W L where T is a constant. At the plate edges where y = ± 2 and x = ± 2 , the boundary conditions are:

ny = T (5.9) µ ¶ ze ze0 nx = −Et − (5.10) Rx Rx0 · 2 ¸ µ ¶ ∂ (ze − ze0) 1 1 my = D 2 + νD − (5.11) ∂y W Rx Rx0 y=± 2 µ ¶ · 2 ¸ 1 1 ∂ (ze − ze0) mx = D − + νD 2 (5.12) Rx Rx0 ∂y L x=± 2 · 3 ¸ ∂ (ze − ze0) nyz = −D 3 (5.13) ∂y W y=± 2

nxz = 0 (5.14)

For an elastic plate with single curvature ( 1 = 0, 1 6= 0) subjects to the Rx0 Ry0

W following boundary conditions at y = ± 2 :

ny = my = nyz = 0 (5.15)

Equations (5.7) and (5.8) reduce to a fourth order, homogeneous ODE, when the radius of primary bending curvature is Rx: d4ze + 4γ4ze = 0 (5.16) dy4 78 4 3(1−ν2) 1 with γ = 2 2 . If the initial primary curvature in non-zero, i.e., 6= 0, a forth Rxt Rx0 order, non-homogeneous ODE is resulted:

4 2 d ze 4 12(1 − ν )ze0 4 + 4γ ze = . (5.17) dy RxRx0t

When the RHS of Equation (5.17) is small, it can be neglected without causing signiﬁcant error. The solution of Equation (5.16), i.e., the transverse (anticlastic) deﬂection, is given as follows:

ze(y) = cosh γy(C1 cos γy + C2 sin γy) + sinh γy(C3 cos γy + C4 sin γy). (5.18)

Due to the symmetric anticlastic proﬁle, i.e., ze(y) is an even function of y, C2 = C3 = 0 and the above solution reduces to

ze(y) = C1 cos γy cosh γy + C4 sin γy sinh γy (5.19)

The integration constants C1 and C4 can be determined from the boundary conditions previously discussed:

· 2 ¸ d ze £ 2 2 ¤ 1 ν ν 2 = 2γ C4 cos γy cosh γy − 2γ C1 sin γy sinh γy = + − dy W Ry0 Rx0 Rx y=± 2 (5.20) · 3 ¸ d ze £ 3 3 ¤ 3 = 2γ (C4 − C1) cos γy sinh γy − 2γ (C4 + C1) sin γy cosh γy = 0. dy W y=± 2 (5.21)

Solve the above two equations for constants C1 and C4 and substitute them back into

Equation (5.19), the anticlastic proﬁle is obtained µ ¶ ze Rx Rx ν = + − 1 p [K1 cosh γy cos γy + K2 sinh γy sin γy] (5.22) t νRy0 Rx0 3(1 − ν2) where the constants K1 and K2 are deﬁned as follows: µ ¶ · µ ¶ µ ¶ µ ¶ µ ¶¸ K 1 γW γW γW γW 1 = sinh cos ∓ cosh sin K2 sinh γW + sin γW 2 2 2 2

79 The normal stresses σ11 and σ22 are known for for an isotropic elastic plate [78]: 6m h σ = ± x 1 − 2ν2(K cos γy cosh γy − K sin γy sinh γy) 11 t2 2 1 q i (5.23) 1−ν2 ± 2ν 3 (K2 sin γy sinh γy + K1 cos γy cosh γy) 6m σ = ± x [ν − 2ν(K cos γy cosh γy − K sin γy sinh γy)] (5.24) 22 t2 2 1 where the plus and minus signs are for tensile and compressive stresses respectively.

6mx Notice that the solution is normalized by t2 , which the longitudinal stress at the outer ﬁber of a beam subjected to plane-stress bending.

5.4.2 Results for Pure Bending of an Initially Flat Plate ( 1 = 1 = 0) Rx0 Rx0

Equation (5.22) maybe used to visualize the anticlastic curvature for a range of plate widths and primary curvatures. Figure 5.5 shows the variation of the normalized anticlastic deﬂection and the normalized transverse stress along the plate width direction, for a rectangular plate (W = 50 and t = 1) bent to various curvatures.

As shown in Figure 5.5, the anticlastic displacement tends to localize toward the plate edges as β increases, while the plate center remains relatively ﬂat. Correspond- ingly, more transverse stress is developed in the central area of the plate, but it decays to zero at the plate edges, Figure 5.5. As β increases, the area with biaxial stress state expands, and the transverse stress drops faster near the edges. In the limit of β = ∞, the center of the plate approaches plane-strain, with σ22 = νσ11, as illustrated by the thin dotted line in Figure 5.5(b). From the elastic plate theory, it is known that a lateral bending moment, my = νmx, exists over the ﬂat portion of the deformed plate to maintain a cylindrical surface. However, my must equal zero at the edges. As pointed out by Fung [106], the near-edge region with localized anticlastic deformation

80 0.12 0.4 W=50, t=1 1/3 0.1 ν=1/3, β=W 2/R t x

0.08 11 0.3 σ / 22

0.06 σ β=5 0.04 β*=13.5 0.2 β=50 0.02 β=400

0 Stress ratio 0.1 β=5 β*=13.5

Normalizeddeflection z/t -0.02 β=50 β=400 -0.04 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Normalized with coordinate y/W Normalized width coordinate y/W (a) (b)

Figure 5.5: Closed-form solution for: (a) normalized anticlastic deﬂection, and (b) normalized transverse stress.

acts as a boundary layer, through which the later bending moment is built up, from zero at edge, to νmx at the center.

The critical value β∗ at which the anticlastic deflection curve starts to change and thus to have an inflection point can be calculated as follows: · ¸ µ ¶ µ ¶ d2ze γW γW 2 = 0 ⇒ tan + tanh = 0 (5.25) dy y=0 2 2 √ p where γW = β 4 3(1 − ν2). The first root of the above transcendental equation

∗ 1 gives the critical value of β: β = 13.5 when ν = 3 . The maximum deﬂection at edges can also be derived. Knowing that sinh(γW ) ≈

1 γW cosh(γW ) → 2 e as γW → ∞, and the constants K1 and K2 become µ ¶ · µ ¶ µ ¶¸ K1 − γW γW γW = e 2 cos ∓ sin (5.26) K2 2 2

81 Near the plate edges, approximations are also made such that sinh(γy) ≈ cosh(γy) →

1 γy 2 e . Therefore, the anticlastic deﬂection becomes µ ¶ ze R R ν = x + x − 1 p e−γy¯ [cos(γy¯) − sin(γy¯)] (5.27) t νRy0 Rx0 3(1 − ν2)

W wherey ¯ = 2 − y is the distance measured from the plate edge toward plate center. For initially ﬂat plate, i.e., 1 = 1 = 0, the maximum deﬂection occurs at the Rx0 Ry0 plate edges and it only depends on Poisson’s ratio: µ ¶ ze ν = p . (5.28) 2 t max 12(1 − ν )

1 For ν = 3 , the maximum deﬂection zemax is 10.2% of sheet thickness.

β Case W (mm) L (mm) R (mm) Mesh (W × L) a 5 8 5 10×32 5 b 10 32 20 10×64 c 20 128 80 10×128 a 25 10 6.25 25×20 100 b 50 40 25 25×40 c 100 160 100 25×80

Table 5.1: Parameters used in ﬁnite element model for elastic bending.

In order to verify the closed-form solution, a series of elastic finite element simulations were conducted for pure bending of an initially flat sheet (t = 1) with various widths and bending radii, as listed in Table 5.1. The finite element meshes are so chosen that the element aspect ratio is 1. Further refinement showed negligible difference in result, as will be shown later. In order to assess the invariance of the FE results

82 for a ﬁxed β, this parameter is rewritten in terms of non-dimensional quantities as follows: W 2 (W/t)2 β = = (5.29) Rt (R/t)

The second form reveals the relationship between β, the thickness-normalized specimen width (W/t), and the normalized bending radius (R/t).

The simulation results for β = 5 are compared with the analytic solutions in

Figure 5.6, using 3 combinations of strip widths and bending radii. The simulated anticlastic proﬁles agree with the closed-form solutions, for this intermediate case: neither plane-stress nor plane-strain.

0.16 0.4 W5, R=5 W=10, R=20 1/3 W=20, R=80 0.12 Analytic 11 0.3 W5, R=5 σ

/ W=10, R=20

22 W=20, R=80

σ Analytic 0.08 0.2

0.04

Stress ratio 0.1 Normalizeddeflection z/t t=1, ν=1/3, β=5 t=1, ν =1/3, β=5 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Normalized width coordinate y/W Normalized width coordinate y/W (a) (b)

Figure 5.6: Anticlastic deﬂection and stress ratio for an elastic material with β = 5.

For larger β, the analytic solution predicts that the anticlastic deflection is more concentrated toward the sheet edges. This is confirmed by finite element results for

β = 100, as shown in Figure 5.7(a). The stress state in the sheet center area is

83 close to plane-strain, with stress ratio σ22 approaching the Poisson’s constant, see σ11 Figure 5.7(b). Mesh reﬁnement shows negligible diﬀerence in displacement and stress solution, as illustrated by the selected case for W = 50mm, Figure 5.7.

0.12 0.4 Theory σ /σ =1/3 W=25, R=6.25 22 11 W=50, R=25 (12x20) 0.08 W=50, R=25 (25x40) 0.3 W=50, R=25 (50x80) Analytic

W=100, R=100 11 W=25, W=6.25 σ

0.04 / 0.2 W=50, R=25 (12x20) 22

σ W=50, R=25 (25x40) W=50, R=25 (50x80) 0 0.1 W=100, R=100 Normalizeddeflection z/t

t=1, ν=1/3, β=100 t=1, ν =1/3, β=100 -0.04 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Normalized width coordinate y/W Normalized width coordinate y/W (a) (b)

Figure 5.7: Anticlastic deﬂection and stress ratio for elastic material with β = 100.

Conventional knowledge usually distinguishes plane-stress bending from plane- strain bending by the ratio of width to thickness [83]. For example, plane-stress bending is assumed for W/t ¿ 1 and plane-strain for W/t À 1. For these two limiting cases, the primary curvature is proportional to the bending moment M, according to elementary bending theory:

½ M 1 EI , plane-stress = M(1−ν2) (5.30) Rx EI , plane-strain where I is the section moment of inertia. Based on the previous analysis, the transition from plane-stress to plane-strain is not only a function of W/t, if the anticlastic

84 deformation is considered. The actual stress state depends not only on W/t, but also on the normalized primary bending radius Rx/t (as reﬂected in the Searle’s parameter

β).

The eﬀect of anticlastic curvature on stress state can be realized as follows. When the anticlastic surface can freely develop (β ¿ 1), it has a constant curvature of

ρˆ = − ν by the elastic Poisson’s effect. When considering the width effect as β y Rx increases, the anticlastic deflection is suppressed in the central area of the plate.

As a result, the transverse curvature varies across width, as can be evaluated from

Ashwell’s closed-from solution (Equation 5.19):

d2ze 2ν ρy = 2 = − [K2 cosh γy cos γy − K1 sinh γy sin γy] (5.31) dy Rx Through Poisson’s ratio, the restrained anticlastic curvature will aﬀect the principal one. Now, Equation (5.30) can be generalized to incorporate the change of the principal bending curvature caused by the anticlastic deformation (for the same bending moment), in terms of a dimensionless parameter φ:

1 M 2 ρx = = (1 − φν ) (5.32) Rx EI Plane-stress and plane-strain are two special cases corresponding to φ = 0 and φ = 1 respectively. Parameter φ has been called anticlastic factor [85], and it depends only on β and the Poisson’s constant ν: · √ √ ¸ 2 cosh(k β) − cos(k β) φ = 1 − √ √ √ (5.33) k β sinh(k β) + sin(k β) p where k = 4 3(1 − ν2).

Figure 5.8 shows that φ is close to zero when β is small, but it saturates for large

β. The transition from beam to plate is no longer abrupt. The fast rise of φ correlates

∗ 1 to the aforementioned critical value β : φ reaches 0.5 when β = 13.5 for ν = 3 . 85 1 φ=1 (plate) 0.8 φ φ=0.5 0.6 β*=13.54

0.4

Primary curvature Anticlastic factor 0.2 ρ = M*(1-φν 2)/EI x φ=0 (beam) 0 0 200 400β 600 800 1000

Figure 5.8: Variation of the anticlastic factor φ with β.

d2ze ν Equation (5.31) reduces to 2 = − , as γ → 0, i.e., the transverse curvature is dy Rx constant along the width direction, and it has opposite sign of the primary one. The ratio between these two curvatures is equal to the Poisson’s ratio, ν.

5.4.3 Bending of Initially Curved Plate

The foregoing model applies to the initial bending stage of the draw-bend test as an initially ﬂat sheet is drawn over the tool radius. The second stage to be considered is the unbending, or straightening, of the primary curvature as the strip leaves contact with the tooling. Considering the contact constraints with the adjacent tooling, the

0 0 initial condition of the plate for this stage can be idealized as Rx = R and Ry = ∞ (i.e., no anticlastic curvature while in contact with the tool), as shown schematically in Figure 5.9. This starting condition is consistent with FE analysis of the draw-bend operation, which shows essentially no transverse curvature near the tool contact.

86 Initial shape y

Thickness t x

Rx0 Length A

L A

z Section A−A

Width W ze(y) Unbend R = ∞ mx x y 2 y W 2 z0(y)= − Straightened shape 2Ry0 12Ry0

Figure 5.9: Bending an initially curved plate.

The curved plate is straightened by applying a uniform moment mx (per unit width) along the transverse edge, until the primary curvature disappears, Figure 5.9.

In order to solve the inhomogeneous ODE (Equation 5.17) in closed-form, the initial plate proﬁle is approximated by parabolic function, as given by Equations (5.5) and

(5.6). To calculate the shape of a cross-section A-A, ze(y), after unbending, the following approximations are utilized:

sinh x ≈ sin x ≈ x, and cosh x ≈ cos x ≈ 1 as x → 0 (5.34)

Then, constants K1 and K2 become

µ ¶ γW γW µ ¶ K1 2 ∓ 2 0 = = 1 (5.35) K2 γw + γw 2

87 and the anticlastic deﬂection reduces to

z νy2 ³z ´ ν W 2 ν = ,, and = = βx0 (5.36) t 2Rx0t t max 8 Rx0t 8

According to the foregoing analysis based on the elastic bending theory, straightening an initially curved plate ( 1 6= 0, 1 = 0) produces signiﬁcant anticlastic Rx0 Ry0 deﬂection. The maximum anticlastic displacement is proportional to the Searle’s

W 2 parameter, βx0 = , and the Poisson’s ratio, ν. Rx0t

6 6 W=5 Theory: (z/t) =β ν/8 W=5 max x0 ν=0.5 W=10 W=10 2 max 5 β =W /R t, t=1, ν=1/3 max 5 W=25 x0 x0 W=25 W=50 W=50 4 W=100 4 W=100 ν=1/3 Theory Theory 3 3

2 2

1 1 Maximum deflection (z/t) Elastic material Maximum deflection (z/t) Elasto-plastic material 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Searle's parameter β Searle's parameter β x0 x0

(a) Elastic (b) Elasto-plastic

Figure 5.10: Maximum anticlastic deﬂection from ﬁnite element simulation.

To validate the closed-form solution, the straightening problem was simulated by

ﬁnite element method, using both elastic and elasto-plastic material models. For the later case, von Miese yield function and isotropic hardening were used, with a Voce’s type of hardening law [6]: σ = 381−215e−9.6εp (Mpa). The FE model is schematically shown in Figure 5.9, where a cylindrical plate is ﬂattened by a uniform moment mx

88 at its transverse edge. The cross-sections at x = 0 after unbending are compared with the analytic solution given by Equation (5.36), Figure 5.10. As will be discussed next, the error caused by the parabolic approximation of a circular cross-section is minimal as x → 0. As shown in Figure 5.10(a), elastic simulation results agree with ¡ ¢ z the closed-form solution. The maximum anticlastic deﬂection, t max, is invariant for a ﬁxed β, regardless of the specimen widths (W =5mm–100mm) and the initial radii

(Rx0 =5mm–2000mm). Both simulation and closed-form solution demonstrate that the maximum anticlastic deﬂection at the plate edges can be a few times the plate thickness, while bending an initially ﬂat plate can only produce zmax = 0.102t when

1 ν = 3 . This explains why signiﬁcant anticlastic displacement happens in the forming step of the draw-bend test, where a sheet is straightened as it leaves the tool surface.

For an elasto-plastic material, simulation results deviate from the elastic solution when β > 20, Figure 5.10(b). This is because shear stress is required to maintain

1 compatibility between the elastic (ν = 3 ) and plastic (ν = 0.5) regions of an plastically bent plate, while the shear eﬀect is neglected in deriving the closed-form solution [92].

At the same β value, larger deviation exists for narrower plates, because the primary bending radius is smaller and more plastic deformation occurs throughout the plate thickness.

5.4.4 Error Analysis of Elastic Theory

As previously mentioned, one major assumption was made in order to solve the fourth order ODE of the elastic bending problem. That is, the initial cross-sections in the longitudinal and width directions, and the distorted cross-section in the longitudinal direction, were all approximated by parabola. Or equivalently, curvature is

89 x2 +(z − R)2 = R2

z R

A b B C h D 2 ∆ z x ∆h = 2R b O x x = −2

Figure 5.11: Error analysis for using parabolic function.

00 z00 calculated by ρ = z instead of ρ = 3 . As shown schematically in Figure 5.11, (1+z02) 2 the parabolic approximation causes signiﬁcant error when the chord length, b, is a signiﬁcant fraction of the radius of a circle, R. The depth of the parabola COD, ∆h,

b ¯ (at x = ± 2 ), and the depth of the arc AOB, ∆h are calculated by

b2 ∆h = (5.37) 2R  s µ ¶ b 2 ∆h¯ = R 1 − 1 −  (5.38) 2R

Apply Taylor’s series expansion to the second equation, µ ¶ µ ¶ b2 R b 4 b ∆h¯ = + + O (5.39) 8R 8 2R 2R where O(•) indicates the higher order terms. Then, the percentage error introduced

∆h¯−∆h by using parabolic function is deﬁned as Err = ∆h × 100%. For various ratios of b R , the computed errors are listed in Table 5.2.

90 b R 0.2 0.28 0.45 0.63 1.0 2.0 Err (%) 1 2 5 10 25 100

Table 5.2: Error introduced by using parabolic function.

b As can be seen from Table 5.2, the error quickly grows as R increases. For draw- bend test, the unbending starts from an initially circular strip that was wrapped

b around a tool of radius R, so that R = 2. Therefore, nearly 100% error is expected if the closed-form solution is applied to a cross-section that is located near the plate

W edges (x = 2 , Figure 5.9).

5.5 Anticlastic Curvature in Draw-Bend Test

Two groups of rectangular 6022-T4 aluminum strips, with 0.9mm thickness, were tested in the current work. Specimens from the ﬁrst group were 50mm wide, and they were tested under various sheet tensions for two tool radii of 12.7mm and 3.2mm. The second group had strip width ranging from 12mm to 50mm, and was tested using

12.7mm tool at two normalized sheet tension: Fb = 0.5 and Fb = 0.9.

5.5.1 Eﬀect of Back Force

As shown in Figure 5.12, the springback angle decreases with the normalized back force, while the anticlastic curvature varies oppositely. As Fb approaches 0.7–0.8, there is a dramatic drop in the springback angle, accompanied by a rapid increase in the anticlastic curvature. The sudden decrease of ∆θ has been attributed to the

91 persistent anticlastic curvature, which substantially increases the section moment of inertia when Fb ≥ 0.7 [69].

100 6 Barlat'96 yield Barlat'96 yield G-W hardening ) G-W hardening -1 80 8x300 S4R 5 8x300 S4R mm -3 4 60 (x10 a 3 R/t=3.5 (Exp.)

(degree) R/t=3.5 (FEM) 40 R/t=14 (Exp.)

∆θ 2 R/t=14 (FEM) R/t=3.5 (Exp.) 20 R.t=3.5 (FEM) R/t=14 (Exp.) 1 R/t=14 (FEM) Curvature1/R 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back forcre F Normalized back forcre F b b

(a) (b)

Figure 5.12: Eﬀect of the back force on: (a) springback angle and (b) unloaded anticlastic curvature. Lines are FEM simulation results and markers are experimental data.

Simulated springback angles agree with the experimental data for both bending radii (R/t = 3.5 and R = 14), Figure 5.12(a). It is also noticed that small bending radius causes more springback, which is contrary to the simple bending results.

However, it is in accord with other stretch-bending experiments [74].

Both experiment and FEM simulation have shown the decrease of anticlastic curvature when Fb ≥ 0.9–1.0, as can be seen from Figure 5.12(b). However, simulation results for small bending radius (R/t = 3.5) shows appreciable deviation from measurement. This is presumably because of the use of shell element which does not

92 work well for small radius bending [82]. When R/t < 5, the general shell assumptions, namely zero through-thickness stress and plane section remaining planar after deformation, are no longer valid [8].

3 1.6 Barlat'96 yield G-W hardening 2.5 t=0.92 mm 1.2

2 h(mm) h(mm) ∆

∆ 0.8 1.5

Loaded 0.4 1 Unloaded R/t=14 R/t=3.5 R/t=3.5 R/t=14 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Normalized front force F Normalized front force F f f

(a) (b)

Figure 5.13: Variation of the maximum anticlastic deﬂection with the front pulling force in the draw-bend test: (a) loaded and (b) unloaded.

The maximum anticlastic deﬂection (i.e., ∆h) from FE simulation is plotted against the normalized front force (Ff ), see Figure 5.13. Before unloading, the maximum anticlastic deﬂection decreases with the front force, and it can be as much as

3 times the sheet thickness when Ff < 0.5. The remaining depth gradually increases after springback, with a abrupt change as the normalized front force approaches unity,

Figure 5.13(b). This indicates that the persistence of anticlastic curvature is determined by the sheet tension.

93 The persistent anticlastic curvature has a signiﬁcant role in reducing springback.

As shown in Figure 5.14, the bending moment continuously decreases with the back force, so does the springback angle if there were no sudden change in the moment of inertia of the sheet cross-section. However, the bending rigidity of the strip dramatically increases because of the persistent anticlastic curvature during springback when

Ff ≥ 1.0. The moment of inertia for a circular cross-section, I, can be calculated using standard formulas [107]. As illustrated in Figure 5.14, the normalized moment of inertia, I/I0, is increased by a factor of about 3 when the front force exceeds yielding.

W t3 Here, I0 = 12 is the reference moment of inertia for a ﬂat rectangular cross-section. Because of this sudden increase of bending rigidity, the springback angle is greatly reduced, Figure 5.12(a).

700 4 R/t=3.5 600

500 Flat section 3 I = Wt 3/12 0

400 I/ I

R/t=14 0 300 2 Moment (N*mm) Moment 200 Barlat'96 Yld 100 G-W hardening 8x300 S4R 0 1 0 0.2 0.4 0.6 0.8 1 1.2 Normalized back force F b

Figure 5.14: Moment and normalized section moment of inertia by ﬁnite element simulation.

94 5.5.2 Eﬀect of Specimen Width

The experimental results for springback angle and anticlastic curvature are plotted in Figure 5.15, for specimens with various width from 12mm to 50mm. For both

Fb = 0.5 and Fb = 0.9, the anticlastic curvature decreases with the sample width.

However, the springback angle, ∆θ, ﬁrst decreases with sample width for the case of

Fb = 0.5, then it increases after a local minimum that happens around W = 25mm,

Figure 5.15(a). For Fb = 0.9, ∆θ monotonically declines with the specimen width,

Figure 5.15(b).

10 30 10 ∆θ ∆θ Curvature (x 10 (x Curvature 44 10 (x Curvature -1 25 -1 Curvature (mm ) 8 Curvature (mm ) 8

20 40 6 6 15 (degree) 4 (degree) 4 -3 -3 36 -3

∆θ 10 ∆θ mm mm

2 -1 2 -1 5 ) 32 ) R/t=14, F =0.5 R/t=14, F =0.9 b b 0 0 0 10 20 30 40 50 60 10 20 30 40 50 60 Width (mm) Width (mm) (a) (b)

Figure 5.15: Springback angle and anticlastic curvature from draw-bend test for (a) Fb = 0.5 and (b) Fb = 0.9.

Finite element simulations were carried out for samples tested by Fb = 0.5. As shown in Figure 5.16, simulation agrees with the measured springback angle and anticlastic curvature. Result of a 2D simulation using beam elements is also presented in Figure 5.16(a) for comparison. Since plane-stress state is assumed in the sheet

95 width direction when beam elements (B21) are used, it gives the highest springback angle. As specimen becomes narrower, ∆θ using 3D shell elements approaches the result predicted by 2D beam elements.

50 10 Exp. R/t=14, F =0.5 b FEM (S4R) )

FEM (B21) -1 8 45 mm

-3 6 40

(degree) 4 ∆θ 35 2 Curvature (x10 Exp. R/t=14, F =0.5 FEM (S4R) b 30 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Width (mm) Width (mm) (a) (b)

Figure 5.16: Compare simulation and measurement for Fb = 0.5 case: (a) springback angle and (b) anticlastic curvature.

5.5.3 Application of the Elastic Bending Theory

To apply the closed-form solution, the draw-bend test procedure is divided into three sequential steps: bending, unbending and springback. The ﬁrst step is trivial, in which the sheet is wrapped around the tool under superposed tension. It is treated as a plane-strain problem, because the strip conforms to the tool surface, and hence transverse displacement can be neglected.

96 The second and third steps are much more complicated 3D problems, because anticlastic deformation is present in both steps. In the unbending step, the strip

1 loses its primary curvature ( R , when it is in contact with the tool) as it slides over the tool surface. Eventually, it becomes straight in the longitudinal direction, i.e,

(2) 1 Rx = ∞. Meanwhile, a transverse curvature, (2) , is developed in the sheet width Rx direction. In the last step, the sample obtains a primary curvature in the side-wall curl region after springback. The radius of this curvature, r0, depends on the sheet tension, tool radius and friction condition, as well as the material properties such as the yield surface shape and the strain hardening behavior [69,82]. During springback,

1 1 the previously developed anticlastic curvature will change from (2) to R . Table 5.3 Ry a summarizes the primary and anticlastic curvatures involved in all three steps.

Step Rx0 Ry0 Rx Ry (1). Bending ∞ ∞ R ∞ (2) (2). Unbending R ∞ ∞ Ry (2) 0 (3). Springback ∞ Ry r Ra

Table 5.3: The radii of primary and anticlastic curvature in draw-bend test.

The elastic theory of plate bending can be applied for the last step, as schematically shown in Figure 5.17. The output of the unbending step from FE is used as input for the analysis in the springback step. Then, the predictions by the elastic theory are compared with the FE results; and, for two cases, with experimental data as well. The goal is to explain why the anticlastic curvature persists after springback

97 only for small sheet tensions. Apparently, the unbending process involves plasticity, and thus the elastic plate theory will not give satisfactory results. Nonetheless, it helps to understand why the maximum anticlastic deﬂection before unloading can be as much as 3 times the sheet thickness in the draw-bend test. For the last step, the application of elastic theory is reasonable, since springback is generally an elastic process.

Springback analysis

R Ry0

≈ ∞ Rx0 nx mx t

W y L z x

mx nx Loaded sample

Figure 5.17: Unbending and springback analysis for draw-bend test.

To apply the elastic theory for the springback analysis, three radii of curvature,

(2) (2) (3) Rx , Ry and Rx from FE simulation, are used as input of the closed-form solution,

98 (3) to calculate the anticlastic curvature after springback, Ry . The predicted maximum anticlastic deflection is then compared with the FE results. As shown in Figure 5.18, the elastic theory prediction agrees qualitatively with the finite element simulations for both tool radii, but it under-estimates the magnitude of the anticlastic deflection.

Both FEM and elastic prediction show that the unloaded depth of the anticlastic profile initially increases with the sheet tension, but decreases after the front force exceeds the yielding force of the strip, which corresponds to the occurrence of the persistent anticlastic curvature after springback. It is also noted that the elastic prediction deviates more from the FE results as the normalized front force is larger than unity. On the other hand, the elastic solution is closer to the finite element simulation for larger tool radius. One possible explanation is that springback also involves non-elastic deformation because of the large plastic deformation accumulated before unloading, and the reduction of the flow stress after a reversed strain path [46].

The calculated cross-section proﬁles are now compared with the FE simulations and the experimental data, for two specimens tested at Fb = 0.4 and Fb = 0.8 with

R/t = 14.0, Figure 5.19. For small back force, the anticlastic deﬂection is localized toward the specimen edges, while the cross-section appears to be circular for Fb = 0.8.

The overall agreement between the elastic prediction, ﬁnite element simulation and experimental measurement is fairly satisfactory. However, the elastic solution tends to under-estimate the magnitude of the anticlastic deﬂection, as previously demonstrated by Figure 5.18.

99 1.6 1.6 Barlat'96 yield Barlat'96 yield G-W hardening G-W hardening R/t=3.5, 8x300 S4R R/t=14, 8x300 S4R 1.2 1.2 h(mm) h(mm)

∆ 0.8 ∆ 0.8

0.4 0.4 Unloaded Unloaded FEM FEM Analytic Analytic 0 0 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.2 1.4 1.6 Normalized front forcre F Normalized front forcre F f f

(a) (b)

Figure 5.18: Comparison of simulated and analytically predicted maximum anticlastic deﬂections for draw-bend tested samples after springback: (a) R/t = 3.5 and (b) R/t = 14.0.

0.05 0 R/t=14 R/t=14 F =0.4 F =0.8 0 b -0.2 b -0.4 -0.05 -0.6

-0.1 -0.8

-1 -0.15 -1.2 Exp. Anticlastic deflectionz Anticlastic deflection z -0.2 Exp. FEM -1.4 FEM Analyitc Analytic -0.25 -1.6 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Noramalized width coordinate y/W Noramalized width coordinate y/W (a) (b)

Figure 5.19: Comparison of measured, simulated and analytically predicted anticlastic proﬁles for draw-bend tested samples: (a) Fb = 0.4 and (b) Fb = 0.8.

100 5.6 Discussion

Based on the previous analysis, the anticlastic deformation in draw-bend tested samples can be characterized by a single dimensionless parameter, β, which combines the inﬂuence of the sheet tension, tool radius and sample geometry. As shown in

Figure 5.20, the cross-section profiles are plotted for two different groups of draw- bend tested specimens. The first group has the same width but the back force is different, Figure 5.20(a); while Figure 5.20(b) shows the effect of the sample width on the cross-section shape. For both cases, it is realized that β uniquley determines the shape of the cross-section. When β < 10–15, all cross-sections are nearly circular. As

β increases, the anticlastic deﬂection tends to localize toward the sheet edges, while the center of sheet is essentially ﬂat.

0.5 β β=11.92, R/t=14, F =0.7 0 =0.28, W=6.3mm 0 b

-0.5 -0.1 β=1.1, W=12.5mm β=4.05, R/t=3.5, F =1.1 -1 b -0.2 b=4.32, W=25mm -1.5 β 0 0 =19.18, W=50mm

-0.1 β=22.25, R/t=14, F =0.4 -0.1 b R/t=14 F =0.5 β b Anticlastic deflection (mm)

Anticlastic deflection (mm) =31.98, R/t=3.5, F =0.2 Unloaded b -0.2 -0.2 Unloaded β=10.48, W=37mm Width=50mm 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Normalized width coordinate y/W Normalized width coordinate y/W (a) (b)

Figure 5.20: Simulated anticlastic proﬁles of draw-bend tested samples: (a) for various back forces and (b) various widths at Fb = 0.5.

101 Figure 5.21 summarizes how β varies with the normalized back force and the specimen width. It is worthwhile to notice the similarity between Figure 5.21(a) and

W 2 Figure 5.12(a). Both the unloaded β (calculated as r0t ) and the springback angle (∆θ) decreases with increasing back force. The rapid drop in β happens around Fb = 0.7–

0.8, which corresponds to the occurrence of the persistent anticlastic curvature.

40 20 β=W 2/(r' t) β=W 2/(r' t) t=0.92mm R/t=14, F =0.5 b 30 15 t=0.92mm

20 10 (unloaded) (unloaded) β β 10 5 R/t=3.5 R/t=14 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 10 20 30 40 50 60 Normalized back force F b Width (mm) (a) (b)

Figure 5.21: Variation of Searle’s parameter β with (a) back force and (b) specimen width.

5.7 Conclusions

Elastic bending theory and ﬁnite element simulation are utilized to investigate the role of forming variables on anticlastic curvature, and springback by draw-bend test.

The following conclusions are reached:

102 1. Springback steadily decreases as sheet tension increases, with sudden decline

of springback angle as the front pulling force approaches yielding. Persistent

anticlastic curvature is identiﬁed as the cause of this rapid change. Larger tool

radius leads to less springback, but in a less important way as sheet tension

does.

2. In draw-bend test, anticlastic curvature is developed in the unbending process

during forming, and it persists after springback when the applied sheet ten-

sion exceeds a critical value. The persistent anticlastic curvature signiﬁcantly

increases the section moment of inertia, and thus dramatically reduces spring-

back.

3. For Fb = 0.5 and R/t = 14, the springback angle ﬁrst decreases, then increases

with strip width, but the anticlastic curvature monotonically decreases with

specimen width. However, both springback angle and anticlastic curvature de-

crease with specimen width for Fb = 0.9.

4. The occurrence and persistence of anticlastic curvature in draw-bend test can

be explained by elastic bending theory. The cross-section shape after unloading

is determined by Searle’s parameter, β, which depends on the specimen geom-

etry (W/t) and sheet tension (via the curl radius Rx). The rapid decrease in

springback angle at Fb = 0.7–0.8 corresponds to a critical β value of 10–15,

above which the anticlastic displacement tends to concentrate toward the sheet

edges.

5. It is worth noting that the stress state in the lateral direction (plane-stress or

plane-strain) cannot be simply identiﬁed by the sheet width-to-thickness ratio,

103 even for simple bending problems. The radius of primary bending curvature can aﬀect deformation mode too, via Searle’s parameter. For draw-bend test, the springback process is closer to plane-stress rather than plane-strain, because of the persistent anticlastic curvature.

104 CHAPTER 6

TIME-DEPENDENT SPRINGBACK

Note: Some of the experimental data was provided by W.D. Carden. A manuscript of this work has been submitted to the International Journal of Plasticity for publication.

Abstract

Draw-bend tests, devised to measure springback in previous work, revealed that the specimen shapes for aluminum alloys can continue to change for long periods following forming and unloading. Steels tested under identical conditions showed no time-dependent springback. In order to quantify the eﬀect and infer its basis, four aluminum alloys, 2008-T4, 5182-O, 6022-T4 and 6111-T4, were draw-bend tested under conditions promoting the time-dependent response (small tool radius and low sheet tension). Detailed measurements were made over 15 months following forming, after which the shape changes were diﬃcult to separate from experimental scatter.

Earlier tests were re-measured up to 7 years following forming. The shape changes are generally proportional to log(time) up to a few months, after which the kinetics becomes slower. In order to understand the basis of the phenomenon, two models were considered: residual stress-driven creep, and anelastic deformation. In the

105 ﬁrst case, creep properties of 6022-T4 were measured and used to simulate creep- based time-dependent springback. Qualitative agreement was obtained using a crude

ﬁnite element model. For the second possibility, novel anelasticity tests following reverse-path loading were performed for 6022-T4, aluminum-killed drawing quality steel (AKDQ) and drawing quality special killed steel (DQSK). Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys.

6.1 Introduction

Springback occurs when sheet metal parts are released from forming tools. If not correctly predicted and compensated for, springback will cause the ﬁnal part shape to deviate from design speciﬁcations and to create assembly problems. For simple forming operations, such as pure and stretching bending [76, 108], U-bending (channel forming) and V-bending [109,110], empirical methods and analytical models have been developed and successfully utilized to predict springback. However, these methods are not suitable for many industrial stamping operations in which complicated deformation paths and evolving contact conditions are present.

The draw-bend test realistically simulates the springback situation encountered in many sheet-forming operations, where bending and unbending occur successively, with simultaneous stretching, as material is drawn over a tool surface [69,74,82]. For most materials and process conditions, the specimen shape is reproducible and static.

However, for aluminum alloys draw-bend tested at certain combinations of back force

Fb (Fb is expressed as a normalized quantity by dividing the controlled back force by the force required to yield the sheet in uniaxial tension) and R/t (tool radius/sheet

106 thickness), the springback shape was observed to change signiﬁcantly with time following the forming and unloading steps [67]. This phenomenon had apparently been unreported previously, although stress relaxation of aluminum alloys is better-known, and is closely linked metallurgically [68]. The possibility of time-varying part shapes has signiﬁcant implications for industries striving to substitute sheet aluminum for sheet steel to reduce mass, while maintaining dimensional tolerances and consistency.

The current work is aimed at clarifying and quantifying the basis of time-dependent springback. Process conditions of the draw-bend test (Fb and R/t) were chosen to maximize the time-dependent springback and thus to improve resolution. Test results are reported for four aluminum alloys, including both heat-treatable and non- heat-treatable types, for periods of up to 15 months following forming. Preliminary analytical results of simulated time-dependent springback based on creep relaxation of residual stress are compared with experimental measurements for one alloy.

The possible mechanisms underlying time-dependent springback are discussed in light of two relevant metallurgical phenomena: creep driven by residual stress [111] and anelastic flow [112]. Both creep and anelasticity are time-dependent deformation processes with slow kinetics at low homologous temperature, but strain still accu- mulates over a long period of time. Creep usually denotes a slow viscous flow of solid under macroscopically non-zero stress, via atomic diffusion (through lattice or along grain boundary) and dislocation motion (glide or climb). The mechanism of creep depends on composition, microstructure features, temperature and stress [113].

Anelastic ﬂow, on the other hand, is often observed after a path change in stress or strain [114]. In uniaxial tensile test, anelasticity follows unloading and results in a

107 hysteresis loop upon re-loading. Anelastic strain is closely related to time-dependent microscopic deformation process [115].

Preliminary analytical results of simulated time-dependent springback based on creep relaxation of residual stress are compared with experimental measurements for one alloy. To identify the possible contribution of anelasticity, tension and tension/compression tests were carried out for 6022-T4, AKDQ (aluminum killed drawing- quality) steel and DQSK steel. For 6022-T4, anelastic strain rate and magnitude were compared to their counterparts calculated from measured time-dependent springback angle and to ﬁnite element simulation based on creep model.

6.2 Experimental

Standard tensile tests, special draw-bend tests, and uniaxial anelastic tests in tension and tension/compression were performed.

6.2.1 Materials

Four aluminum sheet alloys of nominally 1mm thickness were provided by Alu- minum Company of America [116]: 2008-T4, 5182-O, 6022-T4 and 6111-T4. The chemical compositions of these alloys are listed in Table 6.1 [117,118].

Alloy t (mm) Mg Si Cu Mn Fe Al other 2008-T4 0.94 0.41 0.68 0.96 0.06 0.17 Bal. — 5182-O 0.90 4.59 0.07 0.01 0.23 0.17 Bal. — 6022-T4 0.91 0.58 1.24 0.09 0.07 0.13 Bal. 0.02Ti 6111-T4 1.04 0.5–1.0 0.7–1.1 0.5–0.9 0.15–0.45 0.4 Bal. 0.1Cr

Table 6.1: Chemical composition (in weight pct.) and thickness of aluminum sheets.

108 The microstructures of as-received materials were examined at the Alcoa Technical

Center using optical microscopy at 200× magniﬁcation. As shown in Figure 6.1, all the alloys have elongated grains in the longitudinal and long transverse directions (denoted as L and LT, i.e., rolling and transverse directions respectively), but equiaxed grains in the short transverse direction (ST, i.e., thickness direction). Grain counts were taken in all three directions (L, LT and ST), with results reported in Table 6.2.

The average grain sizes are estimated based on ASTM Standard E112 [119]: 50µm for 2008-T4 and 6022-T4, 20µm for 5182-O and 56µm for 6111-T4.

L−LT

LT−ST L−ST m 2008−T4 µ 5182−O 100

6022−T4 6111−T4

Figure 6.1: Optical micrographs revealing grain structures.

109 Alloy grains/mm grains/mm2 grains/mm3 ASTM Direction LT L ST LT-ST L-ST LT-L-ST E112 Grade 2008-T4 15 17 35 530 611 9160 6.8 5182-O 39 39 54 2084 2122 91917 8.0 6022-T4 23 19 27 614 515 11852 6.0–6.5 6111-T4 15 16 29 448 469 7173 5.5–6.0

Table 6.2: Grain size of aluminum alloys.

400 Strain rate = 0.0012/s

300

200 2008-T4 5182-O True stress(MPa) 100 6022-T4 6111-T4

00 0.05 0.1 0.15 0.2 True strain

Figure 6.2: Uniaxial stress-strain curves for the tested aluminum alloys.

Standard uniaxial tensile tests [120] in the rolling direction were performed at room temperature, at a crosshead speed of 0.127 mm/s, corresponding to an initial strain rate of 1.2 × 10−3/s. Axial strain was measured using a 50-mm gage clip-on extensometer. The true stress-strain curves are shown in Figure 6.2. The non-heat- treatable 5182-O alloy exhibits serrated ﬂow, which is associated with the Portevin-Le

110 Chatelier eﬀect caused by the interaction between mobile dislocations and diﬀusing solute atoms [121].

6.2.2 Draw-Bend Experiments

The draw-bend experiments were carried out at the Colorado School of Mines, using equipment designed for friction measurements of coated sheet [71]. Tests were performed on tooling with die radii ranging from 3.2mm to 25.4mm, and the springback results were analyzed as a function of the die radius to sheet thickness ratio

R/t. Details of the test device have been presented elsewhere [69, 72]. Sheet tension was controlled directly and precisely using a second closed-loop controlled hydraulic actuator.

Rectangular strips (500mm×50mm) were sheared with their lengths parallel to the sheet rolling direction. With one end clamped in the upper grip, each strip was hand- formed around the radius to 90 degrees and then clamped in the lower grip. As will be shown later, variation of this hand-formed radius is insigniﬁcant with respect to the

final springback angle. Samples were subjected to a constant back force, then drawn over an unrotating cylindrical tool at 40mm/s. Standard industrial lubricant [81] was brushed on both strip and tool surfaces. After forming and springback, specimens were removed from the test device and their shapes were traced on paper and then recorded digitally. Traces were first taken one minute after forming and unloading, then repeated at intervals up to 15 months. Direct measurement of springback angles from tracings was previously validated by comparing tracing measurements with those from digitized CAD images [69], with a typical difference of approximately 0.2°.

111 (1) θ1 (2) 6022−T4 θ R/t =3.5 r′ R′ 2 Fb =0.5 Time after forming (s): (3) 1.8 × 107 74000 ∆θ 23000 17000 (4) 30 270 690 Unloaded Loaded 3000

(a) (b)

Figure 6.3: Draw-bend specimens: (a) schematic geometry before and after springback; (b) tracings at various times following forming for 6022-T4 aluminum.

The geometry of an unloaded sample is shown schematically in Figure 6.3(a), and a typical shape trace is shown in Figure 6.3(b). Four deformation regions may be distinguished: Regions 2 and 3 deﬁne the important specimen geometry along with

0 0 the corresponding angles ∆θ1 and ∆θ2, and radii of curvature R and r . Regions 1 and 4 remain straight throughout the test. The change of θ1 (i.e.,∆θ1) is caused by springback of Region 2 from the forming tool radius R, while θ2 is a measure of the so-called “sidewall curl” observed in many sheet-formed parts [103]. The springback angle ∆θ is determined from the springback contributions from the two regions:

π ∆θ = θ − θloaded = θ − (6.1) 1 1 1 1 2 loaded ∆θ2 = θ2 − θ2 = θ2 − 0 (6.2)

∆θ = ∆θ2 + ∆θ1 (6.3)

112 ∆θ1, corresponding to the region in contact with the tool at the end of the test, is small and nearly independent of back force and R/t [69], see, for example, Figure 6.4, as would be expected by the large draw distance (127mm) that Region 3 is subjected to as compared to the small spatial extent of Region 2 (i.e., approximately 5mm for R/t = 3.5). Furthermore, simulations (next section) show that ∆θ1 is nearly constant for times up to 1010s following forming. Therefore, it is sufficient to interpret differences in ∆θ, which is directly measured, as differences in ∆θ2 without loss of accuracy.

30 60 6022-T4 6022-T4 R/t=10.5 F =0.9 b 50 20 40

30 ∆θ 10 2 20 ∆θ ∆θ 10

Angle (degree) 2 ∆θ Angle (degree) 0 0 ∆θ ∆θ 1 1 -10 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 -10 0 5 10 15 20 Normalized back force R/t ratio (a) (b)

Figure 6.4: Variation of measured time-independent springback angle on: (a) back force; (b) tool radius.

Multiple tests under identical conditions showed that the standard deviation of θ

(or ∆θ) was about 1.1 degrees for 6022-T4 [69].

113 6.2.3 Anelastic Tests

The possible role of anelastic deformation in time-dependent springback was investigated using two unloading experiments, following either uniaxial tension or compression and tension. The tests were carried out with 6022-T4, AKDQ and DQSK steels. For the tension/unloading, rapid unloading was achieved by opening hydraulic grips during standard tensile test after a prescribed strain ε0 (corresponds to stress σ0) was reached, Figure 6.5(a). After an instantaneous contraction, the time-dependent anelastic strain was measured using an extensometer with 25-mm gauge length, Fig- ure 6.5(b). Strain signals were fed to a National Instrument 4350 A/D digitizer with

24-bit resolution, and then recorded digitally for times up to 1-2 hours, using a data acquisition rate of 3 points per second.

(a) (b) Strain Stress ε0 ε0, σ0

∆ el = − σ0 ε E

εan unloading

60s Strain τ0 Time

Figure 6.5: Schematic of the tension/unloading test: (a) general load path, (b) detail of unloading region.

The compression/tension/unloading test was conducted similarly, except that prior compression was achieved using a special device and specimen, Figure 6.6(a), to

114 avoid buckling. The sample was sandwiched between two flat plates on which lateral pressure was applied [122]. Teflon was used between the sample and plate to reduce friction. Specimens were first compressed to about 0.045, then the loading direction was reversed and specimens were further pulled in tension for 0.011, 0.04 and 0.11 respectively before unloading, Figure 6.6(b). The compression/tension/unloading tests were devised because they more closely reproduce the reverse paths followed by material elements as they pass over the tool radius.

±σ monotonic compression 0.045 tension unload

Side pressure

| True stress ±σ 0.01 0.04 0.11 Effective strain (a) (b)

Figure 6.6: Schematic of compression/tension/unloading test: (a) specimen geometry and stabilization ﬁxture; (b) general load path.

6.3 Results

6.3.1 Static (Time-Independent) Draw-Bend Tests

In order to understand the time-dependent draw-bend results, the static situation following unloading should be characterized. Typical static springback angles

115 for 6022-T4 were shown in Figure 6.4. For large R/t, springback is nearly time- independent. The steady decline of springback angle with increasing normalized back force, Fb, is expected [69, 74], but there is an unusual region of rapid decrease as the back force approaches 0.7–0.8 times the yield force. Tool radius has less inﬂuence, with increasing R/t reducing springback moderately [67, 69]. The variation of ∆θ with respect to process variables is essentially same as ∆θ2; ∆θ1 is small and nearly constant.

The static draw-bend tests were simulated by the ﬁnite element method, for a range of back forces (0.1 ≤ Fb ≤ 1.2) and three R/t ratios: 3.5, 7 and 14. Both

2D and 3D analysis were carried out, using plane-stress beam element (Abaqus type

B21), and 4-node shell element with reduced integration (S4R), respectively [24]. The sheet strip was discretized into 300 elements of non-uniform size in the longitudinal direction. To ensure numerical accuracy, the elements were smaller in the contact areas (Regions 2 and 3, Figure 6.3(a)), with one contact node per 4.5 degrees of turn angle [82]. For 3D analysis, mirror symmetry was utilized, and only half of the sheet sample was modeled with 8 elements in the sheet-width direction. For both beam and shell elements, 51 through-thickness integration points were used to minimize numerical error [82]. To closely represent plastic anisotropy of 6022-T4 sheet, Barlat’96 yield function [43] and a modiﬁed anisotropic hardening model [46] were chosen. A friction coeﬃcient of 0.15 was used for lubricated test conditions [69].

The sudden decrease of ∆θ for Fb near 0.8 is correlated with the appearance of persistent anticlastic curvature [69], the secondary curvature normal to the bending curvature that remains after unloading. The presence of permanent anticlastic curvature increases the section moment of inertia of an initially ﬂat sheet [107], and thus

116 substantially reduces springback. For small back forces, this secondary curvature nearly disappears during unloading and thus has little influence on the final springback angle. For large back forces, plastic deformation occurs throughout the sheet thickness by the sheet tension, thus maintaining much of the anticlastic curvature established during the bending and unbending of the strip. As shown in Figure 6.7, when the front force is just sufficient to yield the sheet in tension (Ff = 1), the corresponding back force is in the range of 0.68 to 0.77, depending on R/t. Thus, the sudden decrease of springback angle occurs when the largest sheet tension (which occurs near the out-most side of the die contact) reaches the yield tension.

1.6 Shell element 1.4 Friction=0.15

1.2 F *= 1.0 1 f

0.8

F =F 0.6 f b

0.4 R/t=3.5 Normalized frontforce 0.2 R/t=7 R/t=14 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Normalized back force

Figure 6.7: Variation of simulated front force with back force for various R/t ratios.

It should be noted that the back force and R/t ratio aﬀect the through-thickness residual stress after forming and springback. Finite element simulation results, Fig- ure 6.8, show that smaller R/t and smaller Fb increase the maximum residual stress.

117 If creep occurs, the areas of peak stress would be expected to dominate the response because of the large stress exponent for typical creep laws at low homologous temperature (T/Tm < 0.3-0.5, Tm is the melting temperature in Kelvin), N = 4–6 [123].

250 300 R/t=3.5 Beam element Tensile Shell element 200 F =0.5 6022-T4 b 200 150 100 100 R/t=3.5 R/t=7.0 50 0 R/t=7.0 R/t=14.0 0 R/t=14.0 Stress (MPa) -100 -50 -200 -100 Compressive Maximum residual stress (MPa) -150 -300 -0.5 -0.25 0 0.25 0.5 0 0.2 0.4 0.6 0.8 1 Through-thickness coordinate (mm) Normalized back force (a) (b)

Figure 6.8: Eﬀect of bending radius and back force on residual stress: (a) through- thickness residual stress for various R/t at Fb = 0.5; (b) variation of maximum tensile and compressive residual stresses with back force for three R/t ratios.

6.3.2 Time-Dependent Draw-Bend Springback

Detailed time-dependent springback measurements were made at various time intervals for the precess conditions expected to maximize the response (based on a residual stress-driven creep model): R/t = 3.5 and low Fb. The results, presented in

Figure 6.9, show that springback angle is proportional to log(time) following forming and unloading for each of the alloys tested. The slopes (m) fall in a range of 0.57 to

118 1.59, with the large slopes generally for Fb = 0.2 and the small ones for Fb = 0.8.

This is consistent with the expected magnitude of residual stress following loading and unloading.

The linearity of the data in Figure 6.9 implies that the dependence of springback angle on time τ as follows: µ ¶ τ ∆θ = ∆θ0 + δθ(τ) = ∆θ0(τ1) + m log (6.4) τ0 where m represents the slopes shown, ∆θ0 is the initial springback (taken approximately 60s after unloading, i.e., τ1 = 60s), and δθ(τ) is the time-dependent part.

At large back force (Fb > 0.7), m is smaller because the formed sample is more re- sistant to springback because of increased section moment caused by the persistent anticlastic curvature. As can be seen in Table 6.3, tool radius aﬀects both the magnitude and rate of time-dependent springback angle with small R/t value promoting time-dependent springback.

Using the maximum and minimum slopes (1.89 and 0.76) from Figure 6.9 allows estimation of the δθ to be disregarded before the ﬁrst measurement at 60s. Assuming the minimum physical time for measurement of 1s (dynamics controls this minimum time), the range of missed δθ is 1.4 to 3.4 degrees. A similar calculation shows that the total time-dependent angle change expected from 1s to 3×108s (10 years) ranges between 6.4 and 16 degrees.

In order to put the magnitude of the angular changes into a material perspective, the change of strain at the outer ﬁber of the specimen may be computed from the change of radius of curvature for Region 3 , r0. As noted earlier, this region dominates the overall angle change. The following equation applies to this maximum strain

119 90 100

80 F = 0.2 90 b F = 0.2 m=1.11 b 70 m=1.29 80 60 F = 0.5 70 1.65 b (degree)

(degree) 50 ∆θ ∆θ 60 40 2008-T4 5182-O R/t=3.5 R/t=3.5 50 F = 0.8 30 1.13 b F = 0.8 0.76 b 20 40 10 100 1000 10 4 10 5 10 6 10 7 10 8 10 100 1000 10 4 10 5 10 6 10 7 10 8 Time (s) Time (s) (a) (b)

90 90 1.86 F = 0.2 1.19 b m=1.89 80 F = 0.2 80 m=1.06 b 70 70 1.67 F = 0.5 F = 0.5 1.58 b 60 1.25 b 60 (degree) (degree) 50 ∆θ ∆θ 50 40 6022-T4 6111-T4 40 R/t=3.5 1.46 30 R/t=3.5 F = 0.8 b F = 0.8 1.39 0.91 b 30 20 10 100 1000 10 4 10 5 10 6 10 7 10 8 10 100 1000 10 4 10 5 10 6 10 7 10 8 Time (s) Time (s) (c) (d)

Figure 6.9: Change of springback angle with time after forming: (a) 2008-T4, (b) 5182-O, (c) 6022-T4, (d) 6111-T4. Multiple tests are diﬀerentiated by open and closed markers. Slopes shown are in degree/log(s).

120 ◦ Radius (mm) R/t Fb ∆θ(τ1) ∆θ(τ2) Change ( ) m† 0.2 81.6 90.0 8.4 1.44 3.2 3.5 0.5 61.9 69.8 7.9 1.36 (Lubricated) 0.8 33.2 39.3 6.1 1.05 0.2 78.8 86.3 7.5 1.29 6.4 7.0 0.5 56.5 63.0 6.5 1.12 (Lubricated) 0.9 7.5 8.8 1.3 0.22 0.2 62.1 65.5 3.4 0.58 12.7 14.0 0.5 43.3 46.8 3.5 0.60 (Lubricated) 0.8 7.0 8.5 1.5 0.26 †m = ∆θ(τ2)−∆θ(τ1) is the average slope. log(τ2)−log(τ1)

Table 6.3: Kinetics of time-dependent springback of 6022-T4 aluminum from mea- 7 surements at τ1 = 60s and τ2 = 4 × 10 s (15 months).

change:

t δθ tδθ −3 δε = 0 = = 3.6 × 10 δθ (6.5) 2r θ2 2L Here, t is the sheet thickness and L is the arc length of Region 3 (127 mm, the drawing distance). Each degree of δθ corresponds to an outer ﬁber strain change of 6×10−5. Since about 8 degrees of total time-dependent springback is typically observed in the ﬁrst year following forming, the expected maximum time-dependent strain is approximately 5×10−4 for 6022-T4 aluminum alloy. This is approximately

20% of the elastic limit strain of this alloy (εe = σY /E = 172MPa/69GPa = 2.5 ×

10−3). Nonetheless, this small strain creates signiﬁcant shape change for long, slender geometries such as for the draw-bend specimen.

121 The following equation may be used to convert the unit of slopes in Figure 6.9 to more usual units based on natural logarithms and radians: µ ¶ µ ¶ µ ¶ d(∆θ) radians π radians d(∆θ) degree = 2.303 × × (6.6) d ln(τ) ln s 180 degree d log(τ) log s

Using the maximum and minimum slopes presented in Figure 6.9 (max. = 1.89 for 6022-T4, Fb = 0.2 and min.=0.76 for 2008-T4, Fb = 0.8) allows estimation of the range of strain rates encountered in these experiments via Equations 6.5 and 6.6. At

60s, the strain rate is 4.6 × 10−6/s for 6022-T4 and 1.8 × 10−6/s for 2008-T4. At

4 × 107s, the strain rates are 6.9 × 10−12/s and 2.8 × 10−12/s for 6022-T4 and 2008-

T4, respectively. It should be noted that the slopes tend to decrease at the highest time (particularly noticeable for 6022-T4, Fb = 0.2), which implies time-dependent springback will eventually stop. However, the accuracy of the data does not justify higher-order ﬁts over these time periods.

Other time-dependent springback results were analyzed more generally in order to verify the dependence on process variables and long times. Table 6.3 summarizes slopes (m in Equation 6.4) obtained for various tests based on ∆θ measured at two

7 times: τ1 = 60s (initial measurement) and τ2 = 3.3×10 s (approximately 1 year after forming). The results verify the earlier expectation that time-dependent springback is larger for small R/t and low Fb. For the most stable case shown (R/t = 14, Fb = 0.8), the angle change after one year is only 1.5 degrees, as compared with 8 degrees for small R/t, low Fb.

In order to examine the very long-time springback behavior, samples of the same alloy used in the current study that were deformed in June 1996 [67] were re-examined for additional shape change after the additional 7 years. The results, Table 6.4, ex- hibited considerable scatter, presumably through handling of some specimens during

122 transit and storage. Nonetheless, three trends can be discussed in the data: smaller time-dependent springback for larger R/t and larger Fb, and generally lower slope over the long time period. For example, for R/t = 3.5 and Fb = 0.5, the long term tests indicate slopes of 0.25–0.34 as compared with slopes of 1.47 (Figure 6.9) and

1.36 (Table 6.3).

To reveal in more detail the kinetics of time-dependent springback, springback angles for specimens tested in the previous and current studies were plotted on a log(time) scale, Figure 6.10. The springback angles are approximately linear with log(time), for times up to 106s (2 weeks) after forming. After that, the slopes decrease and ∆θ’s saturate at ∼ 5×107s (1.5 years). After one month (2.6×106s), the measured time-dependent springback angle (δθ1month) is about 70–80% of the saturation value

(δθ∞). In view of this analysis, it is clear why the slopes in Tables 6.3 and 6.4, which are computed for long times, underestimate the rate of short-time response.

84 R/t=1.8, F =0.3 b Free rotating 78 1 month

(degree) R/t=3.5, F =0.5 b ∆θ Lubricated 6022-T4 66 (1996) (2001) (1996) 60 10 100 1000 10 4 10 5 10 6 10 7 10 8 10 9 Time (sec)

Figure 6.10: Measured time-dependent springback angles for6022-T4 from two studies (Carden, 1996, and this work).

123 ◦ Radius (mm) R/t Fb ∆θ(τ1) ∆θ(τ2) Change ( ) m† 0.3 72.5 82.6 10.1 0.67 1.6 1.8 0.5 51.5 59.8 8.3 0.55 (Free rolling) 0.5 59.1 65.0 5.9 0.39 0.9 14.5 17.9 3.4 0.22 0.5 65.8 71.0 5.2 0.34 3.2 3.5 0.5 70.6 74.0 4.6 0.30 (Free rolling) 0.5 68.3 72.0 3.7 0.25 3.2 (Lubricated) 1.1 5.6 6.5 0.9 0.06 0.5 65.3 70.5 5.2 0.34 6.4 7.0 0.5 58.8 63.0 4.2 0.28 (Free rolling) 0.5 62.8 64.3 1.5 0.10 0.9 29.3 30.3 1.0 0.07 0.5 43.6 48.3 4.5 0.30 9.5 10.5 0.7 19.0 25.0 6.0 0.40 (Dry) 0.9 4.1 3.7 -0.4 -0.026 1.1 2.9 3.0 0.1 0.007 9.5 0.5 57.0 59.5 2.0 0.13 (Free rolling) 10.5 0.7 41.2 47.0 5.8 0.38 0.9 32.5 32.0 -0.5 -0.033 12.7 14.0 0.9 4.5 4.1 -0.4 -0.026 (Lubricated) 0.9 5.7 5.8 0.1 0.007 0.5 33.7 36.0 2.3 0.15 25.4 28.0 0.5 34.8 36.1 1.3 0.086 (Free rolling) 0.5 39.2 38.0 -1.2 -0.08 0.9 7.6 8.3 0.7 0.046 †m = ∆θ(τ2)−∆θ(τ1) is the average slope. log(τ2)−log(τ1)

Table 6.4: Long-time kinetics of springback for 6022-T4 aluminum based on re- 8 measurement of samples at τ1 = 60s and τ2 = 2.2 × 10 s (7 years).

124 6.3.3 Room Temperature Creep Test for 6022-T4

In order to begin to test the hypothesis quantitatively, the room-temperature creep behavior must be known and modeled. Constant-load uniaxial creep tests for the

6022-T4 alloy were conducted at room temperature with initial stresses ranging from

183.6 to 236 MPa. Creep strain was measured using a 1-inch gauge extensometer.

Since the strain change caused by time-dependent springback in draw-bend test is

−4 small (4 × 10 for 6022-T4 when Fb = 0.2), all creep tests were stopped after about

20 minutes so that total creep strain was less than 1×10−3. The creep curves obtained in this manner are shown in Figure 6.11.

0.01 0.008 σ = 236.0 6022-T4 σ = 236.0 6022-T4 0.007 σ = 208.6

0.006 σ = 188.7 σ = 174.9 0.005 σ = 208.6 0.1 1 0.004 σ = 188.7 0.001 σ

0.003 = 174.9 strain Creep Creep strain 0.002 Steady state creep Primary creep 0.001 -20 5.68 -16 ( 6.11-0.0024 σ) 0.1 ε = 3.4x10 ( σ) t ε = 2.374x10 ( σ) t 0.0001 0 200 400 600 800 1000 1200 1400 1 10 100 1000 10 4 Time (s) Time (s) (a) (b)

Figure 6.11: Room temperature creep curves for 6022-T4: (a) steady state creep law; (b) primary creep law.

125 If the primary creep (τ < 200s) is neglected, a simple steady state power law of creep can be ﬁtted from the steady state regime of the creep curves [113], Fig- ure 6.11(a): µ ¶ σ N ε˙ss = K (6.7) σ0 where K = 3.36 × 10−20/s is a material constant, N = 5.68 is the stress exponent and

σ0 = 1MPa. A better ﬁt is obtained by incorporating the primary creep according to the following equation [124]: µ ¶ σ N ε = K0 τ 0.1 (6.8) σ0 where K0 = 2.37 × 10−16/s and N = 6.02 − 0.0024σ, Figure 6.11(b).

4 10 -4 Creep stress=96% of yielding

) -5

-3 3 6022-T4 10 ) -1

2 10 -6 6022-T4 Fit curves Creep rate (s -7

Creep strain (x10 1 10

DQSK DQSK 0 10 -8 0 500 1000 1500 2000 0 500 1000 1500 2000 Time (sec) Time (sec) (a) (b)

Figure 6.12: Room temperature creep behavior for 6022-T4 and DQSK: (a) creep strain; (b) creep rate.

Figure 6.12 compares the room temperature creep curves of aluminum 6022-T4 and DQSK steel. As shown in Figure 6.12(a), the creep strain of 6022-T4 is about

126 10–15 times that of DQSK steel when the creep load is 96% of the yielding force.

In order to compute the creep rate, the following equation is used to ﬁt the room temperature creep curves:

εcreep = A + B ln(τ) (6.9)

From Figure 6.12(b), it is clear that steel creeps at a slower rate than aluminum since steel has a much higher melting temperature (T293K /Tm = 0.16) than aluminum

(T293K /Tm = 0.35).

6.3.4 Creep-Based Springback Simulation

Quantitative evaluation of the residual-stress-creep model of time-dependent springback requires simulation to capture the mechanics of the draw-bend test, to determine the initial residual stress distribution, and to predict the springback angle at various times from a material creep law.

A relatively eﬃcient, approximate ﬁnite element model of the draw-bend test was constructed using Abaqus/Standard [24] to investigate creep-based springback. For simulating the elastic-plastic forming and initial unloading, linear, plane-stress beam elements (Abaqus element type B21) were used with von Mises yield function and isotropic hardening. These approximate material model choices were selected in view of other limitations:

• the time-intensive nature of creep-relaxation simulations at long times;

• the incomplete knowledge of the creep behavior of the alloy, without conﬁdence

in knowledge of creep under changing stress conditions;

127 • the interest in low Fb tests, which maximizes the residual stress. At the same

time, it minimizes the persistent anticlastic curvature, thus closely constraining

the problem to a two-dimensional one.

Fortunately, past work with similar models [82] demonstrated reasonable agreement with measured initial springback angles for Fb ≤ 0.7. This agreement provides con-

ﬁdence that the initial residual stress distribution is well estimated by this model.

Because of the approximations, however, perfect agreement with initial springback is not expected, but the accuracy should be suﬃcient to asses the role of creep on the time-dependent increments of the springback.

The creep laws represented by Equations 6.7 and 6.8 were implemented in Abaqus for the post-unloading simulations. Therefore, three-stage sequential simulations were carried out: 1) time-independent elastic-plastic drawing; 2) time-independent elastic- plastic initial unloading; and 3) creep of the unloaded specimen driven by internal residual stress. The interested, through-thickness stress distribution in region 3 of the specimen after each simulation stage are shown in Figure 6.13(a), for R/t = 3.5 and Fb = 0.5. The corresponding creep-based time-dependent springback behavior is shown in Figure 6.13(b). While the qualitative agreement is good, it is apparent that the measured strain rates are higher than the simulated ones using Equation 6.7 by approximately a factor of 2. The apparent experimental saturation value of δθ

(i.e. δθ as τ → ∞) is similarly higher than the prediction. As can be readily seen in Figure 6.13(a), nearly all of the simulated internal stress has been relieved at a time of 2 × 106s following forming, so there is little retained capacity for further angle change. In view of the approximate material law implemented and the relative lack of

128 information about room-temperature creep for aluminum alloys after reverse loading, the results are in reasonable agreement, but other eﬀects cannot be ruled out.

400 10 R/t=3.5, F =0.5 Loaded b 300 8 200 Unloaded ( τ=0s) 100 6 0 After creep Test1 (degree) 6 (τ=2.0x10 s) Test2 -100 4 Steady state Stress (MPa) δθ(τ) -200 R/t=3.5, F =0.5 2 b -300 Primary Steady state law -400 0 -0.5 -0.25 0 0.25 0.5 10 100 1000 10 4 10 5 10 6 10 7 10 8 Through-thickness coordinate (mm) Time (s) (a) (b)

Figure 6.13: Abaqus simulation results at three stages: (a) through-thickness stress; (b) Comparison of measured and simulated time-dependent springback.

The simulated time-dependent springback angle depends strongly on the form and ﬁt of the creep law used. A parametric study was carried out for diﬀerent creep parameters, with N ranging from 3 to 12 and K from 3.36×10−11 to 3.36×10−20.

A few of the simulation results are reported to save space. As can been seen from

Figure 6.14(a), a higher creep stress exponent leads to a faster initial increase and a larger total angle change. On the other hand, the strength constant K aﬀects only the rate of time-dependent springback, Figure 6.14(b). In spite of the agreement of form between simulation and experimental data, it appears impossible to ﬁt the measured time-dependent springback angle variation by a law of the form represented

129 by Equation 6.7. As shown in Figure 6.13(b), time-dependent springback produced

7 about 8 degrees of shape change between the ﬁrst (τ1 = 60s) and last (τ2 = 4 × 10 s) measurements. However, the closest match with simulation yielded approximately

5.6 degrees with K = 3.36 × 10−19 and N = 5.68.

10 10 -20 N dε/dt = 3.36x10 (σ) K=3.36x10 -18 R/t=3.5, F =0.5 K=3.36x10 -19 8 b N=10 8 K=3.36x10 -20 N=8 Test1 6 6 Test2 N=6 δθ(τ) δθ(τ) 4 4 N=5 R/t=3.5, F =0.5 b 2 Test1 2 dε/dt = K ( σ) 5.68 Test2 10 years 10 years 0 4 6 8 10 0 4 6 8 10 1 100 10 10 10 10 1 100 10 10 10 10 Time (sec) Time (sec) (a) (b)

Figure 6.14: Eﬀect of creep law parameters on simulated time-dependent springback: (a) stress exponent N; (b) strength parameter K.

The quantitative discrepancy between simulated and measured springback kinetics may be attributed in part to the crude material and specimen models employed in the simulations. These diﬀerences should be fairly small for the ﬁrst two simulation stages, elastic-plastic forming and unloading, in view of past success with such models.

Furthermore, plots of the time-dependent springback angle (Figure 6.13(a), e.g.) tend to mask discrepancies in the initial springback simulations.

130 The larger probable source of error lies in the application of the creep laws, which are derived from constant stress tests. The stress state in material element changes rapidly during unloading, include stress reversals. Deformation substructures in aluminum established during plastic deformation, such as dislocation tangles, cell walls and subgrain boundaries, have been shown to be unstable under a strain path change [125]. Dislocations can exhibit directional mobility after plastic deformation, and can run back from obstacles upon load reversal [126]. As a result, it is expected that the creep rate after stress reversal may be signiﬁcantly diﬀerent from that of constant load test. Room temperature creep tests on pure cadmium have shown increased creep rate after shear stress was reversed following accumulated primary creep [127].

6.3.5 Anelastic Deformation after Unloading

Anelastic deformation is usually measured under rapidly falling stress states, and can occur at zero applied load for some time following such a change. In view of the rapidly-changing stress conditions during unloading after draw-bend test, anelasticity may contribute or dominate the observed behavior. The mechanism for anelastic deformation is thought to be viscous dislocation motion driven by local internal stress [128].

In order to test these possibilities, unloading tests following uniaxial tension were

ﬁrst conducted. A standard sheet forming steel, AKDQ (aluminum-killed, drawing- quality), was tested in addition to 6022-T4 aluminum, because the steel alloys did not exhibit time-dependent springback following draw-bend tests [67].

131 Figure 6.15 shows the development of anelastic strain as a function of time after unloading from uniaxial tension. Figure 6.15(a) shows the raw data whereas Fig- ure 6.15(b) has anelastic strain εan normalized by the elastic strain just prior to unloading (∆εel = −σ0/E). Within the precision of the experiment, the anelastic strains appear to saturate within 1-2 hours following the unloading. To evaluate the kinetics of anelastic flow, the following equation is employed to fit the normalized anelastic strain: ε X3 an = A [1 − exp(−B τ)] (6.10) ∆ε i i el i=1 where Ai and Bi (i = 1–3) are best-fit constants. Equation 6.10 indicates that anelastic strain eventually saturates according to the best-fit curves shown in Figure 6.15(b) as solid lines.

0 0.2 AKDQ σ =313MPa σ =361MPa 0 0 ε =0.11 ε =0.19 ) 0 0 -4 -2 0.15

-4 6022-T4 σ =253MPa 0.1 AKDQ, ε =0.11 0 0 ε =0.06 AKDQ, ε =0.19 0 0 6022-T4, ε =0.06 -6 0.05 0 6022-T4, ε =0.13 Anelastic strain (x10 σ =318MPa, ε =0.13 0

0 0 Normalizedanelastic strain 6022-T4, ε =0.21 σ =352MPa, ε =0.21 0 -8 0 0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Time (s) Time (s) (a) (b)

Figure 6.15: Anelastic strain after uniaxial tension.

132 The anelastic strains are very similar for AKDQ and 6022-T4, while considered as a fraction of the total elastic strain, the AKDQ has a large response because it has a higher elastic modulus of 210 GPa. For 6022-T4, recovered anelastic strain within one hour after unloading is 3-7×10−4, or approximately 11% of the elastic strain. Time-dependent springback, on the other hand, is happens in a much slower fashion. For example, the calculated strain (Equations 6.5 and 6.6) at the outer ﬁber of draw-bend tested strip is about 4×10−4, after approximately one year for typical conditions (R/t = 3.5 and Fb = 0.5).

A second series of special anelasticity experiments was conducted to represent more closely the reverse-path deformation characteristics of the draw-bend test. Us- ing special test specimens, fixtures, and processes as described in the Experimental section, uniaxial compression to a strain of 0.045 was first applied, followed by uniaxial tension to strain increments of 0.007, 0.042 and 0.11, then unloading and measuring of anelastic deformation. Thick 6022-T4 sheets (t = 2mm) are used in this test to achieve a higher compressive strain, while 1mm-thick 6022-T4 sheets are used in the draw-bend experiments. The results, Figure 6.16, show similar time constants as the monotonic tests, but the sign of the anelastic strains reverse for tensile strain increments approximately equal to the first stage compression strain. That is, for small tensile strain (0.007–0.01), the anelastic strain is tensile, as would be expected for monotonic compression tests. For large tension increments (∆ε > 0.01), the anelastic strain is compressive, as observed for monotonic tension tests (e.g., Figure 6.15). For intermediate tensile strain increments, the anelastic response nearly disappears.

Note: The correlation between the tensile strain increment where anelas-

ticity disappears and the prior compressive strain may be fortuitous. That

133 2 2 DQSK (t=1.5mm) ε = 0.01 Pre-comp = 0.045 ε = 0.007 ten ten ) ) -4 -4 0 1 ε = 0.042 ten ε = 0.042 -2 0 ten

ε = 0.11 -4 ten -1 Fit curves (Equation 9) Anelastic strain (x10 strain Anelastic Anelastic strain (x10 strain Anelastic Fit curves ε = 0.11 6022-T4 (t=2.5mm) ten (Equation 9) Pre-comp = 0.045 -6 -2 0 1000 2000 3000 4000 5000 0 1250 2500 3750 5000 Time (s) Time (s) (a) (b)

Figure 6.16: Anelastic strain after unloading from compression–tension test: (a) 6022- T4; (b) DQSK steel.

is, the 0.04 tensile increment might produce a neutral anelastic response

even if the initial compressive strain increment was mush larger. This will

be investigated separately.

The observed deformation characteristics in Figure 6.16 indicate that the effect of anelasticity on time-dependent springback is complicated. In draw-bend tests, material fibers are generally subjected to strain paths that depend on the distance from the neutral axis. As shown in Figure 6.17, the bottom fiber (z51) is in tension after about 0.08 compressive strain, while no reversal in strain path occurs for the top one

(z1). Intermediate material ﬁbers have diﬀerent strain increments after unbending.

Therefore, the inﬂuence of anelasticity on time-dependent springback is likely to be less than would be inferred from plots such as Figure 6.15.

134 0.2 6022-T4 z =+0.46mm R/t=3.5 1 0.15 F =0.5 z b z 13 =+0.23mm 0.1 z1 z =0mm 0.05 26 z13 x Strain z26 0 z 39 =-0.23mm z39

z51 -0.05 t z =-0.46mm t =0.92mm 51 -0.1 68 70 72 74 76 78 Drawing distance (mm) (a) (b)

Figure 6.17: Strain paths in draw-bend test: (a) locations of 5 through-thickness integration points (IP); (b) change of strain paths.

In view of the similar time constants for monotonic and reverse-path anelasticity, and in view of the smaller strains attainable for reverse-path tests, it appears unlikely that anelasticity is a signiﬁcant factor in the long-term time-dependent springback kinetics observed. This inference is strengthened considerably by the observation that 6022-T4 and AKDQ have very similar anelastic responses, but time-dependent springback of AKDQ is absent.

6.4 Discussions

The phenomenon of time-dependent springback may have signiﬁcant implications for manufacturing and industrial forming processes, where complex or unpredictable

ﬁnal part shapes are obstacles to consistent assembly and product quality. Based on draw-bend tests, as much as 20% of springback might be of the delayed variety.

135 The draw-bend test provides meaningful and useful data for sheet forming springback because of its sensitive resolution of strain and the similitude with conditions encountered in forming practice. Interpretation of deformation mechanisms, however, is diﬃcult because stress and strain vary continuously through the specimen thickness at all stages of deformation. For mechanistic purposes, the stress and strain ﬁelds may be considered constant within a macroscopic element (with a dimension of 1/10 or 1/100 of the sheet thickness, e.g.), whereas microstructural sources of local stress generally occur on much smaller length scales.

Time-dependent springback after draw-bending provides insight into the physical mechanisms of inelastic deformation. Two related metallurgical phenomena presumably contribute to time-dependent springback under the conditions of the draw-bend test: creep relaxation of residual stress [111] (as simulated here) and anelastic deformation [128]. The mechanistic difference lies primarily in the source(s) of stress to which the mobile dislocations respond. Creep is usually considered driven by a macroscopically uniform stress (such as in a tensile test), although creep relaxation is a related phenomenon where a load is gradually relaxed without external shape change [68]. Anelasticity, on the other hand, is thought to occur by the action of local sources of stress (pile-ups or grain incompatibility effects, for example), even at zero macroscopic stress, often following a significant stress change. For complex strain/stress paths, such as in draw-bending, the distinction between the two macroscopic idealizations is not a simple one. Presumably, both external and internal sources of stress, and static and historically changing stresses contribute to the mechanical situation. Such transient behavior has been measured [46,129].

136 Various models of dislocation interaction with crystal defects were proposed for anelastic deformation, including unbowing from the weak pins [130] and the run-back of pile-up dislocations from obstacles such as grain boundaries [131]. Phenomenolog- ical models, such as the Mechanical Threshold Stress (MTS) model [132], incorporate these ideas in a macroscopic model of relaxation behavior. Predictions from such models can be tested by implementing them into a ﬁnite element program and comparing the simulations with experimental data. This approach has been demonstrated for a simple power-law creep model of the residual stress relaxation [68]. While it cannot be ruled out entirely, it appears that anelasticity is not likely to be the primary mechanism for the long-term time-dependent springback. As shown previously,

6022-T4 and DQSK steel showed similar anelastic ﬂow after unloading, both in terms of the kinetics and the magnitude of the anelastic strain. Furthermore, the recovery of the anelastic strain is a fast process. For example, about 20% of the elastic strain was recovered within one hour after unloading. However, it took more than one year to obtain the same amount of strain due to time-dependent springback in 6022-T4.

The kinetics of time-dependent springback, residual stress-driven creep and anelasticity are compared in terms of the times required to attain 0.2, 0.5 and 0.8 of the corresponding saturation strains or angles (δε∞, δθ∞) respectively, Table 6.5. The kinetics of anelasticity are 1 to 3 orders of magnitude faster than time-dependent springback, with closer matches for short times and smaller strains. The creep simulations are slower than the experiment by a factor of 2 to attain 80% of the saturation.

From these comparisons, it appears that the long-term response after unloading in the draw-bend test is dominated by the residual stress driven creep. The role of anelastic

137 deformation on the time-dependent springback probably contributes to the short-term response only, with some combination of eﬀects producing the overall kinetics.

138 Time (s) δε∞ δθ∞ 0.2 0.5 0.8 Draw-bend, measured† — 8.5° 5×102 1.7 × 104 5.7 × 105 Draw-bend, creep model‡ — 5.7° 3×102 0.9×104 1.3 × 106 Anelasticity (monotonic path)∗ 5.9–6.6×10−4 — 0.7–5×101 1.5–3×102 1.5–2.1×103 Anelasticity (reverse path)∗∗ -1.5–4.9×10−4 — 0.2–4×101 0.008–1.2×103 0.45–4×103

139 † from Test 1 in Figure 6.14(b). ‡ from simulation in Figure 6.14(b), with K = 3.36 × 10−19 and N = 5.68. ∗ from ﬁt curves in Figure 6.15(b), using Equation 6.10. ∗∗ from ﬁt curves in Figure 6.16(a), using Equation 6.10.

Table 6.5: Calculated times to reach fractions (0.2, 0.5 and 0.8) of the saturation strains (δε∞) or springback angles (δθ∞). 6.5 Conclusions

The following conclusions were reached from measurement and analysis of springback following draw-bend testing of four aluminum alloys in the light of separate creep tests for 6022-T4 and novel anelastic tests for 6022-T4, AKDQ and DQSK steels:

1. Springback of aluminum alloys can continue for a period ranging from one

minute to 15 months after forming, with near-saturation after a few months.

This behavior has been conﬁrmed by measurements taken as long as 7 years

following deformation.

2. Time-dependent springback was observed for every aluminum alloy tested, in-

cluding heat-treatable and non-heat-treatable grades: 2008-T4, 5182-O, 6022-

T4, and 6111-T4.

3. The kinetics are similar for all four alloys. Previous work showed that typical

forming steels, in comparison, showed no time-dependent springback under sim-

ilar test conditions [69]. Re-measurement of specimens after 7 years conﬁrmed

this conclusion.

4. Time-dependent springback following draw-bending is maximized for small R/t

values and small back forces. This is correlated to larger residual stresses im-

mediately after forming.

5. The time-dependent springback can be a signiﬁcant portion of total springback.

After 15 months, the largest observed percentages were: 6022-T4, 18%; 6111-

T4, 14%; 2008-T4, 13%; 5182-O, 11%.

140 6. Preliminary simulations of time-dependent springback using a creep relaxation

model correctly reproduced the qualitative form of the phenomenon. However,

the limit and rate of change of the time-dependent springback angle were under-

predicted signiﬁcantly, typically by a factor of 2.

7. Anelasticity does not likely make a signiﬁcant contribution to the long-term

time-dependent springback kinetics, but it may contribute to short-term re-

sponse.

8. Novel tests have been devised and used to reveal the nature of anelasticity,

after reverse strain paths. The direction of anelastic recoveries depends on path

characteristics.

141 CHAPTER 7

CONCLUSIONS

This thesis work utilizes a special draw-bend test to investigate how springback

(static and time-dependent) changes with forming parameters (sheet tension and tool radius) and material properties. Special attention is given to the role of the persistent anticlastic curvature on static springback. The mechanics of this secondary curvature is clariﬁed. Various material constitutive models suitable for modeling plastic anisotropy of sheet metals are employed to simulate draw-bend tests, with new implementations of Barlat 2000 yield function and a nonlinear kinematic hardening law that utilizes three back stress components to account for the Bauschinger eﬀect.

The time-dependent springback of draw-bend tested aluminum alloys, which was discovered by accident in previous work, is carefully re-examined in this study to assess its kinetics and to infer its physical mechanism. The following conclusions are reached:

1. To properly describe the anisotropic yielding of 6022-T4 aluminum, the Bar-

lat’96 and 2000 yield functions are suggested. These two models predict identi-

cal results for the in-plane variation of yield strength and r-values, but the later

has a simpler mathematical form and it has been proven to be convex.

142 2. A nonlinear kinematic hardening model with three back stresses (mNLK ) is

implemented into Abaqus/Standard via UMAT subroutine. Using this model,

the three characteristics of the Bauschinger eﬀect, i.e., reduced yielding upon

load reversal, rapid strain harden and permanent softening after large strains,

are closely reproduced for 6002-T4 sheet after compression/tension and ten-

sion/compression test.

3. The current mNLK model is equally accurate as the previous G-W model [46]

in describing anisotropic hardening after a reversed strain path, but it requires

only 6 ﬁtting parameters and less numerical eﬀort during implementation.

4. Static springback of aluminum alloy 6022-T4 in draw-bend test is dominated

by sheet tension, while tool radius has moderate eﬀect in reducing springback.

5. The sudden increase of springback angle as the back force approaches a critical

value is attributed to the occurrence of persistent anticlastic curvature along

the sheet width direction. The critical back force is approximately 0.7–0.8, and

it slightly depends on tool radius.

6. Sheet orientations (RD or TD) aﬀect the measured springback angle and an-

ticlastic curvature. For transverse strips, the decline of springback angle with

back force is gradual, and the corresponding increase of anticlastic curvature

when back force is near 0.7–0.8 appears to be less abrupt.

7. Simulation results using Barlat 2000 yield function and the mNLK hardening

model are in agreement with experimental data, as well as with simulations

using Barlat’96 and the G-W hardening law.

143 8. Shell element fails to match the decrease of springback when R/t-ratio is less

than 4. Solid element with quadratic interpolation functions is advocated for

bending with small radius, but large computational cost is required. Linear

solid element, however, is not recommended because of its poor performance in

bending applications.

9. In draw-bend test, anticlastic curvature is developed during the forming step,

but it persisted after unloading when front pulling force approaches yielding.

Persistent anticlastic curvature greatly increases the moment of inertia for bend-

ing, and thus it is responsible for the dramatic drop of springback angle near

Fb = 0.7–0.8.

10. The occurrence and persistence of anticlastic curvature in draw-bend test can

be plausibly explained by elastic bending theory.

11. The cross-section shape (i.e., anticlastic proﬁle) of draw-bend tested strip is

solely determined by a dimensionless parameter, β, (the Searle’s parameter),

which includes the sample geometry and forming parameters (through the radius

of curvature of the curled region). When β < 10–15, the anticlastic displacement

is small and localized near the sheet edges. For large β, the maximum anticlastic

deﬂection could be as large as 1.5 times the sheet thickness,and the cross-section

is nearly circular.

12. It is not suﬃcient to diﬀerentiate plane-stress and plane-strain states by using

the ratio of width to thickness only even for pure bending. The radius of bending

curvature has to be considered too, via the Searle’s parameter.

144 13. Time-dependent springback is studied for aluminum alloys 2008-T4, 5182-O,

6022-T4 and 6111-T4. It can continue for a period ranging from one minute

to 15 months after forming, with near-saturation after a few months. This

behavior has been conﬁrmed by measurements taken as long as 7 years following

deformation.

14. The kinetics are similar for all four alloys. Previous work showed that typical

forming steels, in comparison, showed no time-dependent springback under sim-

ilar test conditions [69]. Re-measurement of specimens after 7 years conﬁrmed

this conclusion.

15. Time-dependent springback following draw-bending is maximized for small R/t

values and small back forces. This is correlated to larger residual stresses im-

mediately after forming.

16. The time-dependent springback can be a signiﬁcant portion of total springback.

After 15 months, the largest observed percentages were: 6022-T4, 18%; 6111-

T4, 14%; 2008-T4, 13%; 5182-O, 11%.

17. Preliminary simulations of time-dependent springback using a creep relaxation

model correctly reproduced the qualitative form of the phenomenon. However,

the limit and rate of change of the time-dependent springback angle were under-

predicted signiﬁcantly, typically by a factor of 2.

18. Anelasticity does not likely make a signiﬁcant contribution to the long-term

time-dependent springback kinetics, but it may contribute to short-term re-

sponse.

145 19. Novel tests have been devised and used to reveal the nature of anelasticity,

after reverse strain paths. The direction of anelastic recoveries depends on path

characteristics.

146 APPENDIX A

NUMERICAL ALGORITHM FOR ABAQUS UMAT

Given the total strain increment ∆², the elastic trial stress is ﬁrst calculated by assuming pure elastic response and freezing the internal variables:

trial el σn+1 = σn + C : ∆² (A.1)

trial If plastic deformation occurs in the current increment, σn+1 will locate outside of the

trial trial previous yield surface, i.e., f = σeq(σn+1 − αn) − σ0 − R(qn) > 0. Subsequently, a plastic correction step is applied, and the elastic trial stress is projected onto a properly updated yield surface, using a full implicit, backward Euler algorithm as discussed in Chapter 3.

Due to local convergence nature of the Newton-Ralphson method, the choice of the initial guess is critical in order to achieve a quadratic rate of convergence when solving a nonlinear equation [9]. Instead of using ∆λ(0) = 0 as the ﬁrst guess, a better estimate can be derived from Taloy’s series expansion.

As shown in Figure A.1, a backward Euler stress σC (at point C) can be deﬁned as follows:

trial el σC = σ − ∆λC : a (A.2)

147 The backward Euler stress can be treated as a ﬁrst guess of σn+1. Then, the new back stress at point C is calculated:

ci∆λ αC = αn + (σ − α) − γi∆λαi (A.3) σeq

B σtrial n+1

trial σ2 C f > 0

σn+1

A σn fC ≈ 0

Elastic Region σ1

fn+1 =0

Figure A.1: A ﬁrst guess for backward Euler algorithm.

The yield function at point C (fC ) is expanded with respect to the trial value fB , and the following equation is resulted:

trial ∂σeq ∂σeq ∂σy fC = f + : ∆σ + : ∆α − ∆q ∂σ ∂α ∂q (A.4)

= ftrial + a : (∆σ − ∆α) − hiso∆λ

dR where σy = σ0 + R(q) is the ﬂow stress, hiso = dq is the isotropic hardening modulus.

Let fC ≈ 0 and utilize ∆λ = ∆q, an initial guess of ∆λ can be computed:

trial ¯ (0) f ¯ ∆λ = P el P ¯ (A.5) hiso + ci + a : C : a − a : (γiαi) B 148 Then, the ﬁrst guesses of stress and internal variable increments are obtained:

σ(0) = σtrial − ∆λ(0)Cel : a (A.6) (0) (0) trial ci∆λ α = αi + (σ − α) − γi∆λαi (A.7) σeq ∆²p(0) = −∆λ(0)a (A.8)

The above procedure is graphically illustrated in Figure A.1:

Generally, the initial guess will not satisfy the yield condition, because the return mapping direction is the yield surface normal at the trial stress point B. In order to establish an iterative solution procedure for stress updating, the following residual quantities are deﬁned as the diﬀerences between the updated values and their corresponding backward Euler estimations [48]:

trial el rσ = σ − (σ − ∆λC : a) (A.9) · ¸ trial ci∆λ rαi = αi − αi + (σ − α) − γi∆λαi (A.10) σeq

Here, rσ and rαi are residual stress and residual back stresses. As the iteration proceeds, these residuals are minimized until convergence tolerance is met. As a result, the ﬁnal stress point satisﬁes the yield condition.

In order to construct a Newton-Ralphson iteration, Taylor’s series expansions are

applied to the residuals rσ and rαi , and the yield function:

¡ ¢ new old el el ∂a rσ = rσ + δσ + δλC : a + ∆λC : ∂σ :(δσ − δα) (A.11) new old rαi = rαi + δαi + γi∆λδαi − ∆λ0ci(δσ − δα)

ci ∆λ0ci (A.12) − δλ (σ − α) + δλγiαi + δσeq(σ − α) σeq σeq new old f = f + δσeq − hisoδλ (A.13) ∂σ ∂σ δσ = eq : δσ + eq : δα = a :(δσ − δα) (A.14) eq ∂σ ∂α

149 0 with ∆λ = ∆λ/σeq for simplicity. The superscript “new” and “old” denote that the quantities are evaluated at the beginning and end of the current increment.

After rearranging Equations A.11 and A.12, and combining them with Equa- tions A.13 and A.14, the followings equations are resulted:

£ ¡ ¢¤ ¡ ¢ old el el ∂a el ∂a −rσ − δλC : a = I + ∆λC : ∂σ : δσ − ∆λC : ∂σ : δα (A.15) old · ¸ old d1if ci − d1ihiso D0iδαi − d1iδσ = −rαi + (σ − α) + δλ (σ − α) − γiαi σeq σeq (A.16) old a :(δσ − δα) = −f + hisoδλ (A.17)

The following auxiliary variables are deﬁned for the sake of brevity:

D = 1 + γ ∆λ + ∆λ0c (A.18) 0i | {zi } | {z }i d2i d1i To solve for the iterative changes in stress, back stresses and the plastic multiplier, the three components of Equations A.16 are added together to get the iterative change of the total back stress: µ ¶ µ ¶ µ ¶ XN d XN rold XN d f old δα = 1i δσ − αi + 1i (σ − α) D D D σ i=1 0i i=1 0i i=1 0i eq " µ ¶ µ ¶# XN c − d h XN γ α + δλ i 1i iso (σ − α) − i i (A.19) σ D D i=1 eq 0i i=1 0i old T1f = T1δσ − RESa + (σ − α) + δλ [ T2(σ − α) − VECa ] σeq

150 Again, for brevity the followings are deﬁned:

d11 d12 d13 T1 = + + (A.20) D01 D02 D03 c1 − d11hiso c2 − d12hiso c3 − d13hiso T2 = + + (A.21) σeqD01 σeqD02 σeqD03 γ α γ α γ α VECa = 1 1 + 2 2 + 3 3 (A.22) D01 D02 D03 rold rold rold RESa = α1 + α2 + α3 (A.23) D01 D02 D03 old T1f SavVEC1 = RESa − (σ − α) (A.24) σeq

SavVEC2 = VECa − T2(σ − α) (A.25)

In Equations A.11, A.12 and A.13, all new quantities on the RHS are set to zeroes.

After substituting Equation (A.19) back into (A.15), the iterative change of stress is obtained:

−1 −1 δσ = −Q : rall − δλ Q : sall (A.26) where

el ∂a Q = I + (1 − T1)(∆λC : ) (A.27) · ∂σ ¸ ¡ ¢ old old el ∂a T1f rall = rσ + ∆λC : ∂σ RESa − (σ − α) (A.28) σeq ¡ ¢ el el ∂a sall = C : a + ∆λC : ∂σ [VECa − T2(σ − α)] (A.29)

Plug Equations A.28 and A.29 back into Equation A.19, the iterative change in back stresses are found: · ¸ d rold d f old δα = 1i δσ + − αi + 1i (σ − α) i D D D σ 0i · 0i 0i eq ¸ (A.30) c − d h γ α + δλ i 1i iso (σ − α) − i i D0iσeq D0i

The iterative change of the total back stress is the sum of δα1, δα2 and δα3 :

· old ¸ T1f δα = T1δσ + RESa − (σ − α) + δλ [T2(σ − α) − VECa] (A.31) σeq 151 Substituting Equations A.26 and A.31 back into Equation A.17, the iterative change of the plastic multiplier can be calculated as:

old f old − (1 − T )a : Q−1 : r + a : RESa − ( T1f ) a :(σ − α) 1 all σeq δλ = −1 hiso + (1 − T1)a : Q : sall + T2 a :(σ − α) − a : VECa (A.32) old −1 f − (1 − T1)a : Q : rall + a : SavVEC1 = −1 hiso + (1 − T1)a : Q : sall − a : SavVEC2 After convergence is reached, the incremental changes of stress and the internal variables are obtained:

∆σ(k+1) = ∆σ(k) + δσ (A.33)

∆α(k+1) = ∆α(k) + δα (A.34)

∆λ(k+1) = ∆λ(k) + δλ (A.35)

Finally, the stress, back stress and the plastic multiplier are updated at the end of current increment:

σn+1 = σn + ∆σ (A.36)

αn+1 = αn + ∆α (A.37)

λn+1 = λn + ∆λ (A.38)

Now, the consistent tangent modulus is calculated in order to preserve the quadratic rate of convergence when the Newton-Ralphson method is used for solving the global equilibrium. First, Equations A.2, A.3 and the yield function are written in rate forms:

¡ ¢ ˙el el ˙ el el ∂a σ˙ = σ˙ trial − (∆λC : a) = C : ²˙ − λC : a − ∆λC : ∂σ :(σ˙ − α˙ ) (A.39) 0 0 ˙ ci ∆λ ci ˙ α˙ i = ∆λ ci(σ˙ − α˙ ) + λ (σ − α) − δσeq(σ − α) − λγiαi − γi∆λα˙ i (A.40) σeq σeq ˙ ˙ ˙ f =σ ˙ eq − hisoλ = a :(σ − α) − hisoλ = 0 (A.41)

152 The plastic multiplier parameter, λ˙ , is sloved from the above three equations simultaneously:

˙ (1 − T1) a : R : ²˙ λ = ∂a hiso + (1 − T1) a : R : a + (1 − T1) a : (∆λR : ∂σ ): SavVEC2 − a : SavVEC2 (A.42) where R = Q−1 : Cel. Then, substituting λ˙ back to Equation A.39, ( £ ¡ ¢ ¤) (1 − T )(R : a) ⊗ R : a + ∆λR : ∂a : savVEC σ˙ = R − 1 ∂σ 2 : ²˙ (A.43) Denominator

The consistent tangent modulus is then computed as follows: ¯ δσ ¯ Calg = ¯ δ² n+1 £ ¡ ¢ ¤ (1 − T )(R : a) ⊗ R : a + ∆λR : ∂a : savVEC 1 £ ∂σ ¤ 2 = R − ∂a hiso + (1 − T1) a : R : a + a : (1 − T1)(∆λR : ∂σ ) − I : SavVEC2 (A.44)

It is worthy noting that the Calg is quite diﬀerent from the continuum tangent stiﬀness Cep, which relates the stress and total strain rates as follows:

σ˙ = Cep : ²˙ (A.45)

The consistent tangent modulus is obtained after the discretization of the plastic rate equations over a time interval [tn, tn+1]. Therefore, it depends on the plastic

∂2f multiplier ∆λ and the curvature of the yield surface, ∂σ2 . In the limiting case of ∆λ → 0, Calg approaches to Cep. The use of Calg instead of Cep is essential to achieve a quadratic rate of convergence, if the global nodal equilibrium equation is solved using the Newton-Ralphson method [47]. Mathematically, it is proved that

Calg is stiﬀer than Cep [133], in the sense that

η :( Calg − Cep): η ≥ 0, (η 6= 0) (A.46) where η is an arbitrary second order, symmetric tensor.

153 APPENDIX B

DRAW-BEND TEST DATA

e e Fb †Fb (N) †Ff (N) Ff ∆θ (°) Ra (mm) 0.2 1600±87 3500±117 0.436 80.9±0.7 1754±86 0.22 1770±46 3826±59 0.476 83.0 1588 0.29 2306±38 4460±54 0.555 75.9 1337 0.39 3138±43 5436±59 0.677 67.9 1337 0.5 4004±132 6294±125 0.783 63.4±3.0 1351±39 0.6 4832±155 7345±151 0.914 52.0 1210 0.65 5236±137 7672±125 0.955 47.0 977 — — 8036 1.0 — — 0.7 5631±116 8063±114 1.003 42.4±0.5 851±80 0.75 6052±69 8440±70 1.050 40.5±0.5 908±32 0.8 6450±105 8827±97 1.098 32.4±3.5 724±47 0.83 6650±105 9083±103 1.130 24.6 541 0.85 6851±109 9381±100 1.167 20.5±3.7 445±55 0.9 7257±105 9787±99 1.218 15.1±2.6 304±20 0.95 7653±108 10200±106 1.269 12.6±0.3 293±9 1.0 8066±89 10612±137 1.321 9.0 293 1.1 8894±41 11296±163 1.406 7.0 397 e f † Fb and Ff are actual forces, while Fb and Ff are the normalized values.

Table B.1: Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 3.5 and W = 50mm.

154 e e Fb Fb (N) Ff (N) Ff ∆θ (°) Ra (mm) 0.05† 430±6 771±20 0.0959 71.9 3629 0.09† 721±7 1112±30 0.138 67.0 3175 0.14 1144±73 1516±77 0.189 66.7 2540 0.2 1633±49 2077±76 0.259 62.1 2540 0.31 2465±45 3085±83 0.384 57.1 1814 0.41 3275±28 3987±69 0.495 50.7 1494 0.51 4071±108 4922±155 0.613 43.0±0.3 1232±67 0.61 4930±38 5932±103 0.738 36.0 907 0.66 5329±144 6354±180 0.791 31.4±0.9 838±16 0.71 5721±137 6774±173 0.843 24.6±0.4 719±12 0.74 5915±87 7136±184 0.888 12.8 369 0.77 6162±125 7264±155 0.904 11.8±2.3 332±63 0.82 6561±71 7734±155 0.962 7.5±0.5 255±8 — — 8036 1.0 — — 0.87 6965±74 8184±156 1.018 6.0 227 0.91 7357±81 8646±198 1.076 3.8±1.2 213±5 0.96 7762±116 8999±204 1.120 4.6 202 1.02 8183±118 9447±206 1.176 3.3 195 1.12 8984±55 10575±291 1.316 2.2 204 †—A reduced pulling speed of 10mm/s was used for these two samples.

Table B.2: Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) with R/t = 14 and W = 50mm.

155 e e Fb Fb (N) Ff (N) Ff ∆θ (°) Ra (mm) 0.2 1518±85 3171±91 0.415 74.6±0.6 5880±86 0.31 2400±56 4191±58 0.549 68.8 5080 0.42 3204±64 4894±60 0.641 64.8 4618 0.44 3369±125 5140±106 0.673 61.4 4233 0.47 3577±109 5453±92 0.714 57.8 3908 0.49 3727±143 5425±124 0.711 59.3±1.3 4233 0.54 4126±137 5839±121 0.765 53.9 3908 0.59 4485±153 6131±138 0.803 51.1 3387 0.64 4877±131 6545±118 0.857 49.0 2988 0.69 5235±140 6795±122 0.890 45.9 2988 0.74 5621±125 7218±110 0.946 40.8 2674 0.79 5988±119 7451±108 0.976 37.8±1.0 2322±212 — — 7634 1.0 — — 0.83 6359±79 7835±75 1.026 33.4 1881 0.88 6738±127 8149±118 1.068 29.1 1588 0.93 7110±115 8486±104 1.112 26.2 1373 0.98 7488±121 8756±118 1.147 19.8 941 1.0 8051±144 9597±125 1.257 11.4 726

Table B.3: Measured sheet tension, springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (TD) with R/t = 3.5 and W = 50mm.

156 ¯ ¯ Width Fb = 0.5 Fb = 0.9

(mm) ∆θ (°) Ra (mm) ∆θ (°) Ra (mm) 12.7 38.0±0.1 171±10 14.9±1.8 154±8 19.0 36.1±1.0 202±7 12.0±0.1 171±16 25.4 36.0±1.8 268±11 10.9±0.4 167±4 31.8 35.7±0.6 372±8 8.0±0.2 180±7 38.1 39.2±0.2 631±16 6.2±0.5 190±4 44.5 41.4±0.9 957±27 4.1±0.8 192±8 50.8 43.0±0.3 1232±67 3.8±1.2 213±5

Table B.4: Measured springback angle and the radius of anticlastic curvature for draw-bend tested 6022-T4 sheets (RD) using R/t = 14 tool.

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