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Principles of the Draw-Bend Springback Test

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Principles of the Draw-Bend Springback Test PRINCIPLES OF THE DRAW-BEND SPRINGBACK TEST DISSERTATION Presented in Partial Ful¯llment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By 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 plastic- ity, but they have simpler mathematical forms and require fewer model parameters. The mechanics of the persistent anticlastic curvature were studied by draw-bend experiments, ¯nite element analysis and an elastic plate-bending theory. The trans- verse cross-section shape was dictated by a dimensionless parameter, ¯, which de- pends 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 ¯, 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 ¯ is less than 10{15, and thus greatly reduced the ¯nal 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 pro- moting the time-dependent response. The time-dependent springback angles are ap- proximately 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 us- ing a crude ¯nite 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 under- standing 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., \E®ect 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 Kara¯llis 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 E®ect of Back Force and Tool Radius . 55 4.3.2 E®ect of Specimen Width . 57 4.4 Finite Element Results and Discussion . 58 4.4.1 E®ect of Back Force . 60 4.4.2 E®ect 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 E®ect of Back Force . 91 5.5.2 E®ect 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 ¯nite 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 ¿1 = 60s and ¿2 = 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 ¿1 = 60s and ¿2 = 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 (±"1) or springback angles (±θ1). 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.
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