Determination of Process Parameters for Stamping and Sheet Hydroforming of Sheet Metal Parts Using Finite Element Method

Home , Hydroforming

DETERMINATION OF PROCESS PARAMETERS FOR STAMPING AND SHEET HYDROFORMING OF SHEET METAL PARTS USING FINITE ELEMENT METHOD

DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Hariharasudhan Palaniswamy, M.S. * * * * *

The Ohio State University 2007

Dissertation Committee: Approved by Professor. Taylan Altan, Adviser ______Professor. Gary L. Kinzel Adviser Professor. Henry R. Busby Graduate program in Mechanical Engineering Professor. Jerald R. Brevick

ABSTRACT

Increase in the complexity of the parts and emphasis on the low formability and expensive lightweight materials require the use of multipoint cushion systems in modern presses and optimal blank shapes. Thus, better control of metal flow can be achieved for increasing the drawability and reducing the scrap rate and manufacturing cost in sheet metal forming. The blank holder force (BHF) / cushion pin force /cylinder force required to program the multipoint cushion system for forming a part is difficult and costly to estimate by trial and error FE simulation and die tryouts as there more than one variable to change. Hence, multipoint cushion system available in a modern press is hardly used in practice. Estimation of BHF to program the multipoint cushion system could be best done through structured FE simulations in process design stage so that its advantages can be incorporated in process design and carried forward to die tryout and production to realize its potential.

Successful application of FE simulations in stamping process design depends on the accuracy of the input parameters. Conventionally, properties of sheet materials obtained from tensile test are used in FE simulation. This data is insufficient for the analysis because a) maximum strain obtained in uniaxial tensile test is small compared to strains encountered in stamping operations b) the stress state in tensile test is uniaxial while in regular stamping it is biaxial. Therefore, there is a need for biaxial test to accurately determine the material properties (flow stress and anisotropy constants) over larger strain range for forming FE simulations.

In this study, the issue of appropriate test method to determine material properties for process simulation, the optimal blank shape determination, and the estimation of optimum force to program multipoint cushion system were addressed. Elliptical bulge test was developed to use along with circular bulge test to estimate flow stress and anisotropy of sheet materials over a large strain range, nearly twice that of tensile test for use in process simulation. The developed test was applied to estimate properties of DP600, AKDQ, BH210 steel and A5182-O aluminum alloy. Deep drawing simulations

ii conducted using the material properties from the developed bulge test better correlated with experiments, compared to FE predictions using tensile test data. This result indicated that material properties obtained from bulge test are more appropriate for process simulation compared to tensile test.

Numerical optimization technique coupled with FE analysis of the forming process was developed to predict optimum forces required to program the multipoint cushion system. Four possible modes for application of BHF in multiple-point cushion systems were considered, namely a) BHF constant in space/location and time/stroke, b) BHF variable in time/stroke and constant in space/location, c) BHF variable in space/location and constant in time/stroke and d) BHF variable in space/location and time/stroke. The optimum BHF was predicted by (a) minimizing the risk of failure by tearing (thinning) in the formed part and (b) avoiding wrinkling. The developed technique was applied to predict the BHF to form a) an automotive part (liftgate-inner) from A6111-T4 aluminum alloy and BH210 steel by stamping process, b) IFU-Hishida part from aluminum alloy A5182-O and BH210 steel by stamping process, c) 90 mm diameter roundcup from ST14 sheet material by sheet hydroforming process with punch (SHF-P) and d) a rectangular part from DP600 sheet material by sheet hydroforming with punch (SHF-D). Material properties obtained from biaxial tests, developed as part of this study were used in the FE simulation. Experimental results showed that the FEM based optimization methodology is able to predict the optimum BHF required to program multipoint cushion system for forming the parts without failure and significantly reduce trial and error effort in the investigated stamping and sheet hydroforming operations. BHF variable in space/location significantly improved the formability/drawbility compared to conventional method of BHF constant in space/location for the investigated parts in stamping and sheet hydroforming.

In addition, element shape function based backtracking methodology was developed in this study to predict the optimal blank shape using FE simulation of forming process. The developed methodology “BLANKOPT” was successfully applied to predict optimum blank shape for an industrial part to eliminate post forming operation and reduce scrap, thereby reducing manufacturing cost. .

iii

Dedicated to my family, and

Lingappa Gounder, A.M. Saravana Gounder, V.K.Appachi Gounder and

A.S. Shakthivadivel

ACKNOWLEDGMENTS

I wish to thank my adviser Prof. Taylan Altan for giving me the opportunity to do my graduate research at Engineering Research Center for Net Shape Manufacturing

(ERC/NSM) and for his intellectual support of this work. His advice always encouraged me to give my best effort to my research.

I would like to thank Prof. Gary L. Kinzel, Prof. Henry R. Busby and Prof. Jerald R.

Brevick for accepting to be my dissertation committee members. I would like to thank Dr.

M.Y. Demeri, Dr. C.T.Wang, and John Lowery of USCAR flexible binder project for sponsoring the research on programming multipoint cushion systems and providing helpful suggestions. I am grateful to Prof. H. Hoffmann, Institute for Metal Forming and

Casting (Utg), Technical University, Munich, Germany, and Dr. B.Griesbach, AUDI AG,

Ingolstadt, Germany, for providing invaluable opportunity for me to do an internship at

AUDI AG, Germany. Also, I would like to thank Prof. E. Tekkaya, Institute of Forming

Technology for Light Weight Construction, IUL, University of Dortmund, Germany, Dipl.-

Ing. K. Schnupp of Schnupp Hydraulik, Bogen, Germany for providing press time and facility to do experiments in sheet hydroforming.

I extend my sincere thanks to my colleagues at ERC/NSM, for their discussion and assistance. Special thanks are due to Dr. Gracious Ngaile, Dr. Hyunjoong Cho, Dr.

Yingyot Aue-u-lan, Dr. Ibrahim Al-Zkeri, Mr. Manas Shirgaokar, Mr. Ajay Yadav, Mr.

Hyunok Kim, Dipl.-Ing. Meinhard Braedel, Dipl.-Ing, Ingo Faass, Dipl.-Ing. Micheal

Mirtsch, Dr. Paolo Bortot, Mr. Giovanni Spampinato, Dipl.-Ing. Joerg Witulski, Mr.

Arunkumar Thandapani, and Ms.Vandana Vavilikolane.

Finally, my special thanks are extended to my parents, my brother and my relatives for their continuous support and encouragement during my PhD study at The Ohio State

University, without their support, my graduate studies would not have been possible.

VITA

October 11, 1977……………………….Born - Erode, India

1998……………………………………...B.E., Mechanical Engineering, National Institute of Technology, (formerly, Regional Engineering College), Tiruchirappali, India. 1998-2000……………………………….Engineer, Sheet metal Tool Design TATA Motors Ltd, Pune, India. 2000-2007……………………………….Graduate Research Associate, Engineering Research Center for Net Shape Manufacturing (ERC/NSM). The Ohio State University, Columbus, Ohio 2006-2007………………………………Guest Researcher, AUDI AG, Ingolstadt, Germany

PUBLICATIONS

Peer reviewed journals

1. H.Palaniswamy, G.Ngaile, and T.Altan, 2004, “Optimization of blank dimensions to reduce springback in Flexforming process”, Journal of Material Processing Technology, Vol 146, pp.28-34

2. H.Palaniswamy, G.Ngaile and T.Altan, 2004, “Finite element simulation of magnesium alloy sheet forming at elevated temperatures”, Journal of Material Processing Technology, Vol 146, pp.52-60

3. S.Chandrasekaran, H.Palaniswamy, N.Jain, G.Ngaile and T.Altan, 2005, “Evaluation of stamping lubricants at various temperature levels using the ironing test”, International journal of machine tools and manufacture, Vol 45, pp.379-388

vii

4. H.Palaniswamy, M.Braedel, A.Thandapani, and T.Altan, 2006, “Optimal programming of multipoint cushion system for sheet metal forming”, Annals of CIRP, Vol 1, pp. 215-220

5. M.Braedel, H.Palaniswamy, H.Hoffmann, B.Griesbach, and T. Altan, 2007, “Investigation of die tryout using programmable multipoint cushion system for stamping large automotive body panels”, Manuscript prepared for submission to CIRP 2008

6. I.Fass, H.Palaniswamy, H.Hoffmann, B.Griesbach, and T. Altan, 2007, “ Effect of spacers on material flow for process control in automotive stamping”, Manuscript prepared for submission to CIRP 2008

7. H. Palaniswamy, A.Yadav, M.Braedel, M.Ujevic, T. Altan, 2007, “Process optimization and analysis for cost effective application of sheet hydroforming technology”, Manuscript prepared for review.

8. H. Palaniswamy, J. Witulski, V.Vavilikolane, M.Trompeter, E.Tekkaya, T. Altan, 2007, “Programming of multipoint cushion systems in sheet hydroforming technology”, Manuscript prepared for review.

Non peer reviewed journals and Conferences

1. H.Palaniswamy, A.Thandapani, S.Kulukuru, and T.Altan, 2004, “Prediction of blank holder force in stamping using Finite Element Analysis”, MATERIALS PROCESSING AND DESIGN: Modeling, Simulation and Applications - NUMIFORM 2004, Vol 712, pp 910-915

2. T.Altan, H. Palaniswamy, and H. Cho, 2005, “Process modeling for precision forming of discrete parts-State of the technology and critical issues”, Steel Research International, Vol 76, pp.191-198

viii

3. T.Altan, H.Palaniswamy, and Y.Aue-u-lan, 2005, “Tube and sheet hydroforming – Advances in materials modeling, tooling and process simulation”, Proceedings of SHEMET2005, pp. 1-12

4. H.Palaniswamy, T.Altan, P.Bortot, and M.Mirtsch, 2006, “Determination of sheet material properties using biaxial bulge tests”, Proceedings of SFB 2006, IWU, Fraunhofer institute, Chemnitz, pp.110-120

5. H.Palaniswamy, T.Altan, 2006, “Programming multipoint cushion system, progress and future work”, New developments in sheet metal forming, Edited by M.Liewald, IFU, Stuttgart, pp 120-136

6. T.Altan, H.Palaniswamy, G.Spampinato, A.Yadav, 2006, “New technologies to form light weight automotive panels”, Proceedings of IABC conference, 2006, Vol 3, pp 10-16.

7. T.Altan, H.Palaniswamy, 2007, “Process modeling and optimization in sheet metal forming – application and challenges”, Accepted for publication in Steel Research International.

FIELDS OF STUDY

Major Field: Mechanical Engineering

TABLE OF CONTENTS

ABSTRACT ...... ii ACKNOWLEDGMENTS ...... v VITA ...... vii LIST OF FIGURES ...... xvii LIST OF TABLES ...... xxxii CHAPTER 1 INTRODUCTION ...... 1 1.1 Stamping ...... 1 1.2 Sheet hydroforming ...... 4 1.3 FE simulation in stamping process design ...... 4 1.4 Rationale and motivation of the work ...... 5 1.5 Research objectives ...... 7

CHAPTER 2 APPROACH...... 8 2.1 Research plan ...... 8 2.2 Organization of the dissertation ...... 8

CHAPTER 3 TECHNICAL BACKGROUND ...... 11 3.1 Stamping die engineering ...... 11 3.2 Advances in cushion technology and tool design in sheet metal forming ...... 12 3.2.1 Press cushion ...... 12 3.2.2 Built –in cushion system in dies ...... 16 3.2.3 Tool design ...... 18 3.3 FEM for simulation of stamping and sheet hydroforming process 19 3.4 Description of sheet material in sheet metal forming simulation .. 23 3.4.1 Shell element formulation ...... 23 3.4.2 Material models ...... 24 3.5 Determination of sheet material properties for process simulation ...... 27

3.5.1 Tensile test ...... 27 3.5.2 Shear test ...... 28 3.5.3 Tensile and Bending test ...... 29 3.5.4 Compression test ...... 30 3.5.5 Bulge test ...... 31 3.6 Prediction of stamping defects using FE simulation ...... 31 3.6.1 Tearing ...... 31 3.6.2 Wrinkling ...... 32 3.7 Process optimization in sheet metal forming ...... 33 3.7.1 Blank holder force ...... 33 3.7.2 Blank shape ...... 34

CHAPTER 4 ESTIMATION OF SHEET MATERIAL PROPERTIES USING BIAXIAL BULGE TESTS ...... 35 4.1 Introduction ...... 35 4.2 Background of proposed biaxial tests ...... 36 4.2.1 Hill’s 1948 orthotropic yield criterion [Hill 1956] ...... 36 4.2.2 Effect of the anisotropic coefficients on the yield surface ...... 37 4.2.3 Elliptical bulge test...... 39 4.3 Preliminary FE simulation to validate proposed biaxial tests ...... 42 4.3.1 FE Model ...... 42 4.3.2 FE Results ...... 44 4.4 Tooling design and manufacturing for the proposed bulge test ... 48 4.4.1 FE simulation ...... 49 4.4.2 FE Results ...... 49 4.4.3 Manufacture of the elliptical die ...... 51 4.5 Analysis of the bulge test using membrane theory ...... 53 4.6 Relationship between bulged sheet geometry and properties of the sheet material ...... 56 4.6.1 Thinning at the top of the elliptical dome...... 57 4.6.2 Radius of curvature at the top of the elliptical dome...... 60 4.7 Methodology to estimate flow stress and anisotropy from bulge tests ...... 66

4.8 Estimation of the anisotropy values (r0, r90) and flow stress from circular bulge test and elliptical bulge test 1 and elliptical bulge test 2 ...... 66 4.8.1 Step 1: Estimation of flow stress for a given anisotropy values from the circular and elliptical bulge test experimental data...... 67 4.8.2 Step 2 : Estimation of anisotropy coefficients and flow stress for a given strain hardening coefficient value from the circular and elliptical bulge test experimental data...... 69

4.9 Estimation of the anisotropy value (r45) from elliptical bulge test 3 ...... 70 4.10 Estimation of the flow stress using the developed bulge test ...... 73 4.10.1 Experimental setup...... 73 4.10.2 Experimental results ...... 74 4.11 Comparison of tensile test and bulge test results ...... 81 4.12 Selection of methodology to determine flow stress and anisotropy coefficient for process simulation in stamping ...... 83 4.12.1 Bulge test ...... 83 4.12.2 Deep drawing of round cup ...... 89 4.13 Summary and conclusions ...... 96 4.14 Future work ...... 100

CHAPTER 5 ESTIMATION OF OPTIMUM BLANK SHAPE FOR SHEET METAL FORMING ...... 101 5.1 Introduction ...... 101 5.2 Definition of optimal blank shape ...... 102 5.3 Methodology to determine optimal blank shape ...... 103 5.3.1 Methodology for identification of the element in the final formed part that contains the target nodes ...... 104 5.3.2 Algorithm for back tracking the target node location to the initial blank ...... 107 5.4 Numerical Implementation ...... 110 5.5 Application – Torque converter turbine shell ...... 111 5.5.1 Background ...... 111

xii

5.5.2 Process description ...... 111 5.5.3 FE Model ...... 112 5.5.4 FE results ...... 114 5.5.5 Estimation of optimal blank shape ...... 115 5.5.6 Experimental validation ...... 120 5.6 Summary and conclusions ...... 121

CHAPTER 6 OPTIMIZATION OF BLANK HOLDER FORCE TO PROGRAM MULTIPOINT CUSHION SYSTEM IN SHEET METAL FORMING ...... 123 6.1 Introduction ...... 123 6.2 Methodology to estimate BHF ...... 125 6.3 Objective function ...... 126 6.4 Constraint function ...... 126 6.5 Geometry based wrinkle criterion ...... 127 6.5.1 Flange wrinkle ...... 127 6.5.2 Side wall wrinkle ...... 129 6.6 Design variables ...... 133 6.7 Optimization methodology ...... 133 6.7.1 Optimum BHF constant in space/location and time/stroke/pressure 136 6.7.2 Optimum BHF variable in space/location and constant in time/stroke/pressure ...... 136 6.7.3 Optimum BHF constant in space/location and variable in time/stroke/pressure ...... 142 6.7.4 Optimum BHF variable in space/location and in time/stroke/pressure ...... 144 6.8 Numerical implementation ...... 145 6.9 Summary and conclusions ...... 146

CHAPTER 7 PROGRAMMING MULTIPOINT CUSHION SYSTEMS - APPLICATION – STAMPING ...... 149 7.1 Introduction ...... 149 7.2 IFU-Hishida part ...... 150 7.2.1 Multipoint cushion system ...... 150 7.2.2 Objectives ...... 153

xiii

7.2.3 FE model ...... 153 7.2.4 Estimation of nodal damping coefficient ...... 157 7.2.5 Optimization problem definition ...... 158 7.2.6 Optimization results: Material- A5182-O ...... 160 7.2.7 Optimization results: Material - BH210 ...... 170 7.2.8 Comparison of thinning distribution ...... 174 7.2.9 Experimental validation ...... 175 7.3 Liftgate –inner part ...... 179 7.3.1 Multipoint cushion system ...... 179 7.3.2 Objectives ...... 181 7.3.3 FE simulation of the liftgate-inner stamping process ...... 181 7.3.4 Optimization problem ...... 195 7.3.5 Optimization results – Material: A6111-T4 ...... 197 7.3.6 Optimization results – Material: BH210 ...... 205 7.3.7 Experimental validation ...... 208 7.4 Summary and conclusions ...... 219 7.4.1 IFU-Hishida die ...... 220 7.4.2 GM liftgate-inner part...... 221

CHAPTER 8 PROGRAMMING SINGLEPOINT CUSHION SYSTEM - APPLICATION – SHEET HYDROFORMING WITH PUNCH (SHF- P) ...... 223 8.1 Introduction ...... 223 8.2 Process limits, defects and process window in SHF-P process . 224 8.3 Objectives ...... 226 8.4 Experimental setup ...... 226 8.4.1 Press ...... 226 8.4.2 Tooling ...... 228 8.5 Optimization problem ...... 228 8.5.1 Objective ...... 228 8.5.2 Constraints ...... 228 8.5.3 Design variables ...... 228 8.6 FE model ...... 228 8.7 Optimization results ...... 230

xiv

8.8 Experimental validation ...... 231 8.8.1 Comparison of thinning distribution ...... 233 8.9 Summary and conclusions ...... 234

CHAPTER 9 PROGRAMMING MULTIPOINT CUSHION SYSTEM - APPLICATION – SHEET HYDROFORMING WITH DIE (SHF-D) ...... 236 9.1 Introduction ...... 236 9.2 Process limits, defects and process window in SHF-D process 237 9.3 Objectives ...... 238 9.4 Experimental setup ...... 239 9.4.1 High pressure sheet hydroforming press ...... 239 9.4.2 Tooling ...... 239 9.4.3 Multipoint blankholder system ...... 240 9.4.4 Die geometry ...... 242 9.5 FE model ...... 242 9.6 Optimization problem ...... 244 9.6.1 Objective ...... 244 9.6.2 Constraints ...... 244 9.6.3 Design variables ...... 244 9.7 Optimization results ...... 245 9.7.1 Piston pressure constant in space/location and in time/forming pressure ...... 245 9.7.2 Piston pressure variable in space/location and constant in time/forming pressure...... 247 9.7.3 Comparison of thinning distribution ...... 249 9.8 Experimental validation ...... 250 9.9 Comparison of FE results with experiments ...... 251 9.9.1 Draw-in ...... 251 9.9.2 Thinning distribution ...... 251 9.10 Summary and conclusions ...... 257

CHAPTER 10 CONCLUSIONS AND FUTURE WORK ...... 259 10.1 Conclusions ...... 259 10.1.1 Material properties determination ...... 260 xv

10.1.2 Optimal blank shape determination ...... 261 10.1.3 Programming multipoint cushion system ...... 261 10.2 Future work ...... 263 10.2.1 Material properties determination ...... 263 10.2.2 Optimal blank shape determination ...... 264 10.2.3 Programming multipoint cushion system ...... 264

REFERENCES ...... 265 APPENDIX A ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 1 ...... 279 APPENDIX B ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 2 ...... 285 APPENDIX C ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 3 ...... 287 APPENDIX D BULGE TEST EXPERIMENTAL RESULTS - MATERIAL DDS STEEL ...... 292 APPENDIX E BULGE TEST EXPERIMENTAL RESULTS - MATERIAL DP600 ...... 298 APPENDIX F BULGE TEST EXPERIMENTAL RESULTS - MATERIAL A5754- O ...... 304 APPENDIX G COMPARISON OF BULGE TEST EXPERIMENTAL RESULTS WITH FE SIMULATION PREDICTIONS USING THE MATERIAL PROPERTIES OBTAINED FROM TENSILE TEST AND BULGE TEST...... 308 APPENDIX H COMPARISON OF ROUND CUP DEEP DRAWING EXPERIMENTAL MEASUREMENTS WITH FE PREDICTIONS 318 APPENDIX I OPTIMIZATION RESULTS FOR HISHIDA PART FROM BH210 STEEL ...... 323 APPENDIX J DRAWBEAD SECTIONS EXTRACTED FROM LIFT GATE DIE SCANNED DATA ...... 329 APPENDIX K DRAWBEAD FORCES FOR BH210 SHEET MATERIAL ...... 334

xvi

LIST OF FIGURES

Figure 1.1 Schematic of typical stamping process for manufacturing sheet metal parts [Schuler 1998] ...... 2 Figure 1.2 Schematic of the drawing process [Marciniak et al. 2002] ...... 2 Figure 1.3 Schematic of drawing process in single action press [Muller Weingerten, 1999] ...... 3 Figure 1.4 Schematic of drawing process in double action press [Muller Weingerten, 1999] ...... 3 Figure 1.5 Schematic of the SHF-P process [Aust 2001] ...... 4 Figure 1.6 Schematic of the SHF-D process ...... 4 Figure 3.1 Schematic of the stamping tool construction (A/SP, 2000) ...... 11 Figure 3.2 Layout of single point, four point, six point, and eight point cushion system built in the press (Mueller Weingarten 1999) ...... 13 Figure 3.3 Schematic of the 15 point cushion system with cushion cylinder directly applying force on the blank holder (Mueller Weingarten 1999) ...... 14 Figure 3.4 Example application of 15 point cushion system to form automobile side wall (Mueller Weingarten 1999) ...... 14 Figure 3.5 Customized multipoint cushion system to form a family of parts (stainless steel sinks) introduced by Diffenbacher (Pahl, 1997) .... 15 Figure 3.6 Movable multipoint (24) cushion system developed by Schuler (Gloecker et al. 2004) ...... 15 Figure 3.7 Individual hydraulic cylinder for each cushion pin attached to the press bed (Siegert et al. 2000) ...... 16 Figure 3.8 Conventional cushion system with servo motor controlled height adjustable cushion pins with load cell (Siegert et al. 2000) ...... 16 Figure 3.9 Schematic of the cushion system built in the die (Hausermann 2000) ...... 17 Figure 3.10 Conventional box shaped design for the blank holder and die [Hausermann 2000] ...... 18 Figure 3.11 Concept of segmented elastic blank holder (Hausermann 2000) .. 19 Figure 3.12 Dimensions of the tensile test specimen as per ASTM E517-92 A 27

xvii

Figure 3.13 Locations for measurement of thickness and width strain to calculate anisotropy of the sheet material as per ASTM E517-92 A ...... 28 Figure 3.14 Schematic of the shear test to estimate the anisotropic properties of the sheet materials ...... 29 Figure 3.15 Schematic of the compression test specimen and test setup for estimating sheet material properties [Boger et al. 2006] ...... 30 Figure 3.16 Schematic of the FLD curves and the constant thinning to illustrate the choice of indicator for failure in the part during stamping/sheet hydroforming process ...... 32

Figure 4.1 Schematic illustrating the influence of anisotropic constant (r0) on the Hill’s 1948 yield criteria...... 38

Figure 4.2 Schematic illustrating the influence of anisotropic constant (r90) on the Hill’s 1948 yield criteria...... 39 Figure 4.3 Schematic of the die cavity in the circular and elliptical bulge test 39 Figure 4.4 Schematic illustrating positioning of sheet metal with respect to die cavity in the proposed elliptical bulge test (major axis = 107 mm/ 4 inch, minor axis = 53.5 mm/2 inch) ...... 40 Figure 4.5 Schematic illustrating the stress path in the elliptical bulge test 1 and elliptical bulge test 2 ...... 41 Figure 4.6 FE model for the forming simulation of the elliptical bulge test 1 and elliptical bulge test 2 ...... 43 Figure 4.7 FE model for the forming simulation of the elliptical bulge test 3 .. 43 Figure 4.8 Forming pressure versus dome height in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 obtained from FE simulation for isotropic case (Case A)...... 44

Figure 4.9 Effect of anisotropy constant along the rolling direction (r0) on the forming pressure versus dome height in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 (Case B) ...... 45

Figure 4.10 Effect of anisotropy constant along the transverse direction (r90) on the forming pressure in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 ...... 46 Figure 4.11 Effect of anisotropy constant along the 45o to the rolling direction (r45) on the forming pressure in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 ...... 47 Figure 4.12 Maximum thinning at the top of the dome in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 obtained from FE simulation for isotropic case...... 48 Figure 4.13 Comparison of the maximum difference in the forming pressure among the three elliptical test predicted by FE simulation for xviii

different elliptical die geometries using AKDQ steel sheet material ...... 50 Figure 4.14 Maximum achievable dome height in the elliptical test predicted by FE simulation for different elliptical die geometries using AKDQ steel sheet material ...... 50 Figure 4.15 Engineering drawing of the designed elliptical die...... 52 Figure 4.16 Elliptical die manufactured from A2 tool steel after heat treatment and polishing ...... 52 Figure 4.17 Schematic of the infinitesimal element at the apex of the dome ... 53 Figure 4.18 Schematic descrbing the shape of the bulged specimen in the elliptical die assumed for analytical calculations ...... 54 Figure 4.19 Thinning at the top of the dome obtained from elliptical test 1 FE simulations for different n values ...... 57 Figure 4.20 Thinning at the top of the dome obtained from elliptical test 1 FE simulations for different anisotropy values ...... 58 Figure 4.21 Comparison of thinning at the top of the dome obtained from FE simulation of the elliptical test 1, elliptical test 2 and elliptical test 3 for same anisotropy values (r0=1.878,r90=1.465,r45=1.308) and flow stress ...... 59 Figure 4.22 Comparison of thinning at the top of the dome obtained from FE simulation of the elliptical test 1 for different initial sheet thickness ...... 60 Figure 4.23 Radius of curvature at the top of the dome along the major axis obtained from elliptical test 1 FE simulations for different n values 61 Figure 4.24 Radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1 FE simulations for different n values 61 Figure 4.25 Radius of curvature at the top of the dome along the major axis obtained from FE simulation of the elliptical test 1 for different anisotropy values ...... 62 Figure 4.26 Radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1 FE simulations for different anisotropy values ...... 63 Figure 4.27 Comparison of radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1, elliptical test 2, and elliptical test 3 FE simulations for same anisotropy values and flow stress ...... 63 Figure 4.28 Comparison of radius of curvature at the top of the dome along the major axis obtained from FE simulation of the elliptical test 1 for different initial sheet thickness ...... 64

xix

Figure 4.29 Comparison of radius of curvature at the top of the dome along the minor axis obtained from FE simulation of the elliptical test 1 for different initial sheet thickness...... 65 Figure 4.30 Flow chart illustrating the methodology to estimate the flow stress and anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2 (k is iteration counter) ...... 68 Figure 4.31 Flow chart illustrating the methodology to estimate the flow stress for given anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2 ...... 69 Figure 4.32 Flow chart illustrating the methodology to estimate the anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2 for given n value ...... 71 Figure 4.33 Flowchart illustrating the methodology to estimate the flow stress and anisotropy values (r45) from the elliptical bulge test 3 ...... 72 Figure 4.34 Viscous Pressure Bulge (VPB) test setup ...... 74 Figure 4.35 Deformed AKDQ samples in circular bulge test, elliptical bulge test 1, 2, and 3 ...... 75 Figure 4.36 Comparison of the pressure versus dome height curve obtained for three different sample of AKDQ sheet material in circular bulge. .. 76 Figure 4.37 Comparison of the pressure versus dome height curve obtained from elliptical test 1, elliptical test 2 and elliptical test 3 for AKDQ sheet material...... 76 Figure 4.38 Comparison of the flow stress curve obtained from circular test, elliptical test 1, elliptical test 2 and elliptical test 3 for AKDQ sheet material assuming the material as isotropic...... 77 Figure 4.39 Change in the objective function during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for AKDQ sheet material ...... 78 Figure 4.40 Change in the anisotropy values during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for AKDQ sheet material ...... 79 Figure 4.41 Flow stress along rolling direction obtained from the circular and elliptical bulge tests using the estimated optimum anisotropy values for AKDQ sheet material ...... 79 Figure 4.42 Flow stress along rolling direction obtained from the circular and elliptical bulge tests using the estimated optimum anisotropy values for AKDQ , DDS steel, DP600 and A5754-O aluminum alloy sheet material ...... 80 Figure 4.43 Comparison of AKDQ and DP600 steel sheet materials flow stress obtained from proposed bulge test and conventional tensile test. . 82

Figure 4.44 Comparison of DDS steel and A5754-O aluminum sheet materials flow stress obtained from proposed bulge test and conventional tensile test...... 82 Figure 4.45 Comparison of the pressure versus dome height curve obtained from circular bulge test experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 85 Figure 4.46 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -1 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 85 Figure 4.47 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -2 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 86 Figure 4.48 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -3 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 86 Figure 4.49 Comparison of thinning distribution along the major axis obtained from elliptical bulge test –1 experiment and elliptical bulge test –2 experiment for AKDQ sheet material at dome height of 18.3 mm. 88 Figure 4.50 Comparison of thinning distribution at dome height of 31.3 mm obtained from circular bulge test experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 88 Figure 4.51 Comparison of thinning distribution at dome height of 18.3 mm along the major axis obtained from elliptical bulge test –1 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 89 Figure 4.52 Round cup deep drawing tooling used in the experiments ...... 90 Figure 4.53 Picture of the round cup with the rolling direction oriented at two different angles to the front of the tool to approximately check for uniform blank holder pressure distribution on the blank ...... 91 Figure 4.54 Picture of the drawn round cups from different sheet material used in the experiment ...... 92 Figure 4.55 Schematic of the FE model for the round cup deep drawing simulation ...... 93 Figure 4.56 Comparison of the punch force obtained from round cup deep drawing experiment for AKDQ sheet material with FE predictions

xxi

using the material properties obtained from the tensile test and bulge tests...... 94 Figure 4.57 Comparison of the draw-in obtained from round cup deep drawing experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 95 Figure 4.58 Comparison of the thinning distribution in the formed round cup from AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests...... 96 Figure 5.1 Description of optimal initial blank shape ...... 102 Figure 5.2 Developed methodology to estimate optimal blank shape using FE simulation and element shape function based back tracking procedure...... 103 Figure 5.3 Superimposition of target nodes on the deformed blank ...... 104 Figure 5.4 Methodology to identify the elements in the formed part that contains the target node ...... 106 Figure 5.5 Methodology to determine the presence of target node inside the element by Baycentric technique [Hearn et al. 2000] ...... 107 Figure 5.6 Flow chart describing the backtracking methodology using element shape function ...... 109 Figure 5.7 Schematic of the steps involved in the back tracking of target node ...... 110 Figure 5.8 Torque converter turbine shell used in this study to estimate blank shape ...... 111 Figure 5.9 Process sequence currently used to manufacture turbine shell .. 112 Figure 5.10 First stage FE simulation model ...... 113 Figure 5.11 Second stage FE simulation model ...... 113 Figure 5.12 Comparison of formed part dimensions from FE simulation and prototype part ...... 115 Figure 5.13 Target nodes obtained from CAD data of the turbine shell part for use in estimation optimal blank shape ...... 117 Figure 5.14 Schematic of initial blank used in FE simulation...... 117 Figure 5.15 Blank shape predicted by BLANKOPT for iteration 1 based on the FE results using initial blank without hole...... 118 Figure 5.16 Comparison of the formed part using iteration 1 blank predicted by BLANKOPT with desired target nodes...... 118 Figure 5.17 Comparison of the blank shape predicted by BLANKOPT for different iterations ...... 119

xxii

Figure 5.18 Comparison of the formed part using iteration 5 blank shape predicted by BLANKOPT with the desired target nodes...... 119 Figure 5.19 Optimum blank geometry estimated by BLANKOPT and provided to sponsor for tryout ...... 120 Figure 5.20 Formed part from die tryout using optimum blank geometry estimated using BLANKOPT and FE simulation...... 120 Figure 6.1 Potential failure modes in deep drawing influenced by the blank holder force ...... 124 Figure 6.2 Flowchart describing the overall approach to determine optimum BHF to using commercial FEA code PAMSTAMP 2000 and LSDYNA to program multipoint cushion system...... 125 Figure 6.3 Schematic illustrating the flange wrinkle magnitude calculation for elastic blank holder and rigid die ...... 127 Figure 6.4 Schematic illustrating the method to describe the flange area in the final part for flange wrinkle calculation...... 128 Figure 6.5 Schematic of the open die with no bottom face and closed die with bottom face ...... 128 Figure 6.6 Strategy for sidewall wrinkle detection in closed die with bottom face ...... 129 Figure 6.7 Schematic to illustrate the detection of sidewall wrinkle for open die with no bottom face ...... 131 Figure 6.8 Schematic illustrating the calculation of the sidewall wrinkle in the circular section...... 131 Figure 6.9 Schematic illustrating the calculation of the sidewall wrinkle in the straight section...... 132 Figure 6.10 Flow chart describing the methodology developed to estimate BHF constant in space/location and in time/stroke ...... 135 Figure 6.11 Flow chart describing the methodology developed to estimate BHF variable in space/location and constant in time/stroke ...... 138 Figure 6.12 Flow chart describing the quadratic programming methodology with active constraint set method developed to determine search direction for multidimensional optimization problem with inequality constraints...... 141 Figure 6.13 Schematic illustrating the Bspline and control points used to describe the B-Splines ...... 142 Figure 6.14 Schematic illustrating the optimization performed in steps to obtain BHF varying in time time/stroke/pressure and constant in space/location ...... 143

xxiii

Figure 6.15 Sequential optimization procedure to estimate BHF varying in time/stroke/pressure and constant in space/location ...... 144 Figure 6.16 Sequential optimization procedure to estimate BHF varying in time/stroke/pressure and in space/location ...... 145 Figure 6.17 Skeleton of the optimization program and its interaction with user and FEA code PAMSTAMP 2000 and LSDYNA ...... 146 Figure 7.1 Schematic of the parts stamped by IFU-Hishida die and GM Liftgate-inner die used in this study...... 149 Figure 7.2 IFU-Hishida die with multipoint cushion system developed by IFU Stuttgart in cooperation with Moog and Hydac [Hengelhaupt et al. 2006]...... 151 Figure 7.3 Schematic of the segmented elastic binder/ blank holder (IFU 2003) ...... 152 Figure 7.4 Schematic of the prismatic draw ring (die) with honeycomb structure (IFU 2003) ...... 152 Figure 7.5 Dimensions of IFU Hishida die in mm (Hausermann 2000) ...... 153 Figure 7.6 Schematic of the FE model used for IFU Hishida die drawing simulations ...... 154 Figure 7.7 Flow stress of BH210 steel and A5182-O sheet material obtained from bulge test ...... 155 Figure 7.8 Dimensions and shape of the initial blank geometry for the Hishida part ...... 156 Figure 7.9 Description of boundary condition on the blankholder for modal analysis ...... 158 Figure 7.10 Comparison of blank holder forces obtained from FE simulation with and without mass damping...... 158 Figure 7.11 Locations for side wall wrinkle calculation from FE results for IFU- Hishida part ...... 159 Figure 7.12 Location of cushion cylinders in IFU Hishida part...... 160 Figure 7.13 Evolution of the design variable during the optimization for the estimation of optimum BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material ...... 161 Figure 7.14 Evolution of the objective function during the optimization for the estimation of BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material ...... 161 Figure 7.15 Evolution of the constraint function during the optimization for the estimation of BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material ...... 162

xxiv

Figure 7.16 Optimum BHF constant in all the cylinders and variable in stroke/time estimated by developed optimization methodology for A5182-0 sheet material ...... 163 Figure 7.17 Evolution of the design variables during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material ...... 163 Figure 7.18 Evolution of the objective function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material ...... 164 Figure 7.19 Evolution of the constraint function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material ...... 164 Figure 7.20 Optimum BHF variable in space/cylinders/pins and constant in stroke/time estimated by developed optimization methodology for A5182-0 sheet material ...... 165 Figure 7.21 Evolution of the design variables (pin 1 – 5) during the optimization for the estimation of optimum BHF variable in pins/cylinders/space and constant in stroke/time for A5182-0 sheet material ...... 166 Figure 7.22 Evolution of the design variables (pin 6-10) during the optimization for the estimation of optimum BHF variable in pins/cylinders/space and in constant in stroke/time for A5182-0 sheet material ...... 167 Figure 7.23 Evolution of the objective function during the optimization for the estimation of optimum BHF variable in all the pins/cylinders/space and constant in time/pressure for A5182-0 sheet material ...... 167 Figure 7.24 Evolution of the constraint functions during the optimization for the estimation of optimum BHF variable in all the cylinders/space/pins and constant in stroke/time for A5182-0 sheet material ...... 168 Figure 7.25 Comparison of thinning distribution predicted by FE simulation along section X-X in the 75 mm deep IFU Hishida pan for the estimated optimum forces...... 169 Figure 7.26 Comparison of thinning distribution predicted by FE simulation along section Y-Y in the 75 mm deep IFU Hishida pan for the estimated optimum forces...... 170 Figure 7.27 Optimum BHF constant in all the cylinders and in stroke/time estimated by developed optimization methodology for BH210 sheet material ...... 171 Figure 7.28 Optimum BHF constant in all the cylinders and variable in stroke/time estimated by developed optimization methodology for BH210 sheet material ...... 172

xxv

Figure 7.29 Optimum BHF variable in all the cylinders/space/pins and constant in stroke/time estimated by developed optimization methodology for BH210 sheet material ...... 173 Figure 7.30 Comparison of thinning distribution predicted by FE simulation along section X-X in the 85 mm deep IFU Hishida pan for the estimated optimum forces...... 174 Figure 7.31 Comparison of thinning distribution predicted by FE simulation along section Y-Y in the 85 mm deep IFU Hishida pan for the estimated optimum forces...... 175 Figure 7.32 Formed IFU-Hishida part of depth 75 mm from sheet material A5182-O sheet material of thickness 1.00 mm using the estimated optimum BHF constant in space/pins/cylinders and in stroke/time ...... 176 Figure 7.33 Formed IFU-Hishida part of depth 75 mm from sheet material A5182-O sheet material of thickness 1.00 mm using the estimated optimum BHF constant in space/pins/cylinders and variable in stroke/time ...... 177 Figure 7.34 Formed IFU-Hishida part of depth 75 mm from sheet material A5182-O sheet material of thickness 1.00 mm using the estimated optimum BHF variable in space/pins/cylinders and constant in stroke/time ...... 177 Figure 7.35 Formed IFU-Hishida part of depth 85 mm from sheet material BH210 sheet material of thickness 0.813 mm using the estimated optimum BHF constant in space/pins/cylinders and in stroke/time ...... 178 Figure 7.36 Formed IFU-Hishida part of depth 85 mm from sheet material BH210 sheet material of thickness 0.813 mm using the estimated optimum BHF variable in space/pins/cylinders and constant in stroke/time ...... 178 Figure 7.37 Schematic of the GM lift gate die ...... 179 Figure 7.38 Schematic of the 26 point cushion system used for the GM lift gate tooling ...... 180 Figure 7.39 Schematic of the press with the 26 point cushion system and the GM lift gate tooling ...... 180 Figure 7.40 Sections of the upper die, punch and the blankholder obtained from the scanned surface of the lift gate tooling for FE simulation ...... 182 Figure 7.41 Schematic illustration of the different types of bead geometry and its location in the lift gate tooling ...... 183 Figure 7.42 Schematic of the drawbead 1 cross section extracted from scanned data of the lift gate tooling die and the blank holder ...... 184

xxvi

Figure 7.43 Schematic of the blank geometry used in the FE simulation (All dimensions are in inches)...... 185 Figure 7.44 Schematic of the FE model for the drawbead simulation to obtain the drawbead normal force and drawbead restraining force...... 187 Figure 7.45 Flow stress of the sheet materials A6111-T4 and BH210 steel obtained from bulge test ...... 187 Figure 7.46 Drawbead restraining force obtained from FE simulation for drawbead 1 ...... 188 Figure 7.47 Drawbead normal force obtained from FE simulation for drawbead 1 ...... 189 Figure 7.48 Drawbead restraining force obtained for different clearance from FE simulation for drawbead 1 ...... 189 Figure 7.49 Schematic of the FE model for the forming simulation of the lift gate part in PAMSTAMP ...... 192 Figure 7.50 Comparison of the force applied by the cushion pin on the blank holder, force applied by the drawbead on the outer blankholder and the force applied by the blankholder on the sheet...... 194 Figure 7.51 Comparison of the force applied by the cushion pin, and sum of force applied by the drawbead on the outer blankholder and the force applied by the blankholder on the sheet...... 194 Figure 7.52 Locations for flange wall wrinkle calculation from FE results for liftgate-inner part ...... 195 Figure 7.53 Locations for side wall wrinkle calculation from FE results for liftgate-inner part ...... 196 Figure 7.54 Location of cushion cylinders/pins and their numbering in liftgate- inner part...... 197 Figure 7.55 Optimum BHF constant in space/location and stroke/time estimated by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm ...... 198 Figure 7.56 Thinning distribution predicted by FE simulation for the optimum constant blankholder force of 72 kN in each pin ...... 198 Figure 7.57 Optimum BHF variable in space and constant in time predicted by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm ...... 200 Figure 7.58 Thinning distribution predicted by FE simulation for the optimum blankholder force variable in space and constant in time ...... 201 Figure 7.59 Comparison of optimum BHF constant in space and variable in time with optimum BHF constant in space and in time estimated by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm ...... 202 xxvii

Figure 7.60 Thinning distribution predicted by FE simulation for the optimum BHF constant in space and variable in time/stroke in each pin ... 202 Figure 7.61 Step size used in sequential optimization to predict the blank holder force variable in space/location and time/stroke for lift gate-inner part ...... 204 Figure 7.62 Schematic of the blank holder force variable in space and in time predicted by sequential optimization technique for pin 1 to pin 8 204 Figure 7.63 Schematic of the blank holder force variable in space and in time predicted by sequential optimization technique for pin 9 to pin 15 ...... 205 Figure 7.64 Thinning distribution predicted by FE simulation for the optimum BHF constant in space and variable in time/stroke in each pin ... 205 Figure 7.65 Optimum BHF variable in space and constant in time predicted by optimization coupled with FE simulation for forming the lift gate part from BH210 steel of thickness 0.81 mm ...... 207 Figure 7.66 Thinning distribution predicted by FE simulation for the optimum blankholder force variable in space and constant in time for BH210 sheet material...... 208 Figure 7.67 A6111-T4 liftgate-inner part formed in the tryout using the optimum BHF variable in space and constant in time predicted by FE simulation coupled with optimization ...... 209 Figure 7.68 A6111-T4 liftgate-inner part formed in the tryout using the optimum BHF variable in space and in time predicted by FE simulation coupled with optimization ...... 210 Figure 7.69 Comparison of the predicted force from optimization coupled with FE simulation and the modified force in tryout for pin 13 to form the part from A6111-T4 ...... 210 Figure 7.70 A6111-T4 part formed in tryout using the modified optimum BHF variable in space and in time at pin 13 ...... 211 Figure 7.71 The formed part in tryout using the optimum blankholder force variable in space and constant in time predicted by FE simulation coupled with optimization for BH210 steel ...... 212 Figure 7.72 Location of the sections in the formed part for comparison of the thinning distribution from experiment with FE predictions ...... 213 Figure 7.73 Comparison of the thinning distribution along section A-A from FE simulation with experiments using FE predicted BHF that resulted in split and modified BHF that resulted in split free panel for A6111- T4 sheet material ...... 214 Figure 7.74 Comparison of the thinning distribution along section B-B from FE simulation with experiments using predicted optimum BHF that

xxviii

resulted in split panel and modified BHF that resulted in split free panel for A6111-T4 sheet material ...... 215 Figure 7.75 Comparison of the thinning distribution along section C-C from FE simulation with experiments using predicted optimum BHF that resulted in split panel and modified BHF that resulted in split free panel for A6111-T4 sheet material ...... 216 Figure 7.76 Comparison of the thinning distribution along section A-A from FE simulation with experiments using predicted optimum blankholder force for BH210 sheet material ...... 217 Figure 7.77 Comparison of the thinning distribution along section B-B from FE simulation with experiments using predicted optimum BHF for BH210 sheet material ...... 218 Figure 7.78 Comparison of the thinning distribution along section C-C from FE simulation with experiments using FE predicted optimum BHF for BH210 sheet material ...... 219 Figure 8.1 Schematic of the SHF-P process and example automotive parts manufactured using SHF-P ...... 224 Figure 8.2 Process window for the SHF-P process ...... 225 Figure 8.3 Schematic of the 80 ton hydraulic press used for experiments at Schnupp Hydraulic GmbH, Germany ...... 227 Figure 8.4 Schematic of the die ring and pressure pot for forming 90mm diameter cup ...... 227 Figure 8.5 Schematic of the FE-Model for the SHF-P process ...... 229 Figure 8.6 Flow stress of st14 steel obtained from the VPB test...... 230 Figure 8.7 BHF and forming pressure varying with punch stroke estimated by optimization coupled with FEM...... 231 Figure 8.8 Picture of the deep drawn samples using the optimized pressure and BHF curve from FE simulations ...... 232 Figure 8.9 Schematic of the non uniform material flow (drawin) observed in the formed cup ...... 232 Figure 8.10 Location of sections for comparison of FE predictions with experiments ...... 233 Figure 8.11 Comparison of thinning distribution from experiment and FE simulation along section A-A for optimized pressure and BHF profile ...... 234 Figure 8.12 Comparison of thinning distribution from experiment and FE simulation along section B-B for optimized pressure and BHF profile...... 234 Figure 9.1 Schematic of the sheet hydroforming process [Kleiner 1999] ..... 236 xxix

Figure 9.2 Process window in the SHF-D process ...... 237 Figure 9.3 Schematic of horizontal high pressure sheet hydroforming press at IUL [Kleiner et al. 2001] ...... 240 Figure 9.4 Schematic of tool design by IUL with multipoint cushion system in the die for blankholder force application [Kleiner et al. 2003] ...... 241 Figure 9.5 Schematic of the die for the rectangular part formed by SHF-D process at IUL,University of Dortmund ...... 241 Figure 9.6 Schematic of the FE model of SHF-D process for rectangular part geometry used to estimate blank holder force by optimization .... 243 Figure 9.7 Instantaneous maximum forming pressure versus time curve used as an input to the aquadraw model in the FE simulation...... 243 Figure 9.8 Locations in the FE model monitored for the flange wrinkle ...... 244 Figure 9.9 Evolution of the objective function during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material ...... 245 Figure 9.10 Evolution of the constraint functions during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material ...... 246 Figure 9.11 Evolution of the design variable during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material ...... 246 Figure 9.12 Evolution of the objective function during the optimization for the estimation of piston pressure variable in all the cylinders and constant in time/ forming pressure for DP600 sheet material ...... 247 Figure 9.13 Evaluation of the constraint functions during the optimization for the estimation of blank holder cylinder pressure variable in all the cylinders and constant in time/forming pressure for DP600 sheet material ...... 248 Figure 9.14 Evolution of the design variable during the optimization for the estimation of piston pressure variable in all the cylinders and constant in time/forming pressure for DP600 sheet material ...... 248 Figure 9.15 Comparison of thinning distribution along section B-B predicted by FE simulation for the optimum piston pressure variable in all the cylinders and constant in time/pressure and optimum piston pressure constant in all the cylinders and time/pressure for DP600 material...... 249 Figure 9.16 Schematic of the formed sheet along section B-B predicted by FE simulation at different forming pressure for friction condition of μ = 0.10 ...... 250

xxx

Figure 9.17 Picture of the formed part using the predicted piston pressure from FE simulations coupled with optimization ...... 250 Figure 9.18 Comparison of drawn part boundary in FE simulation with experiments for optimum piston pressure uniform in all the pistons for DP600 sheet material ...... 252 Figure 9.19 Comparison of drawn part boundary in FE simulation with experiments for optimum piston pressure varying with space/location for DP600 sheet material ...... 253 Figure 9.20 Comparison of drawn part boundary in the experiments for optimum piston pressure varying with space/location with optimum piston pressure constant in space/location for DP600 sheet material ... 254 Figure 9.21 Comparison of thinning distribution in the experiment with FE predictions along section A-A for uniform piston pressure in all the pistons (material DP600) ...... 255 Figure 9.22 Comparison of thinning distribution in the experiment with FE predictions along section B-B for uniform piston pressure in all the pistons (material DP600) ...... 255 Figure 9.23 Comparison of thinning distribution in the experiment with FE predictions along section A-A for optimum piston pressure variable in space/ location (material DP600) ...... 256 Figure 9.24 Comparison of thinning distribution in the experiment with FE predictions along section B-B for optimum piston pressure variable in space/ location (material DP600) ...... 256 Figure 9.25 Comparison of thinning distribution along section B-B in the experiment for optimum piston pressure constant in space and optimum piston pressure variable in space (material ;DP600) .... 257

xxxi

LIST OF TABLES

Table 4.1 Material properties of AKDQ steel used in the FE simulations ...... 42 Table 4.2 Simulation matrix to study the influence of anisotropy constants on the forming pressure, bulge height and the thickness at the top of the dome...... 43 Table 4.3 Critical dimensions of the elliptical die ...... 51 Table 4.4 Anisotropy values obtained from the circular and elliptical bulge tests for AKDQ, DDS steel, DP600 and A5754-O aluminum alloy sheet material...... 80 Table 4.5 Comparison of DP600, DDS, AKDQ steel and A5754-O aluminum alloy sheet materials anisotropy values obtained from proposed bulge test and conventional tensile test...... 83 Table 4.6 Constant blank holder force predicted using preliminary FE simulation for different sheet materials ...... 91 Table 5.1 Input parameters for first stage forming simulation ...... 114 Table 5.2 Input parameters for second stage forming simulation ...... 114 Table 5.3 Comparison of the CAD dimensions with tryout part dimensions formed using the blank predicted by BLANKOPT ...... 121 Table 7.1 Process conditions and material properties used in FE simulation of IFU Hishida pan deep drawing ...... 155 Table 7.2 Input parameters for liftgate-inner drawbead FE simulation ...... 187 Table 7.3 Drawbead forces for the liftgate-inner drawbeads obtained from drawbead simulation for A6111-T4 sheet material using two different clearances ...... 190 Table 7.4 Input parameters for the liftgate-inner forming FE simulation ...... 193 Table 8.1 Technical specifications of the SHF-P press at Schnupp Hydraulik ...... 226 Table 8.2 Input parameters used in the FE simulation of round cup hydroforming process ...... 230 Table 9.1 Input parameters for FE simulation of rectangular part hydroforming process ...... 242

xxxii

CHAPTER 1

INTRODUCTION

1.1 Stamping

Stamping is a plastic deformation process in which, the work piece, sheet blank, is plastically deformed between the tools (dies) to obtain the desired final configuration stored in the tool. Depending on the complexity of the formed part, one or more stages of forming operations are required to transform the sheet blank to desired final product by stamping process. The multiple stage stamping process is performed by either progressive die, or transfer die, Figure 1.1. Accordingly, progressive press or press lines / transfer press is used to form the part [Schuler 1998]. Among the several forming stages, the first stage is usually drawing in both the progressive and transfer die forming where the part is formed very close to the final shape. The successive stages involve post forming operations such as restrike, bending, flanging, trimming and piercing. Hence, the first stage (drawing stage) in a stamping operation is the focus of this study. In drawing, the sheet metal is formed against the die by the punch while the blank holder applies a predefined force to control the material flow into the die (Figure 1.2). Insufficient material flow results in failure of the part by tearing while excessive material flow results in wrinkling. Therefore, blank holder force (BHF) when designed correctly can avoid failure by tearing and wrinkling in the formed part. The necessary blank holder force is commonly applied by the press cushion system.

Drawing operation is performed using either a single action or a double action press. In single action press, Figure 1.3, the sheet metal is initially placed on the blank holder, movement of the die, attached to the top ram initially clamp the sheets against the blank holder and draws the sheet against the punch in presence of BHF acting on the sheet. In double action press, Figure 1.4, the sheet metal is placed on the die. The blank holder attached to the outer ram clamps the sheet against the die and applies the predefined blank holder force, while the punch attached to the inner ram moves down and draws

1 the sheet against the die. The drawn part need to be reversed/rotated in double action press before being used in successive stages/stations for post forming operations. Hence, the single action press is commonly used in mass production of sheet metal parts. The necessary blank holder force in single action press is applied by the cushion cylinders that support the cushion table/press box. Cushion pins transmit the force from the cushion table to the blank holder, Figure 1.3.

a) Progressive die forming b) Transfer die forming

Figure 1.1 Schematic of typical stamping process for manufacturing sheet metal parts [Schuler 1998]

Figure 1.2 Schematic of the drawing process [Marciniak et al. 2002]

Pressure table/ Cushion table

Figure 1.3 Schematic of drawing process in single action press [Muller Weingerten, 1999]

Inner ram

Outer ram

Blank Holder Punch

Blank Die Press bed / Bolster a) Initial set up

b) Closing of the die c) Forming the part d) End of drawing

Figure 1.4 Schematic of drawing process in double action press [Muller Weingerten, 1999]

1.2 Sheet hydroforming

Sheet hydroforming process is an alternative to drawing process where either punch or the die is replaced by hydraulic medium, which generates the pressure and forms the part. Sheet hydroforming is classified into two types SHF-P and SHF-D: In Sheet HydroForming with Punch (SHF-P) (Figure 1.5), the hydraulic fluid replaces the die while in the Sheet HydroForming with Die (SHF-D) (Figure 1.6), the hydraulic fluid replaces the punch. In SHF-P and SHF-D, the quality of a formed part is determined by the amount of material drawn into the die cavity during the forming process which is controlled by the applied blank holder force. Absence of either punch or the die in SHF process reduces the tooling cost, however, SHF-P process are characterized with relatively low cycle time compared stamping. Hence, production economics need to be considered before making the decision for SHF process over stamping process. The formed part in SHF-P and SHF-D process is subject to post forming operations in transfer press lines similar to stamping to obtain the final part geometry.

Pi Blank holder Fluid Sheet Die

Figure 1.5 Schematic of the SHF-P Figure 1.6 Schematic of the SHF-D process [Aust 2001] process

1.3 FE simulation in stamping process design

Stamping process design involves planning and estimation of number of stages required, and the operations need to be performed in each stage to form the required finish part. Initial process plan is developed based on the past experience of the designer and past experience by the firm with the similar components. The process plan has one are more drawing stations were the blank is formed close to the desired final part. The first stage is usually the drawing stage. The die designer develops the shape of the die, punch and the blank holder for the drawing stage from the given part geometry based on his 4 experience. This process is commonly referred as draw development in the industry. The draw development is done based on designer’s experience. At this stage, FE simulation is used to check feasibility of forming the part for the designed draw development. Iterative FE simulations are used in this stage of process design to improve the draw development for a robust production process. The results are transferred to the next forming stage in the process plan and the procedure is repeated to validate the designs. In some cases, all the drawing stages in process plan are simulated and repeated several times until the draw development for the drawing stages are finalized. Thus, FE simulation acts as a virtual press in stamping industry to check the feasibility of forming for the draw developments generated by the designer. Once the designed draw development passes formability, FE simulations are used to predict the springback in the formed part after forming and compensate for springback in die design such that the formed part is within the tolerance. FE simulation of stamping process for springback prediction is lately of significant interest in the industry due to the introduction of Advanced High Strength Steels (AHSS) that has higher strength and springbacks lot more after forming compared to mild stees. Also, stamping simulations are carried out in industry to carry strain history in the formed part for crash analysis to check crashworthiness of the vehicle. Thus, FE simulation of sheet metal forming process is an integral part in the simultaneous engineering approach to the product development and stamping process development in automotive, aerospace and appliance industry. In the FE simulation for stamping process, only the surface of the tool is considered as the focus is on the deforming sheet material. The tools are modeled rigid bodies while the sheet is modeled as an elastic-plastic object.

1.4 Rationale and motivation of the work

Conventionally, in single action press, the BHF is applied uniformly on the blank holder surface and held constant during the forming process. Therefore, it is relatively easy to estimate the BHF by trial and error in FE simulation and tryouts. Many investigations have shown that a better designed BHF profile, which varies in time/stroke and space/location, can improve the drawing process for parts with complex geometry by providing a better control of the material flow. Material flows freely along the straight sides into die cavity compared to corners where it is naturally constrained [Manabe et al. 1987, Doege et al. 1987, Ahmetogulu et al. 1993, Siegert et al. 1993, Broek et al. 1995, 5

Thomas et al. 1997, Neugebauer et al. 1997,Obermeyer et al. 1998, Mattiasson et al. 1998, Gunnarsson et al. 1998]. Modern single action presses are equipped with MultiPoint Cushion (MPC) systems that allow the BHF to vary with location and during the stroke [Pahl 1996, Siegert et al. 1993, Mueller Weingertan, 1999, Gloecker et al. 2004]. However this capability is under utilized currently in production because a) it is difficult to estimate the BHF need to be applied by each cushion pin using trial and error FE simulations followed by die tryouts, and b) conventional steel sheets can be formed with existing method of constant blank holder force. However, increase in the complexity of the parts and emphasis to use lightweight materials with low formability requires the use of multi-point cushion capabilities to better control the material flow and expand the processing window.

Programming of multi-point cushion system in practice to control material flow during forming requires a methodology to estimate the necessary blank holder force for each cushion pin. Also the estimated forces need to be proven in die tryout before being used in production to form the parts as any experimentation in serial production results in loss of press time, which is very expensive. Estimation of BHF profiles can be best done through FE simulation in the process design stage so that the advantages of MPC system can be considered in the draw development to design robust stamping/sheet hydroforming process and it is cost effective to change the process vitually than with the physical tool. Also, the estimated forces from process simulation can be provided as in input to die tryout to validate and fine tune the BHF profiles thereby saving time in die tryout to program multipoint cushion system by trial and error to form the part.

Also of importance is the blank shape that affects the material flow in the forming process. The blank shape needs to be estimated in the process design stage to better optimize the material flow so that trial and error in die tryout to estimate blank shape can be avoided.

Successful application of FE simulations to estimate optimum process parameters such as blank shape, forces required for programming multipoint cushion system, depends on the accuracy of the input parameters to FE simulation namely, the tool geometry, friction conditions and the material properties of the sheet material. Conventionally material properties of sheet materials obtained from tensile test are used in FE simulation. This

6 data is insufficient for analysis of stamping process because a) maximum strain obtained in uniaxial tensile test before necking is small compared to strains encountered in stamping operations b) the stress state in tensile test is uniaxial while in regular stamping it is biaxial. Therefore, there is a need for biaxial test to accurately determine the material properties (flow stress and anisotropy constants) over larger strain range for use in forming FE simulations.

1.5 Research objectives

The overall objective of this study is to develop a reliable test method for characterizing the properties of sheet materials and a robust design tool coupled with commercial FE codes PAMSTAMP and LSDYNA to assist process designers in selecting a) the BHF that need to applied depending on the type of cushion and b) shape of the blank, that need to be used to form the part with minimum risk of failure by stamping/sheet hydroforming processes. The specific objectives of this study are to develop: 1. a new test method and procedure to determine the flow stress and anisotropy of the sheet material from the biaxial tests over large strain range for use in stamping/sheet hydroforming process simulation. 2. a new methodology that can be coupled with commercial FE codes to estimate the initial blank shape such that after forming it results in final part with desired part boundary. 3. a methodology to estimate optimum BHF through FE simulations required to form the part by stamping/sheet hydroforming process using multipoint/single point blank holder system. This includes four possible modes of application of BHF namely • BHF constant in space/location and time/stroke (single point/multipoint cushion/nitrogen cylinders) • BHF variable in time/stroke in constant in space/location (single point cushion/multipoint cushion) • BHF constant in time/stroke in variable in space/location (multipoint cushion/nitrogen cylinders) • BHF variable in space/location and time/stroke (multipoint cushion)

CHAPTER 2

APPROACH

2.1 Research plan

To achieve the research objectives, the research approach was divided into three major tasks as follows: Task 1: Development of test procedure and methodology to estimate the flow stress and anisotropy of the sheet material using biaxial tests. Task 2: Development of a methodology to predict the optimal blank shape that will result in a desired final formed part using commercial FE codes PAMSTAMP and LS-DYNA. Task 3. Development of methodology to predict the optimum BHF/forming pressure required to form the part in stamping / sheet hydroforming process using FE simulation for the existing single point cushion/nitrogen cylinders and advanced multipoint cushion systems. Commercial FE software’s PAMSTAMP and LS-DYNA were used in the study.

2.2 Organization of the dissertation

Dissertation has 10 chapters. The contents of each chapter are as follows:

Chapter: 1 Brief introduction to stamping, sheet hydroforming, and use of FE simulation for process design in sheet metal forming is discussed. The rationale and motivation behind the research is explained. The specific objectives of this study are provided.

Chapter: 2 The approach used to meet the objectives of this study is presented.

Chapter: 3 Technical background relevant to this study namely; Advances in press and tooling for stamping and sheet hydroforming, FE formulation in commercial FE codes to simulate sheet metal forming, Modeling the physical and material property of sheet material in commercial FE codes, 8

Test methods used to estimate material properties of sheet metal for sheet metal forming simulation, Methodology to predict potential failure modes from FE results, and Advances in optimization of sheet metal forming process available from literature are discussed.

Chapter 4 This chapter is devoted to task 1 of this study, estimation of sheet material properties by biaxial bulge tests. The elliptical bulge test developed as part of this study. Methodology to estimate the flow stress and anisotropy of sheet materials from this study are described in detail. Results from the developed tests for sheet materials AKDQ steel, DSS steel, DR210 steel and A5754-O are provided. Also, the bulge test results were compared with tensile test for the corresponding material. The bulge test and tensile test data were used in deep drawing process simulation to identify the best test method to estimate sheet material properties for process simulation.

Chapter 5 This chapter is devoted to task 2 of this study, development of methodology to predict optimum blank shape. Shape function based backtracking method was developed to estimate the optimum blank shape using FE simulation results. The developed methodology was applied to estimate optimal blank shape of torque converter turbine shell. Estimated blank shape was validated in die tryouts.

Chapter 6 This chapter covers the optimization methodology developed to estimate four possible cases of the optimum blank holder force, namely: a) BHF constant in space/location and time/stroke b) BHF constant in space/location and variable in time/stroke c) BHF variable in space/location and constant in time/stroke d) BHF variable in space/location and time/stroke.

Chapter 7 In this chapter, the developed methodology to estimate optimum BHF was applied to estimate optimum BHF for IFU-Hishida part and GM liftgate-inner part formed by drawing process using multipoint cushion system. The predicted BHF profiles were later validated in experiments. The results from optimization and experimental validation are presented.

Chapter 8 In this chapter, the developed methodology to estimate optimum BHF was applied to estimate optimum BHF for round cup formed by SHF-P process using singlepoint cushion system. The predicted BHF profiles were later validated in experiments. The results from optimization and experimental validation are presented.

Chapter 9 The developed methodology to estimate optimum BHF was applied to estimate optimum BHF for rectangular pan formed by SHF-D process using multipoint cushion system. The predicted BHF profiles were later validated in experiments. The results from optimization and experimental validation are presented in this chapter.

Chapter 10 This chapter provides the summary and conclusion of the findings and future work for this study.

CHAPTER 3

TECHNICAL BACKGROUND

3.1 Stamping die engineering

In stamping, a number of stages are usually required to form a given part. The first stage is drawing where the part is formed very close to the final shape. Therefore, design of the drawing stage is crucial for forming the part and requires the design of the blank holder, punch and die shape from the part geometry. Design guidelines and past experience with similar parts are used for designing the shape of the draw die commonly known as draw development. FEM simulations are used to check the design and iteratively modify the draw development by trial and error and experience to virtually obtain a part without defects.

Figure 3.1 Schematic of the stamping tool construction (A/SP, 2000)

The tools (punch, die, and blank holder) are constructed based on the FE validated draw development using box type sections as shown in Figure 3.1 to have high stiffness in the drawing direction. The tools are first tried in a tryout press to form the part before being used in production. In the tryout, the process parameters namely BHF, blank shape, 11 drawbead shape; blank holder, and die surface are refined by trial and error until a good part is obtained repeatedly. In the FE simulation, the tools are considered rigid while in reality the tools deflect elastically and the stiffness of the tools is not uniform. Due to the box type construction, tools have high stiffness at the location of ribs and lower stiffness at the other locations resulting in variation in the blank holder pressure distribution on the sheet. Therefore, spotting/grinding is done at high-pressure locations and forming is repeated until the part is formed repeatedly without defects. Balancer blocks are added to die and blank holder at the rib locations to assist in uniform blank holder pressure distribution on the sheet. Thus, currently die tryout occupies a large portion of the die development time and cost despite advances in the process simulation (Dingle et al. 2001). Recent advances in press and tooling such as multipoint cushion system and flexible blank holder system focus on minimizing spotting and reducing the tryout time.

3.2 Advances in cushion technology and tool design in sheet metal forming

3.2.1 Press cushion

In stamping process, the necessary BHF is applied either by cushion in the press or by the hydraulic/nitrogen cylinders mounted on the die. Conventionally BHF is kept constant in all location and not changed with punch stroke due to the limitations in the cushion design. Earlier hydraulic cushion system built in the presses had a single hydraulic cylinder at the center of the cushion table / pressure box that applied the necessary blank holder force. Hence, ideally, the force applied by each cushion pin was same on the blank holder resulting in uniform pressure distribution on the sheet. However, in deep drawing, the material flows easily into the die cavity along the straight edges while at the corners its flow is restrained. Therefore, blank holder pressure varying with location is more helpful for deep drawing. Hence, multipoint cushion were introduced in the press to provide the blank holder force varying with space/location. Multipoint cushion system designs can be broadly classified into two types. In type –I, more cylinders were added to the pressure box/cushion table to provide the blank holder force. Each cylinder is independently controlled hence each cylinder can apply different force thereby realizing variation of force in space/location. Number of cylinders/points added to apply force on the cushion table/pressure box ranges from four to eight as shown in Figure 3.2.

Single point cushion system Four point cushion system

Six point cushion system Eight point cushion system

Figure 3.2 Layout of single point, four point, six point, and eight point cushion system built in the press (Mueller Weingarten 1999)

In type-II, the cushion cylinders directly apply the blank holder force on the blank holder in the absence of pressure box /cushion table and cushion pins as shown in Figure 3.3. Each cylinder is independently controlled thereby allowing the blank holder force to vary with location. Currently such a cushion system with 15 independent cushion cylinders in a Mueller Weingarten press is commercially used in stamping side walls of automobiles, Figure 3.4 [Mueller Weingarten 1999]. Diffenbacher, leading press manufacturer also builds presses with multipoint cushion system where the blank holder force is directly transmitted from the cushion cylinder to the blank holder without any pressure box / cushion table. The individual cushion cylinders in the cushion system are located in the press bed such that it is customized to produce a specific family of part, in this cases kitchen sinks, Figure 3.5. It has a nine point cushion system where the outer rows of cylinders are connected to the same hydraulic power source thereby providing the same pressure. The inner cylinders that are symmetrically placed are also coupled to provide eight additional points to change the blank holder force [Pahl, 1997]. The disadvantage of such a system is that the blank holder force cylinders are fixed and cannot be moved thereby limiting the flexibility of press to form wide range of parts. Recently Schuler, another leading press manufacturer introduced 24 point cushion system where the force is transmitted directly to the blank holder from the cushion cylinder in the absence of the pressure box/ cushion table and cushion pins, Figure 3.6. 13

The 24 cylinder are arranged in 8 columns with three cylinders in each column. Each column forms a block. The blocks can be moved longitudinally thereby allowing the option to customize the cylinder location to better apply the blank holder force depending on the part geometry [Gloecker et al. 2004].

Die

Blank holder

Punch

Cushion cylinder

Figure 3.3 Schematic of the 15 point cushion system with cushion cylinder directly applying force on the blank holder (Mueller Weingarten 1999)

Cushion cylinder location

Figure 3.4 Example application of 15 point cushion system to form automobile side wall (Mueller Weingarten 1999) 14

Figure 3.5 Customized multipoint cushion system to form a family of parts (stainless steel sinks) introduced by Diffenbacher (Pahl, 1997)

Figure 3.6 Movable multipoint (24) cushion system developed by Schuler (Gloecker et al. 2004)

At research institute IFU, Technical university of Stuttgart, two prototype multipoint cushion systems were built to apply blank holder force varying in space in cooperation with press manufacturer’s. In design I, the hydraulic cylinders are attached to the press table and apply force directly on the blank holder, Figure 3.7. In design II, the conventional cushion system with cushion table/pressure box is equipped height adjustable cushion pins, Figure 3.8. Height of each cushion pin can be adjusted by minimum of 0.05 mm using servo drive and each cushion pin is equipped with load cell to measure the force. In this system, force between each pins are varied by changing

15 the height of the pins. The force and the height setting for each pin are recorded thereby minimizing the setting time involved during die change (Siegert et al. 2000).

Figure 3.7 Individual hydraulic cylinder for each cushion pin attached to the press bed (Siegert et al. 2000)

Figure 3.8 Conventional cushion system with servo motor controlled height adjustable cushion pins with load cell (Siegert et al. 2000)

3.2.2 Built –in cushion system in dies

Cushion systems that exists in the press can also be built into the dies as shown in Figure 3.9. The cushion cylinders could be hydraulic or nitrogen cylinder attached to 16 the die shoe and directly applies the blank holder force on the blank holder, Figure 3.9. Built in cushion system allows the flexibility to position the cushion cylinder depending on the part during its construction thereby allowing better control over the material flow compared to cushion system in the press where the cushion pin position in the press bed decides the location of force application on the blank holder. Also, the die set can be used in the press without cushion provided the cushion system is tuned to operate under the operating speed of the selected press. In addition, since the force is applied directly on the blank holder, variation in the force between cylinders due to difference in pin height and slide deflection due to offset loading are avoided. Hence the setup time is less compared to the system using the press cushion. Also, each cylinder in the built-in die cushion system can be independently controlled thereby material flow is efficiently controled as a multipoint cushion system in the press [Hausermann 2000, Hengelhaupt et al. 2006, Metal forming controls, 2007]. However, addition of cushion system increases the cost of the tooling and hence are preferred only if it is economical compared to using the cushion system in the press.

Die

Blank holder

Punch

Cushion cylinder

Figure 3.9 Schematic of the cushion system built in the die (Hausermann 2000)

3.2.3 Tool design

The conventional box type designs, Figure 3.10 for the blank holder and die results in non-uniform blank holder pressure distribution on the sheet. Therefore, several design concepts for blank holder design had been proposed. All the concepts emphasized flexible blank holder, which is a plate that elastically deflects and remain in contact with the non-uniform thickness sheet and apply uniform blank holder pressure during the forming process (Doege et al. 1987, Doege et al. 2001, Siegert et al. 2001, Shulkin et al. 1995, Groche et al. 2005, Haussermann,2000). Among these designs, segmented blank holder design concept from IFU-Sttutgart (Siegert et al. 2001) is being widely pursued. The segmented elastic blank holder design from IFU consists of a pyramidal shape block for each cushion pin and the individual blocks are connected by thin plate of 30 to 40 mm thickness as shown in Figure 3.11. The plate and the blocks are cast together as a single structure. The pyramidal blocks are used to transmit the force applied by the cushion at a small concentrated area (area of the cushion pin) at the bottom of the pyramid to uniform pressure on a larger area on top of the blank holder. The die is designed with honeycomb structure to have high uniform stiffness. Segmented blank holder along with multipoint cushion system would assist in locally applying the force thereby, enhancing the formability of the part and significantly reduce the tryout time. However, successful application of this technology requires the knowledge of the force to be applied by each cushion pin for programming multipoint cushion system to form the part. This study addresses this issue of blank holder force prediction through FE simulation of the forming process for programming multipoint point cushion system.

Figure 3.10 Conventional box shaped design for the blank holder and die [Hausermann 2000] 18

Figure 3.11 Concept of segmented elastic blank holder (Hausermann 2000)

3.3 FEM for simulation of stamping and sheet hydroforming process

Analysis of the physical problem in the field of engineering mechanics with assumptions on the geometry, material behavior, loading, and boundary conditions leads to partial differential equations with boundary conditions that need to be solved. Finite element method is one of the robust techniques being used to solve these problems where closed form solution is difficult to obtain. Analysis of the sheet metal deformation in the stamping and sheet hydroforming at room temperature leads to the following equilibrium equation that need to be satisfied at all the points in the deforming body

σ , b =− ρu&& iijij in V Equation 3-1

where σ ij is the cauchy stress tensor, bi is the body force, ρ , is the density of the

material, and u&& i is the acceleration of the body. The boundary conditions that need to be satisfied are

* on S and * on S Equation 3-2 = uu ii u σ = tn ijij f

* where ui is the prescribed the displacement on the boundary Su as boundary condition and * is the prescribed traction boundary condition on the deforming body boundary S t i f The strain- displacement relationship (Compatibility equations) is given by 1 ε ()+= uu Equation 3-3 ij 2 ,, ijji The sheet material is usually described as orthotropic elastic-plastic body. The constitutive equations that relate stress and strain are as follows; The total strain is divided as elastic and plastic e += εεε p . Elastic stress-strain relationship is given by

e σ = Cijklij ε kl Where C is the tensor of elastic constants. The plastic strain increment ∂φ based on associative flow rule is given by & p = λε . where λ is the constant and φ ∂σ ij is the yield potential/yield criteria that describes plastic yielding of the material. The yield

criteria is expressed as φ = f ij )( − (εσσ ij ) = 0 , where f σ ij )( is the function that relates the multiaxial stress state in the material to corresponding uniaxial stress state such that the work done on the material on multiaxial stress state is equal to the uniaxial stress state. Several of these function proposed to describe the yielding behavior of sheet

material behavior are discussed in next section. (εσ ij ) is the yield stress of the material in uniaxial stress state

The partial differential equation for equilibrium (equation 3-1) with constraints is solved on an average over integral ( weak sense) using weighted Galerkian method as

ψ u (σ b ρ dVu =−−= 0) Equation 3-4 ∫ , && iijiji v Applying the divergence theorem it reduces to

σψ −−= uubu ρ * dsutdVu =− 0 Equation 3-5 ∫ (), && iiiijiij ∫ i v Su In case of elastic-plastic problem the stress-strain relationship changes with location depending on the state of stress being elastic or plastic. Therefore, σ = Cepε , where

Cep is the elastic-plastic modulus. Estimation of elastic-plastic modulus is discussed in Simo et al. 1998.The equation reduces to

= εψ T C ε T −− ubu T ρ − T *dstudVu = 0 Equation 3-6 ∫ ()ep && ∫ v Su In the FEM, the deforming body is discretized into elements with nodes at the intersection of two edges of the element. Therefore, the equation 3-6 reduces to

⎧ ⎫ nelement ⎪ ⎪ ψ = εT C ε T −− ubu T ρ − T *dstudVu = 0 Equation 3-7 ∑i =1 ⎨ ()ep && ⎬ ⎪∫ ∫ ⎪ ⎩ve Suelement ⎭ where nelement is the total number of the elements in the structure , Ve is the volume of the

element and Suelement is the surface area of the element where traction boundary condition is applied. In each element, the state variable, displacement is 20 defined/estimated at the nodal points and are interpolated inside element using the element shape function. Therefore, displacement inside each element is given by = uNu ˆ , where N is the interpolation function. The strain is given by 1 ε ()=+= uBuu ˆ . Therefore, the equation 3-7 reduces to ij 2 ,, ijji

⎧ ⎫ nelement ⎪ ⎪ ψ = ˆ TT ˆ T −− NbNuBCBu T ρ ˆ − ˆT T *dstNudVuN = 0 Equation 3-8 ∑i =1 ⎨ ()ep && ⎬ ⎪ ∫ ∫ ⎪ ⎩ ve Suelement ⎭ After performing numerical integration using the gauss quadrature for each element and assembling we get

&&& =++ FUkUCUM ][]][[]][[]][[ Equation 3-9

⎧ ⎫ n ⎧ ⎫ ⎧ ⎫ where [M]= nelement ⎪ T ⎪ ,[K]= element ⎪ T ⎪, [F] = nelement ⎪ T T * ⎪ N ρ )( dVN ()dVBCB + dstNdVbN ∑i =1 ⎨ ⎬ ∑i =1 ⎨∫ ep ⎬ ∑i =1 ⎨ () ⎬ ⎪∫ ⎪ ⎪ ⎪ ⎪ ∫∫ ⎪ ⎩ve ⎭ ⎩ve ⎭ ⎩ve Suelement ⎭ [C] is the damping matrix added to the system. The equation 3-9 need to be solved at every time instant to get the displacement field over the entire time period [0 ..T] of the forming process. In the FEM, the time axis is also discretized into intervals [0, Δt, 2Δt, … T] and solution to the equation 3-9 are obtained at every time interval Δt. The equation 4-9 can be solved using either static implicit (FE code DEFORM 2D, DEFORM3D, AUTOFORM) or dynamic explicit procedure (PAMSTAMP, LSDYNA).

In static implicit procedure, the process is assumed to be quasi-static hence the effect of inertia is neglected. Therefore the equation 3-9 reduces to

= FUk ][]][[ Equation 3-10 which is solved at [0, Δt, …t+Δt, … T], Δ+Δ+ = FUk ][][][ Δ+ tttttt to get the displacement [U]. Due to large deformation in metal forming, [K] and [F] are non linear function in [U]. Therefore, equation 3-10 is solved using Newton-raphson procedure or direct iteration method that involves assembly of the [K] matrix and triangularizing it.

In the dynamic explicit procedure, the effect of body forces is neglected and the equation is solved using the central difference method. In the central difference method, the

21 equilibrium equation is solved at time t to get displacement at (t+Δt) therefore this procedure is called explicit method. &&& =++ FUkUCUM ][][][][][][][ ttttttt

1 U&& ][ t = {}Δ+ tt +− UUU ][][2][ Δ− ttt Equation 3-11 Δt 2 1 U& ][ t = {}Δ+ − UU ][][ Δ− tttt 2Δt Substituting and rearranging we get

t t ⎧ M C][][ ⎫ Δ+ tt t ⎨ 2 + ⎬ = RU ][][ ⎩ Δt 2Δt ⎭ Equation 3-12 t t t tt ⎧ t M][ ⎫ t ⎧ M C][][ ⎫ Δ− tt ⎨ kFR −−= 2][][][ 2 ⎬ U][ − ⎨ 2 − ⎬ U][ ⎩ Δt ⎭ ⎩ Δt 2Δt ⎭ [R]t includes terms that are known at time t, therefore it can be calculated at the element level and assembled. Hence assembling of large stiffness matrix as in implicit procedure can be avoided in explicit method thereby computational time and memory storage are saved considerably. In the analysis, the damping matrix [C] is approximated using a Raleigh damping as C t α M t += β k][][][ t . In the FE codes for stamping /sheet hydroforming simulation, β = 0, therefore the equation reduces to

t ⎧ 11 ⎫ Δ+ ttt M][ ⎨ + α ⎬ = RU ][][ . Equation 3-13 ⎩Δt 2 2Δt ⎭ Solution to the equation 3-13 requires inverse of the global mass matrix [M], which is not a diagonal matrix. However for isoparametric elements, the mass matrix of each element can be diagonalized within element level using H-R-Z lumping scheme resulting in an global diagonalized mass matrix [M] thereby avoiding inverse calculation and quick solution to the equation. In fact, the solution (nodal displacement) can be calculated for each node by locally assembling the stiffness and mass matrix from the elements that share the same node. Thus, explicit methods solve the equilibrium equation for the unknown displacements with less time and computing storage compared to implicit method and are preferred for stamping / sheet hydroforming simulation at room temperature. However in explicit method, the time increment Δt should be less than the critical value (Δtcritical) to avoid divergence of the solution. The critical time interval for

L each element is given by Δtcritical=0.98* , where L is the characteristic length of the E ρ

element ( min of the length of the two diagonals in the element for sheet forming simulation) , E is the Young’s modulus of the deforming material, and ρ is the density of the deforming material. In the FEM, critical time interval for each element at every time increment is calculated and the minimum time interval among all the elements is used for the calculation [Bathe 1997, Ghosh 2002, Zienkwicz et al. 2000, Hughes 2000, Cook et al. 2000].

3.4 Description of sheet material in sheet metal forming simulation

3.4.1 Shell element formulation

In the FEM, the sheet metal is discretized into elements and the variation of the state variable (displacement, strain, stress) within the element is assumed. In case of sheet materials, the dimension of the sheet along the thickness is negligible compared to other two dimensions but are thick enough to support bending loads. Therefore, the sheet metal is represented by shell elements. The isoparametric shell element was formulated by combining a plane stress element (σ zz = 0 ) that supports the axial force and Mindlin type plate elements that supports the bending stress and the transverse stress due to bending . The element represents the sheets mid plane and each node of element has five degrees of freedom that includes displacement in X, Y and Z direction (u,v,w) and

rotation about X and Y axis ( θ x , θy ) that varies in the element plane (X,Y). The displacement in the element is given by

( )(11 −− ης ) ( )(11 +− ης ) ς −= 1...0...1 ⎧u ⎫ ⎧u ⎫ N1 = , N 2 = , n i n ⎧θ ⎫ 4 4 ⎪ ⎪ ⎪ ⎪ t xi η −= 1...0...1 ⎨v ⎬ = N i ⎨v i ⎬ + N i ξ ⎨ ⎬ ∑∑ 2 θ ()()11 ++ ης ()()11 −+ ης ⎪w⎪ i =1 ⎪w ⎪ i =1 ⎩ yi ⎭ N = , N = ξ −= 1...0...1 ⎩ ⎭ ⎩ i ⎭ 3 4 4 4 Equation 3-14

⎧zθ ,xx ⎫ ⎧ u, x ⎫ ⎧ε xx ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ zθ , yy v, y ⎪ ⎪ ε yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪=stretching strain +bending strain Equation 3-15 ⎪ ⎪ ⎪ +vu ,, xy ⎪ ⎪ z()+θθ ,, xyyx ⎪ ⎨ε xy ⎬ = ⎨ ⎬ + ⎨ ⎬ 2 ⎪ε ⎪ ⎪ 2 ⎪ ⎪ ⎪ xz 0 ⎪ ⎪ ⎪ ⎪ ⎪θ x ⎪ ⎪ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yz ⎭ ⎩ 0 ⎭ ⎩⎪θ y ⎭⎪ The strain – displacement relationship is given by equation 3-15. Thus the shell elements accounts for both bending and stretching strain in the deformation of the sheet.

In the calculation of the stiffness matrix for each element, K = T εε dVC , element ∫ ()ep ve integration is performed numerically using the gauss quadrature along the X-Y plane. While the strain (ε ) is a linear function in Z (thickness direction) and can be directly

integrated in case of elastic material, as Cep is a constant. However, bending causes plastic stress in the extreme fibers of the sheet thereby resulting in non linear variation of the stress along the thickness direction. Therefore, numerical integration needs to be performed along thickness direction as well with more integration points to accurately calculate the nonlinear bending stresses [Zienkwicz et al. 2000, Cook et al. 2000, Hughes et al. 2000].

3.4.2 Material models

Continuous cold/hot working of the sheet material during manufacturing in the rolling process causes orientation of the grains and grain boundaries along the rolling direction resulting in anisotropic behavior. Therefore, the sheet material behavior is assumed to be orthotropic implying the mechanical properties vary only along the three direction namely 1) rolling direction b) transverse to the rolling direction and c) thickness direction. Also in the analysis, the sheet material is assumed to be plane stress along thickness direction. Hence, plane stress orthotropic yield criteria are used to study the plastic deformation behavior of sheet metal. Several plane stress orthotropic yield criteria [Hill 1948, Hill 1979, Hosford 1979, Barlat 1989, Hill 1990, Barlat 1991 (YLD91), Hill 1993, Karafillis 1993, Barlat 1994 (YLD94), Barlat 1996 (YLD96), Barlat 2000 (YLD2000), Cazacu Barlat 2003, Banabic Balan 2003 (BBC 2003)] have been proposed till date to phenomologically represent the anisotropy yielding behavior exhibited by F.C.C and

B.C.C crystal structure sheet metal alloys. Among the yield criteria Hill’s1948, Barlat 1991 are widely used.

3.4.2.1 Hill’s 1948 yield criteria

Hill’s yield criteria for orthotropic material is given by

2 2 2 2 2 2 ij Hf XY G XZ F YZ XY YZ LMN τττσσσσσσσ XZ =+++−+−+−= 1222)()()()(2 Equation 3-16 where

σ σ ,, σ zyx are the normal stress in X,Y,Z orthotropic axis

τ τ ,, τ yzxzxy are the shear stress F, G, H, L, M, N are the anisotropy constants in the yield criterion

Assuming plane stress condition along thickness direction (σ τ ,, τ xzyzz = 0) as commonly used in analyses of sheet materials. Hill’s yield criteria reduces to

2 2 2 2 ij Hf XY )()(2 X Y NFG τσσσσσ XY =+++−= 12 Equation 3-17 Using the flow rule and from stress states in tensile test it can be found that H H H 1 r0 r90 ,, === Equation 3-18 G F 2N ⎛ 11 ⎞ ⎜ ⎟ ()r45 12 ⎜ ++ ⎟ ⎝ rr 900 ⎠ where r0,r90,and r45 are the plastic strain ratio obtained from tensile test along rolling direction, transverse to rolling direction and 45o to the rolling direction From equation 3-17 and equation 3-18 we get rr σσ )( 2 r σ 2 r σ 2 )12( ( +++++− rrr )τ 2 900 XY 90 X 0 Y 45 900 XY = σ 2 Equation 3-19 ()rr 090 +1 Therefore either equation 3-19 or equation 3-17 can be used to describe the behavior of sheet material in the FE simulations by the Hill’s 1948 yield criteria. Hill’s 1948 is most widely used for steel and its alloys that has normal anisotropy 2 ++ rrr r value( r = 0 9045 )greater than one and ratio of anisotropy ( 0 ) and ratio of yield 4 r90 σ stress ( 0 ) are either greater or less than one. σ 90

3.4.2.2 Barlat’s 1991 yield criteria

Barlat et al. 1991 extended the isotropic yield criteria proposed by Hosford for anisotropic material. The yield criteria is given by

m m m m φ 21 32 SSSSSS 13 =−+−+−= 2σ Equation 3-20 where ,, SSS 321 are the principle deviatoric stress, σ is the yield stress m = constant depending on the material (m=6 for BCC alloys and m=8 for FCC alloys)

Barlat et al. 1991 introduced anisotropy constants in the deviatoric stress tensor as shown below

⎡2 −− σσσ ⎤ zyx τ τ ⎡ − BC ⎤ ⎢ XY XZ ⎥ ⎢ GH ⎥ ⎢ 3 ⎥ 3 2 −− ⎢ ⎥ ⎢ σσσ zxy ⎥ − CA S = τ XY τ YZ = ⎢ H F ⎥ ⎢ 3 ⎥ ⎢ 3 ⎥ ⎢ 2 −− σσσ ⎥ − AB xyz ⎢ FG ⎥ ⎢ τ XZ τ YZ ⎥ ⎢ ⎥ ⎣⎢ 3 ⎦⎥ ⎣ 3 ⎦ Equation 3-21 Replacing A, B, C, G, H, F in equation 3-21 with aA, bB, cC, gG, hH, fF, respectively and assuming plane stress conditions (σ τ ,, τ xzyzz =0).

⎡ − bBcC ⎤ hH 0 ⎢ 3 ⎥ ⎢ − cCaA ⎥ S = ⎢ hH 0 ⎥ Equation 3-22 ⎢ 3 ⎥ ⎢ − aAbB ⎥ 0 0 ⎣⎢ 3 ⎦⎥

where a, b, c, and h are the anisotropy constants in the yield criterion while f=g=1 for

plane stress case. Principle deviatoric stresses (S1, S2, S3) in equation 3-20 can be obtained by solving the characteristic equation of equation 3-22. The constants are determined using the yield stress and anisotropy from tensile test along the rolling direction (σ 0 , r0), transverse direction (σ 90 , r90), and 45 degree to rolling direction (σ 45 , r45) and yield stress from biaxial stress state (σ b ) [Barlat et al. 1991].

3.5 Determination of sheet material properties for process simulation

Flow stress along rolling direction and anisotropy coefficients are the sheet material properties required for process simulation. Tensile test, bulge test, shear test, compression test, bulge test, and bending test are the test methods reported in literature to obtain sheet material properties for process simulation.

3.5.1 Tensile test

Standardized procedure to estimate the flow stress and anisotropy of the sheet material from tensile test is given in ASTM standards (E517-92 A). In the tensile test, the tensile test coupon (Figure 3.12) cut from the sheet material along the three direction (0o, 45o, and 90o to the rolling direction) is subjected to displacement along length of the specimen. The load required and the extension within the gauge length is measured to estimate the flow stress. Anisotropy values along each direction were obtained from the ratio of the width strain to the thickness strain. Thickness and width strain are calculated from the average of the thickness measurement and width measurements at three locations (Figure 3.13) before and after testing the specimen to an elongation corresponding to value of 2-3% before necking.

Figure 3.12 Dimensions of the tensile test specimen as per ASTM E517-92 A

Danckert et al. 1998, Hoffmann et al. 1996 further enhanced the tensile test measurement and calculation techniques to estimate the anisotropy of the material in real time as a function of strain during the test. During the test, the elongation and the change in the width of the tensile specimen are measured in real time and the thickness strain is calculated based on plastic incompressibility. The measured width strain and the calculated thickness strain were compensated for elastic strains to get anisotropy in real time as function of axial strain.

Figure 3.13 Locations for measurement of thickness and width strain to calculate anisotropy of the sheet material as per ASTM E517-92 A

3.5.2 Shear test

Rauch 1998, Bouvier et al. 2006 conducted shear tests to estimate the anisotropy in sheet material. In the shear test, the sheet material is held at the edges and pulled in opposite direction as shown in the Figure 3.14. The load (F) and the displacement ( ΔL ) were measured to estimate the flow stress (τ) and strain (γ). Rauch 1998 suggested shear tests using the specimen cut from sheet material along two directions θ and 90-θ to the rolling direction of the sheet for estimation of the anisotropic properties in orthotropic materials such as sheet metals. However, it should be noted that the shear test is not feasible for thin sheet material currently being used in automotive industries because the compressive stress in the test induces wrinkling in the sheet material at higher strains. 28

Figure 3.14 Schematic of the shear test to estimate the anisotropic properties of the sheet materials

3.5.3 Tensile and Bending test

Malo et al. 1998 estimated the anisotropic constants in the Hill’s1948 yield criteria and Barlat 1991 yield criteria for the sheet material A7108-T76 using the yield stress of the material and not using the flow rule (plastic strain ratio). Malo et al. 1998 conducted 6 tests, 3 tensile tests using the specimens cut from the sheet along three directions 0o, 45o, and 90o to the rolling direction and the 3 bending tests using the specimens cut from the sheet along three directions 0o, 45o, and 90o to the rolling direction. In the plane stress yield criteria of Hill’s1948 and Barlat 1991, there are only 4 unknown constants. Thus, yield stress from four different tests are required to estimate unknown parameters. The estimated anisotropic coefficients using different combinations of the test varied significantly for both the yield criteria indicating that the constants in the yield surface vary depending on the test used to estimate the constants. This indicated that the yield surfaces Hill’s1948 and Barlat 1991 cannot accommodate these inherent variations in the material properties in different test methods. It should be noted that Malo et al 1998 used just the initial yield stress obtained from four different tests to fit the yield surface while, the anisotropy coefficients change with strain during the forming process.

Therefore, anisotropy coefficients estimated over the entire strain range observed in the test would better describe the anisotropy in the material and could eliminate significant deviations observed in the anisotropy coefficients depending on the test method being used.

3.5.4 Compression test

Boger et al. 2006 developed compression test to estimate the flow stress of sheet material over large strain range. The schematic of the compression test is shown in Figure 3.15. The compression test specimen is supported either side by a plate subjected to transverse force to avoid early buckling of the specimen due to compressive loading. The geometry of the compressive test specimen (dimensions G,B,W,and L) were obtained as function of sheet material and thickness using FE simulation such that achievable strain before initiation of buckling is maximized. The flow stress was obtained using analytical equations from the axial load, stroke and the transverse load applied on the plates to avoid buckling. In the developed compression test, maximum plastic strain of 0.1 could be obtained. The test fixture could be used to load the specimen in both tensile and compressive state and therefore was used to study the Bauschinger effects and cyclic loading effects in sheet materials.

Figure 3.15 Schematic of the compression test specimen and test setup for estimating sheet material properties [Boger et al. 2006]

3.5.5 Bulge test

In the bulge test, the sheet metal clamped at its edges is stretched against circular die using oil/viscous as a pressurizing medium. The sheet metal bulges into a hemispherical dome and eventually it bursts. Flow stress of the sheet material can be calculated from bulge test using analytical equations assuming sheet metal as thin membrane. Calculation of the flow stress from the bulge test requires a) fluid pressure, b) dome height, c) radius of curvature at the top of the dome, and d) thickness at the top of the dome to be measured real time in the experiment (Gerhard et al. 2000, Hoffmann et al. 2006, Bleck et al. 2005). Hoffmann et al. 2006, Bleck et al. 2005 used optical methods to measure the strain, radius of curvature, and dome height to calculate the flow stress. Gerhard et al. 2000, used FE simulation based inverse technique to estimate the flow stress from forming pressure and dome height without measuring the thickness and radius of curvature at the top of the dome. In this study, the bulge test was extended to measure flow stress and anisotropy of sheet materials in biaxial stress state.

3.6 Prediction of stamping defects using FE simulation

Excessive thinning leading to tearing and wrinkling are the common failure modes for the sheet metal in the stamping and hydroforming process. These defects need to be identified in the process simulation and corrected to avoid any potential failure of the part in production. Methods currently used to detect these defects in the FE results are described below.

3.6.1 Tearing

Potential failure of the part by tearing in FE simulation is detected by using either percentage thinning or Forming Limit Diagram (FLD). FLD [Figure 3.16] indicates the limiting in-plane strain (ε1-ε2) beyond which potential failure would occur for different combination of the strains. Therefore, the strain in the formed part calculated by FE simulation is compared with FLD diagram for the corresponding sheet material. Any point in the formed part that lies above the forming limit curve indicates potential failure. In case of thinning as failure criteria, the locations of failure are identified when the thinning in the part exceeds a critical value. FLD’s are generated from Nakazima tests, Hazek tests (Banabic et al. 2000) where the sheet material is subjected to linear strain 31 path. Thus, they are valid to predict failure in sheet material subjected to linear strain path. In the forming process, the material is subjected to non-linear strain paths hence the FLD cannot accurately predict the failure in sheet material. Keeler 2003 improved FLD based failure prediction by adding modification to the FLD to accommodate the change in the strain path when the sheet flows over the draw bead and the die corner radius. Arrieux, 1995 introduced Forming Limit Stress Diagram (FLSD) based on in- plane stress to detect failure. FLSD is path independent but difficult to obtain experimentally from sheet material. Therefore, thinning is a better indicator for failure of the sheet metal during the forming process and can be easily measured

ε1

F o rm in g urve Limit C

0% Thinning ng si ea cr in ng ni in ε2 Th

Figure 3.16 Schematic of the FLD curves and the constant thinning to illustrate the choice of indicator for failure in the part during stamping/sheet hydroforming process

3.6.2 Wrinkling

Wrinkling occurs in the sheet metal forming process due to excessive compressive stress resulting in instability of the deforming material. In the FE simulation of the forming process, the instability can be detected when tangent stiffness matrix [K] (equation 3-10) for the entire system is singular [Riks 1979, Kim et al. 2001]. However in case of explicit finite element method that is commonly used in stamping/sheet hydroforming simulation, the stiffness matrix [k] (equation 3-10) for the entire system is 32 not assembled therefore, this procedure cannot be used. Baodeu et al. 1996, Chu et al. 1996, Kim et al. 2000, Tuguc et al. 2001, Correia et al. 2002, used plastic bifurcation theory for shells proposed by Hutchinson and Neale, while Cao et al. 2000, Xie et al. 2002 used Timoshenko’s buckling theory in the explicit FEM to calculate the critical stress that would result in wrinkles. In both the methods, critical stress for wrinkling at each step in the FE simulation is calculated for every element based on its current state. The calculated critical stress is compared with the stress in the sheet material at the end of next step to check for potential wrinkle. The calculation of critical stress in both the methods involves assumptions on the boundary conditions and the possible displacement field that would result in wrinkling. Therefore both the methods have only limited success for axisymmetric part geometries (Cao et al. 2003, Gelin et al. 2002). Cao et al. 2003, Thomas 1999, Sheng et al. 2004, Jirathearanat 2004, Yang et al. 2002 successfully used geometry based wrinkle methods to detect the wrinkling in formed part from stamping, sheet and tube hydroforming process. In geometry-based methods, wrinkle in the final formed part is obtained by calculating the deviation of the formed part from the tool surface.

3.7 Process optimization in sheet metal forming

3.7.1 Blank holder force

Optimization technique and adaptive simulation technique have been used with FE simulation to estimate the process parameters in sheet metal forming process for single point cushion systems. In optimization technique, classical optimization technique coupled with FE simulation was used to predict the BHF that varies during the stroke for single point cushion systems to minimize the thinning/spring back in the formed part (Yang et al. 2002, Ghauti et al. 1999, Kim et al. 2002, Roy et al. 2002, Gantar 2002, ohata 1999).

In adaptive simulation technique, the forming process is considered as a control problem in which the process parameter that is input to the FE simulation is adjusted during the FE simulation to avoid potential defects in the formed part. Thus adaptive solution is relatively a quick method to obtain the solution compared to optimization. Cao et al. 2003, Cao et al. 1997, Thomas 1999, Strano et al. 2004, Sheng et al. 2004, Geiger et al.

2002 used adaptive simulation technique to predict the BHF for single point cushion system where the BHF is constant in space but variable in time. Adaptive simulation technique was also used to predict the process parameters in hydroforming process where the maximum inputs required to the forming process are only two (axial feeding on both sides of the tube) (Jirathearanat et al. 2004, Aue-u-lan et al. 2004, Gelin et al. 2002). However for multipoint cushion system, where the number of inputs to the control system is more than one, several assumptions need to be made on the relationship between the BHF applied at different locations and potential defect at different locations of the part and therefore it is part dependent [Thomas 1999]. Also, the parameters (gains) in the controller used to predict the BHF in adaptive simulation is part and material dependent and requires several simulation runs to fine tune the parameters (gains).

3.7.2 Blank shape

Several researchers have attempted to design the blank geometry using the FE analysis of the forming process. Their methods are briefly discussed here. Chung et al. 1992, Lee et al. 1998 used ideal deformation theory to predict the initial shape of the blank. In the ideal deformation theory, the final geometry is provided as input, the intermediate stages and the initial blank shape are calculated such that the elements undergo minimum plastic work. Similar techniques such as one step FE analysis to estimate the blank shape are available in commercial FE code PAMSTAMP 2000/2G, AUTOFORM. In this analysis, the effect of process parameters, interface friction conditions are neglected and therefore are approximate. Kim et al. 2000, Park et al. 1999, Shim 2002 introduced backward tracing scheme to design the initial blank geometry from FE simulation results with several assumptions on the modes of deformation in the flange, cup wall, die corner radius thereby restraining their method for simple geometries such as round cup and rectangular pan and in plane deformation. Shim 2002 and Pegeda et al. 2002 introduced optimization based method to estimate the initial blank shape through FE simulations. Optimization based approach required several FE simulations to estimate the optimum blank. Therefore in this study, a robust method was developed that used FE simulation results to determine the initial blank shape, boundary and shape of the holes if any in the initial blank such that after forming it conforms to the desired shape overcoming the limitations of the exiting methods. 34

CHAPTER 4

ESTIMATION OF SHEET MATERIAL PROPERTIES USING BIAXIAL BULGE

TESTS

4.1 Introduction

Properties of the sheet materials at large strains that occur in stamping is essential information for successful use of FE simulations for process design in stamping industry. Conventionally, properties of the sheet metal obtained from uniaxial tensile test are used in FE simulation. This data is insufficient for stamping FE simulations because a) maximum strain obtained in uniaxial tensile test before necking is small compared to strains encountered in stamping operations b) the stress state in tensile test is uniaxial while in regular stamping it is biaxial. Therefore, it is necessary to obtain the material properties from the biaxial test, which provides data for large strain range relevant to stamping operations to accurately model the stamping process through FE simulations.

In analysis of sheet metals, the material behavior is assumed to be orthotropic implying the mechanical properties vary only along the three direction namely 1) rolling direction b) transverse to the rolling direction and c) thickness direction. Also in the analysis, the sheet material is assumed to be plane stress along thickness direction. Therefore, in the FE simulations for sheet metal forming, the material behavior of sheet metal during plastic deformation is represented by plane stress orthotropic yield criteria and the flow stress (equivalent stress – strain relationship) along the rolling direction.

Several plane stress orthotropic yield criteria [Hill 1948, Hill 1979, Hosford 1979, Barlat 1989, Hill 1990, Barlat 1991 (YLD91), Hill 1993, Karafillis 1993, Barlat 1994 (YLD94), Barlat 1996 (YLD96), Barlat 2000 (YLD2000), Cazacu Barlat 2003, Banabic Balan 2003 (BBC 2003)] have been proposed till date to phenomologically represent the anisotropy exhibited by F.C.C and B.C.C crystal structure sheet metal alloys. Among the yield criteria Hill’s 1948 is most widely used for steel and its alloys that has normal anisotropy 35

r σ value ( r ) greater than one and ratio of anisotropy ( 0 ) and ratio of yield stress ( 0 ) r90 σ 90 are either greater or less than one. Hill’s 1990 yield criterion is used for steel and its alloys that violate the limitations of Hill 1948 yield criteria. In case of aluminum alloys where the yield stresses are same in rolling and transverse direction while anisotropy in rolling and transverse are different, Hill’s 1990 cannot be used [Banabic et al. 2000]. Therefore, Barlat’s 1991 yield criterion (YLD91), Barlat’s 1996 yield criterion (YLD96), and Barlat’s 2000 yield criterion (YLD2000-2d) are being widely used for modeling aluminum alloys. Among the yield criteria, Hill’s 1948, was selected because it is commonly used in metal forming simulations for steel and aluminum alloys in commercial FE codes such as PAMSTAMP 2000, PAMSTAMP 2G, LS- DYNA,AUTOFORM and DEFORM 3D. The results can be extended to other yield criteria’s.

Initially, circular bulge test using viscous medium (VPB test) was developed to estimate the flow stress of the material in biaxial stress state through inverse analysis. In the inverse analysis, the material is assumed to be isotropic and follow von Mises yield criterion. In order consider anisotropy in the sheet material, in this study, new biaxial test was developed along with the existing circular bulge test to estimate the flow stress and the anisotropy constants for the Hill’s 1948 yield criteria in biaxial stress state. The developed test was used to estimate the flow stress of sheet materials DP600, DR210, DDS steel, AKDQ steel and aluminum alloy A5754-O. Also, material properties of the sheet materials were obtained from tensile test. The flow stress and anisotropy obtained from the bulge test were compared with tensile data and used in FE simulation of round cup deep drawing process to make a choice between the test methods to determine sheet material properties for process simulation.

4.2 Background of proposed biaxial tests

4.2.1 Hill’s 1948 orthotropic yield criterion [Hill 1956]

Let X, Y, Z be the orthotropic axis of the sheet metal. X along rolling direction of the sheet metal, Y along the transverse direction of the sheet metal and Z along the thickness direction of the sheet metal 36

Plastic yielding for orthotropic material under plastic stress condition along thickness as in sheet metal is given by

rr σσ )( 2 r σ 2 r σ 2 )12( ( +++++− rrr )τ 2 900 XY 90 X 0 Y 45 900 XY = σ 2 Equation 4-1 ()rr 090 +1

where

σ ,σ yx are the normal stress in X,Y orthotropic axis

τ xy are the shear stress r0,r90,and r45 are the anisotropy obtained from tensile test along rolling direction, transverse to rolling direction and 45o to the rolling direction

The equivalent stress-strain relationship ( − εσ ) developed from equation 3-1 such that the incremental work done is given by

dw σd ∂== εσε ijij Equation 4-2 Accordingly, Hill effective stress is

2 2 2 2 2 3 ⎡ H σσ XY G σ X F Y +−+−+− 2)()()( Nτσ XY ⎤ σ = ⎢ ⎥ Equation 4-3 2 ⎣ ++ HGF ⎦

Hill’s effective plastic strain increment is given as

2 2 2 ⎡ ⎛ p p ⎞ p p p p p 2 ⎤ 2 2 G ε yy H∂−∂ ε zz ⎛ G ε H∂−∂ ε ⎞ ⎛G ε H∂−∂ ε ⎞ (∂γ ) ε ()++=∂ ⎢FHGF ⎜ ⎟ + G⎜ ZZ xx ⎟ + H⎜ XX YY ⎟ + 12 ⎥ p 3 ⎢ ⎜ ++ HFGHFG ⎟ ⎜ ++ HFGHFG ⎟ ⎜ ++ HFGHFG ⎟ 2N ⎥ ⎣ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ Equation 4-4

Where

H 1 = H H 2N ⎛ 11 ⎞ = r = r ()r45 12 ⎜ ++ ⎟ 0 90 ⎜ rr ⎟ G F ⎝ 900 ⎠ Equation 4-5

4.2.2 Effect of the anisotropic coefficients on the yield surface

Figure 4.1 , Figure 4.2 show the effect of the anisotropic constant along rolling direction

(r0) and transverse direction (r90), respectively, on the elliptical shape of Hill’s 1948 yield

surface (Equation 4-1) . Increase in anisotropic constant (r0) decreases the yield stress 37 along the transverse direction (Y) while the yield stress along the rolling direction (X) remains the constant. Also, along the equibiaxial stress state/path the yield surface does not change with change in the anisotropic constant (r0) indicating that circular bulge test alone cannot be used to estimate the anisotropy of the sheet material. Along other biaxial stress paths, the anisotropic constant (r0) results in change in the yield surface.

Increase in anisotropic constant (r90) increases the yield stress along the transverse direction (Y) while the yield stress along the rolling direction (X) remains the constant at the input value of 100 MPa. The yield stress in all the biaxial stress states/paths changes

with the change in the anisotropic constant (r90). This indicates that test method that induces biaxial stresses that are not equal need to be considered to estimate the anisotropic coefficients of sheet material from biaxial tests. Also, stress in the test should not be compressive to avoid buckling/wrinkling of the sheet metal.

σ y r0=1.2, r90=1.0

r0=2.1, r90=1.0

σ x

Stress path in circular bulge test r0=1.6, r90=1.0

r0=1.0, r90=1.0

Figure 4.1 Schematic illustrating the influence of anisotropic constant (r0) on the Hill’s 1948 yield criteria.

r0=1.0, r90=2.1 σ y

r0=1.0, r90=1.6

σ x r0=1.0, r90=1.0 Stress path in circular bulge test

r0=1.0, r90=1.2

Figure 4.2 Schematic illustrating the influence of anisotropic constant (r90) on the Hill’s 1948 yield criteria.

Y Y

d b X X a

d = Diameter of the circular die cavity a = Length of major axis of ellipse

Shape of the die cavity in circular bulge b = Length of minor axis of ellipse

test (Existing VPB test) Elliptical test 1

Figure 4.3 Schematic of the die cavity in the circular and elliptical bulge test

4.2.3 Elliptical bulge test

Bulging of the sheet using elliptical dies (Figure 4.3) introduces stress and strains that are tensile, biaxial but not equal. The ratio of the stress and strains in the elliptical bulge test is a function of the ratio of the major axis to minor axis of the ellipse. Therefore, the 39

elliptical bulge test was chosen in this study to estimate the anisotropy of sheet materials in the biaxial stress state. The circular bulge test is indeed a special case of elliptical bulge test with the ratio of the major axis to minor axis of the ellipse equal to one.

Die cavity Sheet material

Rolling Rolling Rolling Direction Direction Direction

Elliptical bulge test 1 Elliptical bulge test 2 Elliptical bulge test 3

Figure 4.4 Schematic illustrating positioning of sheet metal with respect to die cavity in the proposed elliptical bulge test (major axis = 107 mm/ 4 inch, minor axis = 53.5 mm/2 inch)

In the elliptical bulge test, different stress path for a given die geometry can be obtained by changing the relative position of the sheet material. Initially two positions of the sheet with respect to the die were considered namely a) Elliptical test 1 rolling direction of the sheet coincides with major axis of ellipse (Figure 4.4) corresponding to stress path (Figure 4.5), b) Elliptical test 2, transverse direction of the sheet coincides with major axis of ellipse (Figure 4.4) corresponding to stress path (Figure 4.5). In both the tests, the principle stress in the sheet coincides with anisotropy axis therefore shear stress

( σ xy = 0) and the equation 4-1 reduces to

rr )( 2 2 ++− rr σσσσ 2 900 XY 90 X 0 Y = σ 2 Equation 4-6 ()rr 090 +1

In the equation 4-6 anisotropy coefficient r45 is not present and cannot be determined.

The effect of the shear stress (σ xy ) can be studied by placing the sheet such that the rolling direction of the sheet is at an angle to the major and minor axis of the die. The

magnitude of the shear stress (σ xy ) in the sheet depends on the angle and is maximum

when the angle is equal to 45o (can be derived from stress transformations). Therefore, elliptical test 3 (Figure 4.4) was proposed with the rolling direction of the sheet 45 o to the major and minor axis of the ellipse.

r0=1.4, r90=2.1 σ y r =1.0, r =1.0 0 90 r0=1.0, r90=1.6

r0=1.6, r90=1.0

σ x Stress path in elliptical bulge test 2

Equibiaxial stress path r0=1.2, r90=1.0 Stress path in elliptical bulge test 1

Figure 4.5 Schematic illustrating the stress path in the elliptical bulge test 1 and elliptical bulge test 2

In the elliptical bulge test, depending on the location of the sheet, the stress path changes as shown in Figure 4.5. It could be observed that from Figure 4.5, the yield surface is not symmetric about the equibiaxial stress path. Hence, the stress

components (σ ,σ yx ) that cause yielding of the material are different in the elliptical bulge test 1 and 2. Therefore, the pressure required to bulge the specimen to a particular dome height, which depends on the yield stress of the material, would be different for bulge test 1 and test 2 depending on the anisotropic value. However this hypothesis of variation in forming pressure in the elliptical test 1 and elliptical test 2 depending on the anisotropic constants need to be further validated. Therefore FE simulation of the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 41 were conducted to study the influence of the anisotropic value on the forming pressure, dome height, thinning in the apex of the dome and thinning distribution along the major axis and minor axis of the ellipse.

4.3 Preliminary FE simulation to validate proposed biaxial tests

4.3.1 FE Model

Preliminary FE simulation to study the effect of the anisotropy constants on the forming pressure were conducted for the elliptical die geometry of major axis to minor axis ratio of 2.0. The major axis was assumed equal to 107 mm (4 inches) equal to the diameter of the bulge test tooling used to determine the flow stress. The material properties for the AKDQ sheet of thickness 0.83 mm were used in the simulation (Table 4.1). A quarter of the die geometry was considered for the FE simulation of the elliptical test 1 and elliptical test 2 as the material properties, die geometry and the boundary conditions are symmetric along two planes (X-Z and Y-Z) (Figure 4.6). In elliptical test 3, the die geometry is symmetric about the two planes (X-Z and Y-Z) while the sheet material properties are not symmetric about the two planes (X-Z and Y-Z) as the sheet is oriented such that rolling direction is 45o to the major axis. Therefore, the entire die geometry with sheet was modeled for the FE simulation of the elliptical bulge test 3 (Figure 4.7). Table 4.2 shows the simulation matrix considered for this study. Four cases were considered for this study. The values of each anisotropy coefficient were incremented by 0.2 for each case. In each case, FE simulations of elliptical tests 1, 2 and 3 were conducted

Material properties for AKDQ steel Thickness [mm] 0.83 K-value [Mpa] 495 n-value 0.183

ε0 0.005

Table 4.1 Material properties of AKDQ steel used in the FE simulations

Anisotropy coefficients Simulations R0 R45 R90 Case A 1.0 1 1 Case B 1.2 1 1 Case C 1 1 1.2 Case D 1 1.2 1

Table 4.2 Simulation matrix to study the influence of anisotropy constants on the forming pressure, bulge height and the thickness at the top of the dome.

Die

Z Y

Sheet X Figure 4.6 FE model for the forming simulation of the elliptical bulge test 1 and elliptical bulge test 2

Die

Z Y Sheet

Figure 4.7 FE model for the forming simulation of the elliptical bulge test 3

4.3.2 FE Results

4.3.2.1 Forming pressure

In case A, as expected, the forming pressure necessary to reach a dome height was the same for all the tests, Figure 4.8. In case B, a higher forming pressure was predicted in the elliptical test 2 when rolling direction of the sheet was kept parallel to the minor axis of the die (Figure 4.9). In case C, higher forming pressure was observed in the elliptical test 1 and elliptical test 3 compared to elliptical test 2 when rolling direction of the sheet was kept parallel to the major axis of the die (Figure 4.10). In case D, the forming

pressure predicted by FE simulation was the same for both test 1 and test 2 as r45 does not influence the yielding behavior of the material when the principal stress coincides with major and minor axis. The difference in the forming pressure between the test 3 and test 1/ test 2 was not significant (Figure 4.11). The results obtained from these preliminary FE simulations of the various circular and elliptical bulge tests, with different anisotropy coefficients, validated the hypothesis of elliptical test and indicated that it is possible to inversely calculate the anisotropy coefficients from the proposed tests.

12 Elliptical Test 1 Elliptical Test 2 10 Elliptical Test 3

Pressure [MPa] 4

0 0 4 8 12162024 Dome Height [mm]

Figure 4.8 Forming pressure versus dome height in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 obtained from FE simulation for isotropic case (Case A).

12 Elliptical test 2

8 Elliptical test 1 6 Pressure [MPa] Pressure 4 Elliptical test 3

0 0 4 8 12162024 Dome Height [mm]

r0=1.0, r90=1.0 σ y

r0=1.2, r90=1.0

Stress path in σ elliptical bulge test 2 x

Stress path in elliptical bulge test 3 Stress path in elliptical bulge test 1

14 Elliptical test 3 12

8 Elliptical test 1

6 Pressure [MPa] 4 Elliptical test 2

0 04812162024 Dome Height [mm]

σy r0=1.0, r90=1.0

r0=1.0, r90=1.2

σ x Stress path in elliptical bulge test 2

Stress path in elliptical bulge test 3

Stress path in elliptical bulge test 1

Figure 4.10 Effect of anisotropy constant along the transverse direction (r90) on the forming pressure in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3

14 Elliptical test 3 12 10 8 6 Elliptical test 1and 2 4 Pressure [MPa] 2 0 04812162024 Dome Height [mm]

σ y

r0=1.0, r0=1.0, r45=1.0

Stress path in σ elliptical bulge test 2 x

Stress path in r0=1.0, r90=1.0, r45=1.2 elliptical bulge test 3 Stress path in elliptical bulge test 1

Figure 4.11 Effect of anisotropy constant along the 45o to the rolling direction (r45) on the forming pressure in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3

4.3.2.2 Thinning at the top of the dome

Figure 4.12 show the maximum thinning at the apex of the dome at different dome height for the case A. In isotropic condition, the deformation during the elliptical bulge test is independent of the orientation of the sheet with respect to the die therefore the same maximum thinning was obtained in all the three elliptical tests for a given dome 47 height for isotropic material. In case B, C and D, thinning at the top of the dome at a given dome height did not change in elliptical test 1, 2, and 3. In elliptical tests, the sheet material is stretched without material flow into the cavity. Therefore, the sheet deforms the same way in all the three elliptical test. Hence, the thinning at the top of the dome for a given dome height is same for all the three elliptical test in case B, C, and D.

40%

35% Elliptical Test 1 Elliptical Test 2 30% Elliptical Test 3 25%

20%

the dome the 15%

10%

5% Percentage thinning at the top of of top the at thinning Percentage 0% 0 4 8 12162024 Dome Height [mm]

Figure 4.12 Maximum thinning at the top of the dome in the elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 obtained from FE simulation for isotropic case.

4.4 Tooling design and manufacturing for the proposed bulge test

Preliminary FE simulations of the proposed elliptical bulge tests indicated that the pressure required to bulge the specimen (pressure versus dome height curve) in each elliptical test was different for anisotropic materials and is sensitive to anisotropy coefficients of the material. Maximum difference in the pressure of 4 bar (0.4 MPa) was observed between the proposed elliptical tests for the change in anisotropy by 0.2 with elliptical die of major axis to minor axis ratio 2.0 in the preliminary FE simulation. The difference in the forming pressure observed between the proposed elliptical bulge tests depends on the stress path and the anisotropy coefficients of the sheet material. The stress path in turn depends on the ratio of the major axis to minor axis in the elliptical die geometry and the orientation of the sheet with respect to the die. Therefore FE 48 simulations were conducted to obtain optimum die geometry that gives maximum difference in the forming pressure between the proposed elliptical bulge tests for anisotropic materials and obtains higher dome height before failure.

4.4.1 FE simulation

The elliptical die geometry for FE simulations was modeled to have length of the major axis equal to the diameter of the exiting circular die so that the designed die can be accommodated in the existing VPB tooling at ERC/NSM. The die corner radius was assumed equal to the die corner radius in the circular die (6.35 mm) currently being used at ERC/NSM. The minor axis of the elliptical die was varied to get major axis to minor axis ratio of 1.5, 2.0 2.25 and 2.5. The anisotropy coefficients of AKDQ steel were used in the study. The flow stress of the material is provided in the Table 4.1. The FE model discussed in the section 4.3.1 was used in this study but with different die geometry for each case.

4.4.2 FE Results

Figure 4.13 shows the maximum difference in the forming pressure between the elliptical bulge tests for the different die ratio. Figure 4.14 shows the maximum achievable dome height before failure for different die ratio. Difference in the forming pressure between the proposed elliptical tests increases with increase in the die ratio indicating that the die geometry with higher die ratio is good for the elliptical test to determine anisotropy. However, increase in the die ratio decreases the maximum achievable dome height. Also, it is apparent from the forming limit diagram that the maximum plastic strain decreases with increase in the die ratio. Research conducted by Rees et al. 1999 indicated that the maximum achievable strain decreases rapidly beyond the die ratio of 2.0. Therefore, the die ratio of 2.25 that could give a maximum difference in the forming pressure of at least 5 bar and maximum possible dome height of ~ 20 mm for AKDQ material was selected as optimum die geometry in this study.

among the elliptical tests [bar] 1 Maximum forming pressure difference pressure difference forming Maximum 0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 Elliptical die ratio

Figure 4.13 Comparison of the maximum difference in the forming pressure among the three elliptical test predicted by FE simulation for different elliptical die geometries using AKDQ steel sheet material

36 32 28 24 20

[mm] 16 12 8 4 Maximum possible dome height dome height possible Maximum 0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 Elliptical die ratio

Figure 4.14 Maximum achievable dome height in the elliptical test predicted by FE simulation for different elliptical die geometries using AKDQ steel sheet material 50

4.4.3 Manufacture of the elliptical die

Elliptical die was designed with major axis to minor axis ratio of 2.25 as obtained from FE simulations. In the FE simulations, higher strain of 0.20 was observed along the minor axis near the die corner radius (6.35 mm) compared to 0.50 at the apex of the dome. Higher strain in the die corner could result in failure of low formability materials at the die corner radius. Therefore, the die corner radius of 12 mm was chosen for the elliptical die. Preliminary FE simulation conducted for die geometry with ratio 2.25 and the die corner radius of 12 mm indicated a maximum strain was less than 0.10 at the die corner for sheet of thickness 1 mm thereby reducing the risk of failure in the die corner. The critical dimensions for the finalized elliptical die are given in Table 4.3, Figure 4.15. The elliptical die was designed as an insert to add to the exiting tooling without modification. The designed elliptical die has the provision for the lock bead to lock the material flow during the test.

Elliptical Die Geometry Major axis [mm] 104.7 Minor Axis [mm] 46.53 Die corner radius [mm] 12

Table 4.3 Critical dimensions of the elliptical die

The elliptical die insert was manufactured using the A2 tool steel and hardened to 56 HRC. The engineering drawing for the designed elliptical die is shown in Figure 4.15. The tool material was purchased from Crucible Specialty Metals, P.O.Box 977, 575 State Fair Blvd, Syracuse, NY 13201 www.crumetals.com. The tool steel was machined at manufacturing laboratory (Mary’s Lab) in ISE department at The Ohio State University. The machined part was heat treated at Metal Improvements Company, 1515 Universal Road, Columbus, Ohio. The finished die after heat treatment and polishing is shown in Figure 4.16.

Figure 4.15 Engineering drawing of the designed elliptical die

Figure 4.16 Elliptical die manufactured from A2 tool steel after heat treatment and polishing 52

4.5 Analysis of the bulge test using membrane theory

Bulging of the sheet in the circular die and the elliptical die was analyzed using the closed form solutions available in the literature based on membrane theory [Hill 1956, Rees 1995]. Considering a infinitesimal element at the apex of the dome (Figure 4.17), the strain along three different directions (φ, θ, t ) is obtained from thickness strain and volume constancy in plasticity as given in equation 4-7.

δε θ δε φ Minor axis Major axis

Figure 4.17 Schematic of the infinitesimal element at the apex of the dome

ε length of major axis of elliptical die φ = = β ε length of minor axis of the elliptical die θ t − ε − βε ε ln t t t = , ε θ = ,ε φ = Equation 4-7 t0 1 + β 1 + β t = is the instantaneous thickness at the apex of the dome, t0 = is the initial sheet thickness at the apex of the dome, β = strain ratio

The stress along major and minor axis of the ellipse can be obtained from the force equilibrium normal to the infinitesimal element (Figure 4.17) and assuming the deformed shape of the sheet along the major axis and minor axis fits to radius of sphere, Figure 4.18.

Figure 4.18 Schematic descrbing the shape of the bulged specimen in the elliptical die assumed for analytical calculations

The stress along the major and minor axis is given by equation 4-8.

−1 −1 P ⎛ 1 α ⎞ αP ⎛ 1 α ⎞ σ φ σ ⎜ ⎟ ,σ ⎜ ⎟ = α θ ⎜ += ⎟ φ = ⎜ + ⎟ , Equation 4-8 ⎝ Rt Rθθ ⎠ ⎝ Rt Rθθ ⎠ σ θ

Where Rθ is the radius of curvature along the major axis, Rφ is the radius of curvature along the minor axis, p is the forming pressure.

The relationship between the stress ratio (α) and strain ratio (β) can be derived from the associative flow rule. The equivalent stress and strain for the Hill’s 1948 yield criteria can be calculated using the membrane stress (equation 4-8) and membrane strain (equation 4-7) . The equivalent stress and strain equation for elliptical test 1 is as follows:

σ = ()/ ac ()2 ()σσ 2 rrrr (−++ σσ )2 c = 90 θ 0 φ 090 φθ rr 090 + )1( 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ ⎜ ⎟ Equation 4-9 3 ⎝ rr 900 ⎠

r0 = Anisotropy rollingin direction

r90 = Anisotropy in transev erse direction

ε p = ()/ bha 2 2 2 ⎛ 11 − βε ⎞ ⎛ 11 − ε ⎞ ⎛ 1 − ε 1 − βε ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜ t t ⎟ h = ⎜ − ε t ⎟ + ⎜ − ε t ⎟ + ⎜ − ⎟ ⎝ rr 090 1+ β ⎠ ⎝ rr 900 1+ β ⎠ ⎝ 90 1+ β rr 0 1+ β ⎠ 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠ 2 ⎛ 111 ⎞ b ⎜ ++= ⎟ ⎝ rrrr 900090 ⎠ t ε t = ln Equation 4-10 t0 t = Current sheet thickness

t0 = Initial sheet thic kness

r0 = Anisotropy rollingin direction

r90 = Anisotropy in transev erse direction

Detailed derivations of these formulas are provided in Appendix A, B, C for elliptical test 1, elliptical test 2, and elliptical test 3, respectively. Circular test is a special case of elliptical test with strain ratio (β) equal to 1.0.

The analytical equations can be used to calculate the flow stress from elliptical / circular bulge test is as follows Experimental measurements: • Forming Pressure, P, and thickness at the top of the dome t at different dome

heights, radius of curvature at the top of the dome along major Rθ and minor Rφ axis • Die dimensions (length of major axis and length of minor axis and the anisotropy values of the sheet material

• Step 1: Calculate the strain ratio β from die geometry

• Step 2: Calculate the strain components from the thickness at the top of the dome and strain ratio. • Step3:Calculate effective strain from strain components using equation 4-10 • Step 4: Calculate the stress ratioα from the strain ratio β

• Step 5: Calculate the individual stress components using the pressure, radius of curvature at the top of the dome from experiments by equation 4-8. • Step 6: Calculate the effective stress by equation 4-9. • Step 7: Repeat step 1 to step 5 for different dome height to obtain the equivalent stress versus strain curve.

The calculations (equation 4-9 and equation 4-10) require the anisotropy values of the materials to be known. However in our case, anisotropy of the material is unknown therefore we would use three tests to simultaneously determine the flow stress and the anisotropy coefficients. Also, the calculations require the thickness of the sheet at the top of the dome and radius of curvature be known at different dome heights. The sheet thickness at the top of the dome and radius of curvature cannot be measured in experiment in real time and can be only obtained by manually measuring the test specimen bulged to different heights. This requires several experiments to be performed at different dome heights. However, this information can be obtained from FE simulation [Gutcher et al. 2004]. Therefore, FE simulations were conducted to study the relationship between the thickness at the top of the dome, radius of curvature at the top of the dome at different dome height to the unknown material properties namely anisotropy values and flow stress (strength coefficient and strain hardening exponent) in the proposed elliptical bulge tests.

4.6 Relationship between bulged sheet geometry and properties of the sheet material

Gutcher et al. 2004 investigated the relationship between thickness and the radius of curvature at the top of the dome to the dome height and the flow stress (strength coefficient and strain hardening exponent) for circular bulge test (which is a special case of elliptical bulge test). The radius of curvature and the thickness at different dome heights in circular bulge test were found to be dependent on the strain hardening exponent and independent of the strength coefficient. In this study, the relationship between the thickness and the radius of curvature at the top of the dome at different dome height in elliptical bulge test, to anisotropy values and flow stress (strain hardening exponent) was studied for elliptical bulge test 1, elliptical test 2 and elliptical bulge test 3. In the FE simulation, the flow stress of the sheet material was represented by 56

n k 0 += εεσ p )( (k= strength coefficient, n = strain hardening index and ε 0 is the

prestrain) and anisotropy values in the Hill’s 1948 were given by r0, r45 and r 90.

4.6.1 Thinning at the top of the elliptical dome.

4.6.1.1 Effect of n value

Figure 4.19 shows the thinning at the top of the dome for different dome heights obtained from FE simulations for different strain hardening coefficient (n) in elliptical test 1. Increase in strain hardening coefficient (n), decreases the thinning at the top of the dome for the same dome height as higher n value allows more uniform strain hardening in the material than the lower n values thereby postponing the excessive thinning at the top of the dome. Hence, n value significantly influences the thinning of the sheet at the top of the dome in the elliptical bulge test. Similar FE results were also observed for elliptical test 2 and 3. Also, it was observed that the thinning at the top of the dome

40% n=0.05 35% n=0.05 n=0.2 n=0.1 30% n=0.1 n=0.2 25% n=0.3 n=0.3 20% n=0.4 dome n=0.5 15% n=0.4

10%

5% n=0.5 Percentage thinning at the top of the the of top the at thinning Percentage

0% 0 2 4 6 8 10121416182022242628 Dome Height [mm]

Figure 4.19 Thinning at the top of the dome obtained from elliptical test 1 FE simulations for different n values

4.6.1.2 Effect of anisotropy values (r0,r45, and r90)

Figure 4.20 shows the thinning at the top of the dome for different dome heights obtained from elliptical test 1 FE simulations with different anisotropy values. Influence of anisotropy values on the thinning at the top of the dome in elliptical test 1 was found negligible. In the elliptical test, the material is purely stretched in and the thinning at the top depends on the geometry and the strain hardening of the material. Similar FE results were also obtained for elliptical test 2 and elliptical test 3. Also it was observed that the thinning at the top of the dome do not change between the elliptical tests for same dome height, Figure 4.21, as the die geometry is same in elliptical test 1, 2 and 3 for a given material.

40%

35% R0=1.2 R90=1.0 R0=1.0 R90=1.2 30% R0=1.475 R90=1.878 25% R0=1.5 R90=2.1 20% R0=1.9 R90=1.3 dome 15%

10%

5% Percentage thinning at the top of 0% 0 2 4 6 8 10121416182022 Dome Height [mm]

Figure 4.20 Thinning at the top of the dome obtained from elliptical test 1 FE simulations for different anisotropy values

40%

35% Elliptical Test 1 Elliptical Test 2 30% Elliptical Test 3 25%

20%

the dome the 15%

10%

5% Percentage thinning at the top of of top the at thinning Percentage 0% 0 2 4 6 8 10121416182022 Dome Height [mm]

Figure 4.21 Comparison of thinning at the top of the dome obtained from FE simulation of the elliptical test 1, elliptical test 2 and elliptical test 3 for same anisotropy values (r0=1.878,r90=1.465,r45=1.308) and flow stress

4.6.1.3 Effect of initial sheet thickness

FE simulation of the elliptical test 1 was conducted with different initial sheet thickness to study its influence on the thinning at the top of the dome. Figure 4.22 shows the thinning at the top of the dome for different dome heights obtained from FE simulations for sheet material of different thickness in elliptical test 1. It could be observed that the difference in the thinning at the top of the dome for different initial sheet thickness at given dome height is negligible indicating that thinning at the top of the dome is independent of the initial sheet thickness up to sheet thickness of 1.8 mm used in this FE simulation.

40%

35% t=0.4mm 30% t=0.83mm t=1.5mm 25% t=1.8 mm 20%

the dome 15%

10%

5% Percentage thinning at the top of of top the at thinning Percentage 0% 0 2 4 6 8 10121416182022 Dome Height [mm]

Figure 4.22 Comparison of thinning at the top of the dome obtained from FE simulation of the elliptical test 1 for different initial sheet thickness

FE simulations indicated that thinning at the top of the dome in elliptical tests depends only on the strain hardening of the material and is independent of anisotropy coefficients

(r0,r45, and r90) , elliptical tests 1, 2 and 3, and the initial sheet thickness.

4.6.2 Radius of curvature at the top of the elliptical dome.

4.6.2.1 Effect of n value

Figure 4.23, and Figure 4.24 shows the radius of curvature at the top of the dome along the major and minor axis for different dome heights obtained from FE simulations of elliptical test 1 for different strain hardening coefficient (n). Increase in strain hardening of the material results in more uniform strain distribution. Hence, the radius of curvature at the top of the dome along the major axis of the ellipse increases with increase in n value. However, the influence of strain hardening was negligible on the minor axis as the deformed geometry of the sheet is more dictated by the geometry because the length of the minor axis is small.

500 450 400 350 300 250 n = 0.05 n = 0.5 n=0.4 [mm] 200 150 100 n = 0.1 50 elliptical dome along the major axis axis major the along dome elliptical Radius of curvature at the top of the the of top the at curvature of Radius n=0.2 n=0.3 0 0 4 8 12 16 20 24 Dome Height [mm]

Figure 4.23 Radius of curvature at the top of the dome along the major axis obtained from elliptical test 1 FE simulations for different n values

200 180 n=0.05 n=0.1

160 n=0.183 n=0.2

140 n=0.3 n=0.4 120 n=0.5

ong the minor axis axis minor ong the 100 [mm] 80 60 40 20 elliptical dome al Radius of curvature at the top of the the of top the at curvature of Radius 0 0 2 4 6 8 1012141618202224 Dome Height [mm]

Figure 4.24 Radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1 FE simulations for different n values 61

4.6.2.2 Effect of anisotropy values (r0, r45, r90)

Figure 4.25, and Figure 4.26 shows the radius of curvature at the top of the dome along the major axis and minor axis, respectively at different dome heights obtained from elliptical test 1 FE simulations with different anisotropy values. The anisotropy values have negligible influence on the radius of curvature on the top of the dome along the major and minor axis in elliptical test 1 as their effect on the geometry of the formed part is negligible when the material is stretched without draw-in as in elliptical bulge tests. Similar results were observed for elliptical test 2 and 3. Also, radius of curvature at the top of the dome along the major axis was identical in elliptical test 1, 2 and 3 for a given material properties , Figure 4.27 as die geometry is same in all the tests and anisotropy valves have negligible influence on the radius of curvature at the top of the dome.

800 R0=1.2 R90=1.0 700 R0=1.0 R90=1.2 600 R0=1.475 R90=1.878

500 R0=1.5 R90=2.1

400 R0=1.9 R90=1.3

300

200

100 Radius of curvature at the topof the the topof the at curvature of Radius elliptical dome along major axis [mm] axis major along dome elliptical 0 0 2 4 6 8 10 12 14 16 18 20 22 Dome Height [mm]

Figure 4.25 Radius of curvature at the top of the dome along the major axis obtained from FE simulation of the elliptical test 1 for different anisotropy values

500 450 400 R0=1.2 R90=1.0 350 R0=1.0 R90=1.2 300 R0=1.475 R90=1.878

250 R0=1.5 R90=2.1 200 R0=1.9 R90=1.3 150 100 50 dome along the minor axis [mm] axis minor the along dome

Radius of curvature at the top of the the of top the at curvature of Radius 0 0246810121416182022 Dome Height [mm]

Figure 4.26 Radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1 FE simulations for different anisotropy values

200 180 160 Elliptical Test 1 140 Elliptical Test 2 120 Elliptical Test 3 100 [mm] 80 60 40 20 elliptical dome along the minor axis axis minor the along dome elliptical Radius of curvature at the top of the the of top the at curvature of Radius 0 0 2 4 6 8 10121416182022 Dome Height [mm]

Figure 4.27 Comparison of radius of curvature at the top of the dome along the minor axis obtained from elliptical test 1, elliptical test 2, and elliptical test 3 FE simulations for same anisotropy values and flow stress 63

4.6.2.3 Effect of initial sheet thickness

Figure 4.29 and Figure 4.30 show the radius of curvature along the major axis and minor axis, respectively, at the top of the dome for different dome heights obtained from FE simulations for sheet material of different initial sheet thickness in elliptical test 1. Radius of curvature at the top of the dome along the major axis and the minor axis is same for different initial sheet thickness at given dome height after dome height of 5 mm. The deviation in the radius curvature at smaller dome heights could be due the bending effect that increases with increase in sheet thickness. Therefore in the calculations, for dome height beyond 5 mm, that radius of curvature at the top of the dome is independent of the initial sheet thickness up to sheet thickness of 1.8 mm used in this study.

800 t =0.83 mm

700 t=0.4mm 600 t=0.83mm 500 t=1.8mm 400 t = 0.4 mm [mm] 300 t =1.8 mm

200

100 elliptical dome along the major axis Radius of curvature at the top of 0 0246810121416182022 Dome Height [mm]

Figure 4.28 Comparison of radius of curvature at the top of the dome along the major axis obtained from FE simulation of the elliptical test 1 for different initial sheet thickness

200 t = 0.4 mm 180 t=0.4mm 160 t =0.83 mm t=0.83mm 140

120 t=1.8mm

100

60 t =1.8 mm

Radius of curvature at the top of the the of the top at curvature of Radius 20 elliptical dome along the minor axis[mm] minor the along dome elliptical 0 0 2 4 6 8 10 12 14 16 18 20 22 Dome Height [mm]

Figure 4.29 Comparison of radius of curvature at the top of the dome along the minor axis obtained from FE simulation of the elliptical test 1 for different initial sheet thickness.

FE simulations indicated that radius of curvature at the top of the dome in elliptical tests along the major axis and minor axis depends only on the strain hardening of the material and is independent of anisotropy coefficients (r0,r45, and r90) , elliptical tests 1, 2 and 3, and the initial sheet thickness.

The thickness at the top of the dome, radius of curvature at the top of the dome along the major axis, and radius of curvature at the top of the dome along the minor axis is function of only strain hardening index (n value) of the material. However, the function is non linear and cannot be be approximated by a function. Therefore a database containing value of thickness/thinning value at the top of the dome, radius of curvature at the top of the dome along the major axis, and radius of curvature at the top of the dome along the minor axis, was generated for different n value of the material with increments of 0.05 from n=0.05 to n=0.30 using FE simulation of elliptical test1. This database could be used in the calculation of the stress and strain increments for a known strain hardening index (n value). Hence, several tests that need to be conducted at different dome heights to obtain thickness at the top of the dome radius of curvature at the top of

65 the dome along the major axis, and radius of curvature at the top of the dome along the minor axis was avoided by using FE simulation.

4.7 Methodology to estimate flow stress and anisotropy from bulge tests

In the test, the flow stress of the sheet material was described by the commonly used power law ( σ = kε n ) and the Hill 1948 yield criteria was used to describe yielding behavior of sheet material in multiaxial stress state. Hence the parameters that need to be identified to describe material behavior are: Strength coefficient k, strain hardening

exponent, n, and anisotropy values r0,r90, r45. In the elliptical bulge test1, elliptical bulge test 2 and circular bulge test, the principle stress axis in the material coincides with the

anisotropy axis of the material therefore anisotropy value r45 did not influence the stress and strain calculations. This simplifies the problem to estimate the flow stress and anisotropy values (r0, r90) from elliptical test1, elliptical test 2 and circular bulge test. The

estimated flow stress and anisotropy values (r0, r90) was used to determine the

anisotropy value (r45) from elliptical test 3.

4.8 Estimation of the anisotropy values (r0, r90) and flow stress from circular bulge test and elliptical bulge test 1 and elliptical bulge test 2

The problem of estimation of flow stress (assumed to be σ = kε n ) and anisotropy coefficients from circular and elliptical bulge test was formulated as an optimization problem with the design variables as the unknown parameters namely the strain

hardening coefficient (n value) and the anisotropy values (r0, r90) required to analytical calculate the flow stress from equations given in section 4.2 for each test separately. The n value was added as a design variable because the radius and the thickness at the top of the dome required for calculation of flow stress could not be measured in real time in experiment but could be obtained from the database generated from FE simulation as a function of n value and dome height as discussed in section 4.6.

The flow stress (effective stress- strain relation ship) along a particular direction for the sheet material is the same irrespective of the test method used to obtain the flow stress. Therefore, the objective function for the optimization problem was defined as the least

66 square difference between the flow stress along the rolling direction estimated from the circular bulge test, elliptical bulge test 1 and elliptical bulge test 2 given by

N 2 2 2 f = − σσ −+ σσ + − σσ ∑i =1 (()circular elliptical1 ( elliptical1 elliptical 2 ) ( elliptical 2 circular ) ) Equation 4-11 Where

σ circular = Flow stress along rolling direction estimated from circular bulge test experiment data

σ elliptical 1 = Flow stress along rolling direction estimated from elliptical bulge test 1 experiment data

σ elliptical 2 = Flow stress along rolling direction estimated from elliptical bulge test 2 experiment data at the same effective strain values N = Number of strain values used in the least square fit

In the optimization, the objective function is minimized to obtain the flow stress and the anisotropy values of the material. The schematic of the optimization procedure is illustrated in the Figure 4.30. The procedure involves two steps a) estimation of flow stress of the circular bulge test and elliptical test from experimental measurements for known anisotropy values, and b) Estimation of anisotropy values (r0, r90) by minimizing the objective function. These two steps were repeated until the n value and the anisotropy value (r0, r90) converge. Initially anisotropy values were assumed 1.0 in the optimization.

4.8.1 Step 1: Estimation of flow stress for a given anisotropy values from the circular and elliptical bulge test experimental data.

The flow chart describing the procedure to estimate the flow stress for each test using the corresponding experimental measurements (pressure and dome height) is shown in the Figure 4.31. In the calculation of the flow stress, the radius of curvature along the major and minor axis of the ellipse and the thinning at the top of the dome for different dome height obtained from the database generated by FE simulation was used. The thinning and the radius of curvature obtained from the database is a function of n value,

therefore, initially n value of the material (no) was assumed to calculate the flow stress. The calculated flow stress (effective stress and strain) along the rolling direction for the

n assumed n value was curve fit to the power law ( σ = kε ) and the n value (ni) is

67 obtained. The calculation of the flow stress is again repeated using the obtained n value

(ni). This is repeated until the absolute difference between the ni value and ni-1 is less than 0.001 to get the flow stress from each test from the experimental data. It should be noted that in the step 1, anisotropy values are not changed.

Start

k=0

Assume kkk 900 ,, nrr

k=k+1

Calculate the flow Calculate the flow stress Calculate the flow stress stress from circular from elliptical bulge test 1 from elliptical bulge test bulge test for a given for a given anisotropy 2 for a given anisotropy anisotropy values 1 kk −− 1 1 kk −− 1 1 kk −− 1 values ( 0 ,rr 90 ) values ( 0 ,rr 90 ) ( 0 ,rr 90 )

k σ ellipse1,nellipse1 σ ,nk k circular circular σ ellipse2 ,nellipse2

Minimize the least square difference between the flows stress from elliptical and circular k bulge tests for a given n value ( n )

kkk circular ellipse1 σσσ ellipse ,,,,, rrn 9002

If kk −1 kk −1 ξ, rrnn 00 ≤−≤− ξ,

kk −1 rr 9090 ≤− ξ

End

Figure 4.30 Flow chart illustrating the methodology to estimate the flow stress and anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2 (k is iteration counter)

Start

j=0

Assume j n

Experimental data from Calculate the flow stress FE simulation Database for Circular/elliptical bulge along the rolling direction Circular/elliptical bulge test test using analytical equations Radius of curvature and Pressure, Dome height (refer section 5.2) Thickness

Least square fit of the analytically calculated flow stress to power law (σ = kε n )

σ,n j +1

If No jj +1 j=j+1 nn ≤− ξ

Yes

End

Figure 4.31 Flow chart illustrating the methodology to estimate the flow stress for given anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2

4.8.2 Step 2 : Estimation of anisotropy coefficients and flow stress for a given strain hardening coefficient value from the circular and elliptical bulge test experimental data.

The flow chart describing the procedure to estimate the anisotropy values from the flow stress obtained in each of the three tests using the corresponding experimental measurements is shown in the Figure 4.32. This problem of estimating the anisotropy

69 values and the flow stress of the material is treated as a minimization optimization problem. During the optimization, the objective function (Equation 4-11) that describes the difference between the flow stress obtained from each test is minimized to estimate the unknown parameters namely the anisotropy values (r0, r90). The standard modified Newton method was used to solve the optimization problem [Arora 2000]. The gradient and the Hessian of the objective function were estimated by central difference method. The golden search method was used to determine the step length along the direction provided by Newton’s method. In order to avoid any possible divergence as n values of the material was not changed in this step of the calculation; the step length in line search was limited to maximum value of 0.01. The optimization routine was converged when the magnitude of the objective function gradient is negligible small and the anisotropy value converges.

4.9 Estimation of the anisotropy value (r45) from elliptical bulge test 3

The anisotropy value r45 is obtained after the anisotropy values r0, r90 and flow stress of the material is estimated. The procedure to estimate r45 is similar to r0 and r90 except the experimental measurement from elliptical test 3 is used. The procedure is shown in Figure 4.33. The objective function for the minimization is given by

N 2 f = − σσ Equation 4-12 ∑i =1 ()material elliptical 3

Where

σ material = Flow stress theof sheet material along rolling direction obtained from circular test,bulge elliptical bulge test 1 and elliptical bulge test 2 (Section 4.8.)

σ elliptical 3 = Flow stress along rolling direction estimated from elliptical bulge test 3 strain val effective same at the data experiment experiment data at the same effective strain val ues N = ofNumber strain val ues used in the squareleast fit

Start

i=0

Assume ii X= ( ,rr 900 )

Calculate the objective function f(xi)

Calculate the gradient of ∂f objective function ( ) ∂x i (Central difference method)

If Yes ∂f End Magnitude of <ε ∂x i

Calculate the Hessian of ∂ 2f objective function ( ) 2 ∂x i (Central difference method)

i = i+1 Calculate search direction −1 ⎡ ∂ 2f ⎤ ∂f d=- ⎢ ⎥ 2 ∂x ⎣⎢∂x i ⎦⎥ i Newton method

Xi+1 =Xi+αd Calculate step length α Golden section method

Figure 4.32 Flow chart illustrating the methodology to estimate the anisotropy values (r0,r90) from the circular and elliptical bulge test 1 and elliptical bulge test 2 for given n value

Start

k=0

Assume kk 45 ,nr

k=k+1

Calculate the flow stress from elliptical bulge test 1 for a given k −1 anisotropy values ( r45 )

k σ ellipse3 ,nellipse3

Minimize the least square difference between the flows stress from elliptical test 3 and flow stress from k circular bulge tests for a given n value ( n )

kk σ ellipse3 ,, rn 45

If No nn kk −1 ≤− ξ

kk −1 rr 4545 ≤− ξ

Yes

End

Figure 4.33 Flowchart illustrating the methodology to estimate the flow stress and anisotropy values (r45) from the elliptical bulge test 3

4.10 Estimation of the flow stress using the developed bulge test

4.10.1 Experimental setup

Figure 4.34 shows the sketch of the single action tooling that was used for the circular and elliptical bulge test. The upper die was connected to the slide and the die cushion supports the lower die. The punch in the lower die was fixed to the press table and therefore stationary. At the beginning, the tooling was open and the viscous material was filled into the area on top of the punch. The sheet was placed between the upper and lower die. When the tooling closed, the sheet was clamped between the upper and lower die. The slide then moved down together with the blank holder and upper die. Consequently, the viscous medium was pressurized by the stationary punch and the sheet was bulged into the upper die by the viscous medium.

With this tooling, it is possible to measure the dome height and the internal pressure during the experiment. The displacement potentiometer is integrated into the upper die to measure the dome height. The pressure transducer is mounted on the piston to measure the pressure of the viscous medium. The upper die insert has circular cavity for circular bulge test and a lock bead is added to the upper die insert to restrain the material flow during the test. The elliptical die insert designed for this study was added to the upper die insert in the existing location of the lock bead. The lock bead was then added on to the elliptical die insert. In addition to lock bead, a clamping force necessary to avoid drawing the sheet into the cavity was applied by blank holder. Commonly used sheet materials AKDQ steel of thickness 0.83 mm, DDS steel of thickness 0.77 mm, DP600 steel of thickness 0.6 mm and A5754-O of thickness 1.3 mm were used in this istudyto estimate the flow stress and anisotropy values using the proposed bulge test. For the sheet materials tested, a clamping force of 800 kN (176,493 lbf) was used to avoid material flow.

Cap

Die Shoe Mounting for Distance Position Plate Transducer Position Upper Die Transducer

Upper Insert Protection Cup Lockbead Bulged Sheet

Lower Insert Pressure Transducer Viscous Medium Punch Lower Die

Cushion Pin Die Shoe

Figure 4.34 Viscous Pressure Bulge (VPB) test setup

4.10.2 Experimental results

In this section experimental results for material AKDQ is presented in detail along with summary of flow stress and anisotropy values obtained for other materials. Detailed experimental results for other sheet materials DP600, DDS steel and A5754-O are available in Appendix D, Appendix E and Appendix F, respectively.

4.10.2.1 Material :AKDQ

Experiments were conducted using the circular die and elliptical die corresponding to the proposed circular bulge test, elliptical bulge test 1, elliptical bulge test 2 and elliptical bulge test 3 for AKDQ sheet material of thickness 0.83 mm. At least three samples of the sheet material were tested in each of the test until the sheet fails by bursting due to

74 excessive thinning. Figure 4.35 shows the deformed specimens in the circular test and elliptical tests.

R.D. R.D.

Circular bulge test Elliptical bulge test 1

R.D. R.D.

Elliptical bulge test 2 Elliptical bulge test 3

Figure 4.35 Deformed AKDQ samples in circular bulge test, elliptical bulge test 1, 2, and 3

The pressure versus dome height curve obtained from three different samples in circular bulge test was consistent (Figure 4.36). Similar behavior was also observed in elliptical tests. This indicates consistency in the quality of the sheet material. Maximum dome

75 height of ~ 34 mm and ~ 20 mm was observed in circular and elliptical bulge tests, respectively.

120 Experiment-2 100 Experiment-3 Experiment-1 80

40 Pressure (bar) Pressure

0 0 10203040 Dome height (mm)

Figure 4.36 Comparison of the pressure versus dome height curve obtained for three different sample of AKDQ sheet material in circular bulge.

160 Elliptical test 1 140

120 100

80 Elliptical test 2 Elliptical test 3 60 Pressure (bar) Pressure 40

0 0 5 10 15 20 Dome height (mm) Figure 4.37 Comparison of the pressure versus dome height curve obtained from elliptical test 1, elliptical test 2 and elliptical test 3 for AKDQ sheet material. 76

4.10.2.1.1 Comparison of the pressure vs dome height curve from circular test, elliptical test –1, elliptical test-2 , elliptical test –3

The pressure versus dome height obtained from elliptical test 1, elliptical test 2 and elliptical test 3 were different (Figure 4.37). Maximum difference of ~10 bar was observed between the elliptical test 1 and elliptical test 2 at the dome height of 18 mm. This indicates that there is anisotropy in the material and the anisotropy values in the rolling direction, transverse and diagonal are not equal.

4.10.2.1.2 Calculation of flow stress assuming isotropic material

700

Elliptical bulge test 1 and 3 600

500 Circular bulge test

400

Elliptical bulge test 2 300 Stress (Mpa) 200

100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

Figure 4.38 Comparison of the flow stress curve obtained from circular test, elliptical test 1, elliptical test 2 and elliptical test 3 for AKDQ sheet material assuming the material as isotropic.

The flow stress of the material was estimated from the circular and elliptical tests assuming the material as isotropic (Figure 4.38) using Hill’s 1948 yield criteria. The flow stress obtained from the three elliptical tests were different as the pressure versus dome height curve from three elliptical tests were also different. However, flow stress obtained

77 from the circular test was higher than the elliptical test indicating that the material is not isotropic as assumed using the Hill’s 1948 yield criteria.

4.10.2.1.3 Calculation of flow stress along rolling direction and anisotropy assuming anisotropic material property

The flow stress and the anisotropy coefficients of the sheet material for the Hill’s 1948 Yield criteria were estimated from the circular bulge test, elliptical bulge test 1, 2, 3 using the developed optimization methodology with FE simulation database (section 4.8 and 4.9). In the optimization, the anisotropy values were initially assumed 1.0. The anisotropy values along with the flow stress were iteratively estimated during the calculation to minimize the least square difference between the flow stresses estimated from the four tests (Figure 4.39). Optimum anisotropy values (r0=1.26, r90=1.58, r45=1.0) and the flow stress obtained at the end of optimization are given in Figure 4.40 and Figure 4.41, respectively. It could be observed that at the end of optimum calculation, the flow stress data obtained from four tests was identical.

1200

1000

800

600

400

Objective function 200

0 01234 Iteration number

Figure 4.39 Change in the objective function during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for AKDQ sheet material

1.8 1.6 1.4 1.2 1 0.8 r0 0.6 r90 0.4 r45 Anisotropy values Anisotropy 0.2 0 01234 Iteration number

Figure 4.40 Change in the anisotropy values during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for AKDQ sheet material

600 500

400 Circular bulge test 300 Elliptical bulge test 1 200

Stress(MPa) Elliptical bulge test 3 100 Elliptical bulge test 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

Figure 4.41 Flow stress along rolling direction obtained from the circular and elliptical bulge tests using the estimated optimum anisotropy values for AKDQ sheet material

4.10.2.2 Summary flow stress and anisotropy values

Detailed experimental results for the tested sheet materials DDS steel, DP600 steel and A5754-O aluminum alloy are available in Appendix D, E, F, respectively. The

79 methodology developed in section 4.6 was used to estimate the flow stress and anisotropy of the material in biaxial stress state. The flow stress along the rolling direction and anisotropy of the tested material obtained from bulge test are shown in Figure 4.42 and Table 4.5, respectively. The anisotropy value at 45 degree to rolling direction obtained from the bulge test was close to initial guess used in the optimization procedure. This indicated that the pressure versus dome height curve of elliptical test 3

that was used to obtain r45 was sensitive to change in the anisotropy value r45. Similar results were observed during the design of the test as well (refer section 4.3).

900 800 DR210 700 600 DP600 500 400 AKDQ 300 Stress [MPa] Stress 200 AL5754-0 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

Figure 4.42 Flow stress along rolling direction obtained from the circular and elliptical bulge tests using the estimated optimum anisotropy values for AKDQ , DDS steel, DP600 and A5754-O aluminum alloy sheet material

Table 4.4 Anisotropy values obtained from the circular and elliptical bulge tests for AKDQ, DDS steel, DP600 and A5754-O aluminum alloy sheet material Bulge test

Material r0 r45 r90 AKDQ 1.27 1 1.59 DDS steel 1.32 1 1.41 DP600 1.38 1.2 1.42 AL5754 1.05 1 1.06 80

4.11 Comparison of tensile test and bulge test results

Tensile test based on ASTM standards were conducted for the selected sheet materials AKDQ steel, DP600 steel and DDS steel to obtain the flow stress and the anisotropy values in three directions. Figure 4.43 and Figure 4.44, show the comparison of AKDQ, DDS, DP600 steel and A5754-O aluminum alloy sheet material flow stress along rolling direction obtained from tensile test and bulge test. Flow stress over a large strain range nearly twice that of tensile test can be obtained from the bulge test compared to tensile test. In tensile test, the geometry of the test specimen results in stress concentrations leading to early necking at small strains. However in bulge test, there is no effect of geometry and the necking is due to excessive thinning of the specimen at the top of the dome at large strain. Thus, flow stress over large strain range is possible from bulge test compared to tensile test. Flow stress along the rolling direction obtained from bulge test for all the sheet materials was higher than the tensile test for a given strain value. Ideally, the flow stress, which is a property of material, should be independent of the test. In bulge test, the material is subjected to biaxial stress/strain state and the flow stress along the rolling direction which is a data in uniaxial stress state is obtained using the Hills’s yield criteria that describe the plastic yielding of the material and corresponding equivalent stress and strain relationship that relates the multiaxial stress state to equivalent uniaxial stress state. The difference in the flow stress from bulge test and tensile test indicates that yielding behavior of the tested material is not accurately represented the Hill’s 1948 yield criteria at different stress states.

Anisotropy values obtained from bulge test were less than tensile test for all the tested sheet materials, Table 4.5. In general, the anisotropy values decrease with increase in the plastic strain of the material [Hoffmann et al. 2006.Dankert et al. 1998]. The smaller anisotropy values were expected from bulge test compared to tensile test because, the anisotropy obtained from bulge test is an average value over the entire range of strain available for flow stress determination which is much larger compared to tensile test for the tested sheet materials.

900 800

700 DP600 Bulge test 600 DP600 Tensile test 500 400 AKDQ Bulge

Stress [MPa] Stress 300 tt 200 AKDQ tensile tt 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

Figure 4.43 Comparison of AKDQ and DP600 steel sheet materials flow stress obtained from proposed bulge test and conventional tensile test.

600 DDS bulge test

500 DDS tensile test

400

300 AL5754-0 Bulge tt Stress [MPa] Stress 200 AL5754-0 tensile test 100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Strain

Figure 4.44 Comparison of DDS steel and A5754-O aluminum sheet materials flow stress obtained from proposed bulge test and conventional tensile test.

Bulge test Tensile test

Material r0 r45 r90 r0 r45 r90 AKDQ 1.27 1 1.59 1.9 1.4 2.45 DDS steel 1.32 1 1.41 1.83 1.75 1.84 DP600 1.38 1.2 1.42 0.72 0.94 0.95 AL5754 1.05 1 1.06 0.96 1.006 0.78

Table 4.5 Comparison of DP600, DDS, AKDQ steel and A5754-O aluminum alloy sheet materials anisotropy values obtained from proposed bulge test and conventional tensile test.

4.12 Selection of methodology to determine flow stress and anisotropy coefficient for process simulation in stamping

Flow stress and anisotropy coefficients of the sheet materials DP600, AKDQ, DDS steel and A5754-O aluminum alloy obtained from the bulge tests in biaxial stress/strain state were different from those obtained from the tensile test in uniaxial stress/strain state. Hence a choice of test method to determine the material properties for process simulation could not be established. Therefore, material properties obtained from tensile and bulge test were used in FE simulation of elliptical bulge test and circular bulge tests to predict 1) pressure versus dome height curve, and 2) thinning distribution for comparison with the experimental measurements to a)validate the developed bulge test method for estimation of the flow stress and the anisotropy coefficients, and b) identify the difference in FE predictions for using bulge test data and tensile test data to model the sheet material in FE simulation of bulge test. Also, round cup deep drawing simulations were conducted using the material data from tensile test and bulge test. The FE results namely, a) Load versus stroke, b) Draw-in, and c) thinning were compared with experiments to choose a test method for determination of the sheet material properties for process simulation.

4.12.1 Bulge test

In this section, comparison of FE results with experimental results for material AKDQ is presented in detail. Similar comparison for other sheet materials DP600, DDS steel and A5754-O are available in Appendix G.

4.12.1.1 Comparison of pressure versus dome height curve for AKDQ sheet material

In circular bulge test (Figure 4.45), the pressure versus dome height curve predicted using the material properties from bulge test and tensile test were identical with experiment until dome height of 22 mm. Beyond 22 mm, pressure versus dome height predicted using the material properties from tensile test were lower compared the experimental measurement while pressure versus dome height predicted using the material properties from bulge test was identical with the experimental measurements. Similarly, in elliptical test 1, 2, and 3 (Figure 4.46, Figure 4.47,Figure 4.48), pressure versus dome height curve predicted by FE simulation using the tensile test data and bulge test data were identical with experiment until dome height of 18 mm. Beyond 18 mm, pressure versus dome height predicted using the material properties from tensile test were lower compared to the experimental measurement while pressure versus dome height predicted using the material properties from bulge test was identical with the experimental measurements. Also, maximum pressure in circular and elliptical bulge test was predicted at dome height of ~27 mm and 18 mm respectively by the FE simulation using tensile test material properties compared to ~32 mm and 22 mm, respectively in the experiment and FE simulation using bulge test data. In the FE simulation of the circular bulge test and elliptical bulge test, maximum strain of ~0.4 was observed at the dome height of 22 mm and18 mm, respectively compared to maximum strain of 0.22 for which flow stress was available from the tensile test. This indicates that the drop in the pressure at higher dome heights in circular and elliptical bulge test for material properties from tensile test is because of the error in the extrapolation of the input flow stress data from tensile test by the FE code for larger strain. Also, the slope of the extrapolated flow stress from tensile test is lower compared to the actual thereby resulting in early necking and decrease in the pressure as predicted by the FE simulation. Similar observation were also made for other tested materials in the comparison of FE predicted pressure versus dome height curve using bulge test and tensile test data with experimental measurements for circular bulge test and elliptical bulge tests, Appendix G.

120 Experiment-2 Experiment-3 Experiment-1 100 Simulation- Tensile test data Simulation - Bulge test data 80

Pressure (bar) 40

0 0 5 10 15 20 25 30 35 40 Dome height (mm)

Figure 4.45 Comparison of the pressure versus dome height curve obtained from circular bulge test experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

180 Experiment-1 160 Experiment-2 140 Experiment-3 Simulation- Tensile test data 120 Simulation - Bulge test data 100 80

Pressure (bar) Pressure 60 40 20 0 0 5 10 15 20 25 Dome height (mm)

Figure 4.46 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -1 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 85

180 Experiment-1 160 Experiment-2 140 Experiment-3

120 Simulation- Bulge test data Simulation- Tensile test data 100

Pressure (bar) 60

0 0 5 10 15 20 25 Dome height (mm)

Figure 4.47 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -2 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

180 Experiment-1 160 Experiment-2 140 Experiment-3 Simulation- Tensile test 120 Simulation - Bulge test 100

Pressure (bar) Pressure 60

0 0 5 10 15 20 25 Dome height (mm)

Figure 4.48 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -3 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 86

4.12.1.2 Comparison of thinning distribution

Thinning distribution along the major axis in elliptical test –1 and elliptical test –2 at same dome height in the experiment (Figure 4.49) were identical indicating that the thinning distribution in elliptical test does not change with the orientation of the specimen as predicted in the FE simulation (refer section 4.6). Thinning distribution along the test sample in circular test at dome height of 31.3 mm (Figure 4.50) and elliptical test-1 at dome height of 18.3 mm (Figure 4.51) were compared with the FE predictions using the tensile test and bulge test material properties. It could be observed that at higher dome heights, the thinning distribution predicted by FE simulation using the bulge test data agrees well with experimental measurements. FE simulation using tensile test data predicts higher thinning distribution at the apex of the dome compared to the experimental measurements. This is due to the error in the extrapolation of the input flow stress data from tensile test by the FE code for larger strain. The slope of the extrapolated flow stress from tensile test was less compared to the actual thereby resulting in higher thinning using tensile test material properties compared to the experiment and the FE simulation using the bulge test material properties. Similar observation were also made for other tested materials in the comparison of FE predicted thinning using bulge test and tensile test data with experimental measurements for circular bulge test and elliptical bulge tests.

Comparison of FE prediction using material data from tensile test data and bulge test with bulge test experiments indicated that flow stress over small strain from tensile test when used in FE simulation results in extrapolation by FE code beyond the available range. Extrapolation of material data from tensile test resulted in under prediction of the pressure, and over prediction of thinning in the FE simulation compared to experimental measurements and FE predictions using bulge test data. This indicated that bulge test data obtained over large strain range is best suited for FE simulation compared to tensile test.

35 Experiment-Elliptical test1 30 Experiment-Elliptical test 2 25

20 0

15 Thinning % 10 65 90 5

0 0 10203040506070 Curvilinear length (mm)

Figure 4.49 Comparison of thinning distribution along the major axis obtained from elliptical bulge test –1 experiment and elliptical bulge test –2 experiment for AKDQ sheet material at dome height of 18.3 mm.

60 Experiment Simulation-Bulge test 50 Simulation-Tensile test 40

30 0 20 Thinning % Thinning 10 65 90 0 0 102030405060708090 -10 Curvilinear length (mm)

Figure 4.50 Comparison of thinning distribution at dome height of 31.3 mm obtained from circular bulge test experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

35 Experiment

30 Simulation-Bulge test Simulation-Tensile test 25

15 0 Thinning % Thinning 10

5 65 90 0 0 1020304050607080 Curvilinear length (mm)

Figure 4.51 Comparison of thinning distribution at dome height of 18.3 mm along the major axis obtained from elliptical bulge test –1 experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

4.12.2 Deep drawing of round cup

4.12.2.1 Deep drawing experimental setup

The schematic of the round cup tooling at ERC/NSM used in the experiments is shown in Figure 4.52. The punch is mounted to the lower shoe with a load cell to measure the punch force during the deep drawing process. The blank holder slides around the punch in the lower shoe. The upper shoe consists of the die. In the press, the lower shoe is mounted on the press bed and the upper shoe is attached to the press ram. The blank holder is supported by the cushion pins that apply the blank holding force on the blank holder during the forming process. During the deep drawing process, the blank holder is raised to top most level. The blank is placed/positioned on the blank holder. The die attached to the press ram moves down and initially clamps the sheet against the blank holder. The clamped sheet with the die and the blank holder moves further down and forms the sheet against the stationary punch under the action of the blank holder force through the cushion pins. 89

Die

Punch

Blank holder

Cushion pin

Punch diameter 152.4 mm Punch corner radius 20 mm Die corner radius 16 mm Die diameter 158.4 mm Blank diameter 316 mm

Figure 4.52 Round cup deep drawing tooling used in the experiments

Due to the continuous use of the tooling over the years and due to wear and tear, the cushion pins are generally not of the same height and therefore could apply non- uniform blank holder force resulting in a non-symmetric drawn cup. Hence the earing observed in the cup flange due to non-uniform blank holder force could not be distinguished from the earing due to anisotropy of the material. Therefore several trial experiments were conducted by adjusting the height of the pin using shims (thin sheet metal) such that the blank holder force is approximately uniform. The uniform pressure distribution was checked by forming the round cup with rolling direction oriented at 0o and 45o to the front and back of the tooling. It could be observed that despite different orientation of the sheet to the tooling, same earing pattern was observed after addition of necessary shims indicating that the blank holder pressure acting on the sheet is approximately uniform ( Figure 4.53). Constant blank holder force obtained from preliminary FE simulations were used in the experiments for the sheet materials AKDQ, DDS steel, DP600 steel and A5754-O, Table 4.6. Ecoform PST 2717 synthetic lubricant supplied by ECOFROM was used in the experiments for all the tested sheet materials. The lubricant was sprayed

90 initially on the sample and applied uniformly on the either side of the sheet using brush. After completion of the test for each sample, the tooling was cleaned by degreaser to remove any accumulated lubricant on the tool surface for good reproducibility of the experiments.

Rolling direction Rolling direction

Front of tool Front of tool Rolling direction parallel to the Rolling direction 45o to the front- front-back of the tool back of the tool

Figure 4.53 Picture of the round cup with the rolling direction oriented at two different angles to the front of the tool to approximately check for uniform blank holder pressure distribution on the blank

Sheet material Blank holder force (kN) AKDQ steel 130 DDS steel 70 DP600 steel 200 A5754-O 130

Table 4.6 Constant blank holder force predicted using preliminary FE simulation for different sheet materials

4.12.2.2 Deep drawing experimental results

Deep drawing experiments were conducted for the sheet materials AKDQ steel, DDS steel, DP600 steel, and A5754-0. Initial blank of diameter 316 was drawn to round cup of height ~100 mm using the ECOFORM lubricant. The blank holder force predicted by preliminary FE simulation was used in the experiment, Table 4.6. Round cups from 91

AKDQ steel, DDS steel, A5754-O could be successfully drawn using the predicted blank holder force without any failure and negligible wrinkle at the die corner , Figure 4.54. During the test, the punch force and the stroke of the upper die were recorded using data acquisition system.

AKDQ Sheet material DP600 Sheet material

DDS Sheet material A5754-O Sheet material

Figure 4.54 Picture of the drawn round cups from different sheet material used in the experiment

Round cups drawn from DP600 sheet material resulted in wrinkles at the die corner radius (Figure 4.54) due to the large clearance between the die and the blank holder and large die radius for sheet of thickness 0.6 mm. Even higher blank holder force that resulted in failure of the cup during the forming process could not eliminate the wrinkles. Hence, round cup from DP600 of thickness 0.6 mm could not be drawn using the existing tooling at ERC/NSM.

4.12.2.3 FE model

FE simulation of the round cup deep drawing process for sheet materials DP600, AKDQ, DDS steel, and A5754-O aluminum alloy were conducted using commercial FE code PAMSTAMP-2G. The flow stress and the anisotropy coefficients of the sheet materials AKDQ, DDS steel, and A5754-O aluminum alloy obtained from bulge test and tensile test were used in the FE simulation for comparison of FE predictions with experiments. A quarter of the die geometry was considered for the FE simulation of the round cup deep drawing process as the material properties, die geometry and the boundary conditions are symmetric along two planes (X-Z and Y-Z) (Figure 4.55). The tooling dimensions for the ERC/NSM’s round cup tooling given in Figure 4.52 were used in the FE model. Interface friction of 0.06 corresponding to the lubricant ECOFORM was used in the simulations [Chandrasekeran et al. 2005]. The blank holder force used in the experiment was used in FE simulation for the corresponding sheet materials.

Punch

Blank holder

Sheet Z Y

Die X

Figure 4.55 Schematic of the FE model for the round cup deep drawing simulation

4.12.2.4 Comparison of FE predictions with experiments

FE results namely; punch force, draw-in and the thinning distribution were compared with experiments to validate FE results. In this section, comparison of FE results with

93 experimental results for material AKDQ is presented in detail. Similar comparison for other sheet materials DDS steel and A5754-O are available in Appendix H.

4.12.2.4.1 Comparison of punch force

The punch force predicted by FE simulation using the bulge test data was higher compared to the experiment. While the punch force predicted using the tensile test data was less compared to the experiment (Figure 4.56). This indicates that material properties obtained from both bulge test and tensile test cannot accurately describe the behavior of the material in multiaxial stress/strain state that occurs in forming of round cup. FE prediction using the bulge test data resulted in higher force compared to tensile data because a) the flow stress from bulge test is higher compared to tensile test for same strain (Figure 4.43), and b) the anisotropy values obtained from tensile test were higher compared to bulge test (Table 4.5) thereby allow easy material flow/deformation in the flange area resulting in less punch force.

160

140

120

100

80 Experiment-1 Experiment-2 60 Experiment-3 Punch Force kN Force Punch 40 Experiment-4 Simulation - Tensile test 20 Simulation - Bulge test 0 0 20 40 60 80 100 120 Stroke mm

Figure 4.56 Comparison of the punch force obtained from round cup deep drawing experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

4.12.2.4.2 Comparison of flange draw-in

The earing in the cup flange predicted by FE simulation using the bulge test and tensile test data matches well with experiments along the rolling and transverse direction (Figure 4.57). Along the 45o to rolling direction, FE prediction using the tensile test and the bulge test indicated slightly more draw-in (less flange) compared to experiment (Figure 4.57). This indicates a) the coulomb friction coefficient of 0.06 used in the FE simulation agrees with the lubricant used in the experiment, b) the anisotropy along the rolling direction and transverse direction could obtain the same draw-in despite the values are different, and c) anisotropy along the 45o to rolling direction obtained from both bulge test and tensile test do not describe the behavior of the material in multiaxial stress/strain conditions.

150

100

Punch Rolling diameter direction 0 -150 -100 -50 0 50 100 150

Experiment-1 -50 Experiment-2

Experiment-3 -100 Simulation-Bulge test

Simulation-Tensile -150 tt

Figure 4.57 Comparison of the draw-in obtained from round cup deep drawing experiment for AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

4.12.2.4.3 Comparison of thinning distribution

Thinning distribution along the rolling direction and 45o to rolling direction predicted by FE simulation using the tensile data was less compared to experiment while the FE predictions using the bulge test matches with the experiment better compared to tensile test (Figure 4.58). Less thinning and thickening was predicted for tensile data compared to bulge test data because a) lower punch force for tensile data compared to bulge test data (Figure 4.56), and b) higher anisotropy values for tensile test data compared to bulge test data reduces thinning/thickening of the sheet during deformation (Table 4.5).

5 160 180 0 0 25 50 75 100 125 150 175 200 -5 145

-10 Thinning % Thinning Experiment Simulation-Bulge -15 Simulation - Tensile 75 -20 0 55 -25 Curvilinear length (mm)

Figure 4.58 Comparison of the thinning distribution in the formed round cup from AKDQ sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

Similar results were also observed in comparison of FE predictions for deep drawing process namely a) punch force, b) draw-in and c) Thinning distribution with experimental measurements for other sheet materials DDS steel and A5754-O , Appendix H.

4.13 Summary and conclusions

96 in FE simulation. This data is insufficient for stamping FE simulations because a) maximum strain obtained in uniaxial tensile test before necking is small compared to strains encountered in stamping operations b) the stress state in tensile test is uniaxial while in regular stamping it is biaxial. Therefore, it is necessary to obtain the material properties from the biaxial test, which provides data for large strain range relevant to stamping operations to accurately model the stamping process through FE simulations. Therefore, in this study, elliptical biaxial tests were developed to supplement the circular bulge test to estimate the flow stress and the anisotropy of the sheet material. The developed bulge test and conventional tensile test was used to determine the flow stress and anisotropy of AKDQ steel, DDS steel, DP600 steel and A5754-O sheet material. The estimated flow stress from bulge test and tensile test was used in FE simulation of a) circular and elliptical bulge tests and b) round cup deep drawing process. FE predictions were compared with experiments to make a choice on the methodology to determine the flow stress and anisotropy of sheet materials for process simulation. Summary and conclusions drawn from this study are

1. Preliminary investigations indicated that along biaxial stress paths other than equibiaxial stress state (circular bulge test), Hill’s yield surface changes with the

change in the anisotropy coefficient (r0, r90, r45). Therefore, elliptical bulge test that induces biaxial stress path other than equibiaxial was selected for this study to estimate the flow stress and the anisotropy coefficients.

2. Preliminary FE simulations of the proposed elliptical bulge test with three different orientations (0o, 45o and 90o) of the sheet rolling direction to the major axis of the elliptical die for an anisotropic sheet material indicated that the forming pressure changes with the orientation of the sheet. This indicted that, the forming pressure in the elliptical bulge test is sensitive to anisotropy in sheet material.

3. FE simulations also indicated that increase in the ratio (major axis to minor axis) of the elliptical die increased the sensitivity of the forming pressure to different orientation of the sheet with respect to die. However, maximum achievable strain and dome height decreased with increase in the ratio. Within the dimension limitations of existing tooling at ERC/NSM, elliptical die geometry with major axis to minor axis ratio of 2.25 was selected. This die gave maximum dome height of ~19 mm, 97

maximum strain of 0.4, and maximum difference in the forming pressure of 5 bar in the FE simulation.

4. Forming pressure, thickness at the top of the dome, and radius of curvature at different dome heights were needed to calculate the flow stress and anisotropy coefficients by analytical equations from elliptical bulge test. Thickness and the radius of curvature cannot be measured in real time in the experiment. FE simulations of the circular and elliptical test indicated that the thickness at the top of the dome and radius of curvature at different dome height is function of the strain- hardening index of the material and is independent of the anisotropy values. Therefore a database of thickness and radius of curvature at the top of the dome for different dome heights and strain hardening index of the material were generated from FE simulations for use in the estimation of flow stress.

5. Inverse methodology based on optimization technique using a) Experimental measurements (pressure and dome height), b) the thickness and radius of curvature from FE simulation data base and c) analytical solution of the elliptical/circular bulge

test, was developed to estimate the flow stress and anisotropy coefficients (r0 and

r90) from circular and elliptical bulge test 1 and elliptical bulge test 2.

6. The anisotropy coefficient (r45) was estimated by inverse methodology using the estimated flow stress and the experimental results from the elliptical test 3 and FE simulation database.

7. Circular and elliptical bulge test experiments were conducted to determine the flow stress and anisotropy values of sheet materials AKDQ steel, DDS steel, DP600 steel and A5754-O using the developed methdology.

8. Experimental results indicated the elliptical test 3 is not very sensitive to change in

anisotropy value r45 of the material. Therefore, the proposed methodology is not

suitable to estimate anisotropy value r45 of the material.

9. Flow stress obtained from bulge test was higher compared to flow stress from tensile test for same strain for all the tested sheet materials. Also, flow stress over large

strain range can be obtained from tensile test compared to bulge test. Difference in flow stress between tensile test and bulge test is attributed to inability of the Hill’s yield criteria and corresponding equivalent stress and strain relationship to describe a) plastic yielding in multiaxial stress state for tested sheet materials, and b)relate multiaxial stress state to corresponding equivalent uniaxial stress state.

10. Anisotropy obtained from bulge test was less compared to tensile test. Anisotropy of materials generally decreases with increase in plastic straining of the material. Smaller anisotropy values were expected in bulge test as anisotropy obtained from bulge test is an average value over an entire strain range, which is nearly twice compared to tensile test.

11. FE simulation of the circular and elliptical bulge test process using bulge test material data (flow stress and anisotropy) accurately predicted the forming pressure and thinning distribution for the tested sheet materials. While FE simulations using tensile test data under predicted forming pressure and resulted in excessive thinning compared to experiment. This attributed to insufficiency of the tensile data to represent material behavior at higher strains.

12. FE simulation of the round cup deep drawing process using bulge test material data over predicted forming force and accurately predicted the thinning distribution and draw-in. While FE simulations using tensile test data under predicted forming force, thinning distribution and predicted draw-in reasonably well for the tested sheet materials.

13. FE prediction for bulge test and round cup deep drawing process indicated that flow stress and the anisotropy coefficients estimated by bulge test were better compared to tensile test.

4.14 Future work

1. The developed methodology needs to be applied to several steel alloys of different thickness to further validate this concept of biaxial test for determination of anisotropy and flow stress. 2. The developed methodology needs to be extended for aluminum alloys using Barlat 1996 yield criterion.

100

CHAPTER 5

ESTIMATION OF OPTIMUM BLANK SHAPE FOR SHEET METAL

FORMING

5.1 Introduction

Sheet metal forming process in modern industries deal with complicated shapes and the forming process consists of several successive operations until the final shape is formed. Process design for sheet metal products involves design of forming sequence (die and punch geometry for each stage), followed by estimation of various process parameters such as initial blank shape, blank holder force, lubrication, etc, to form parts without any defects (tearing and wrinkling). Generally, forming sequence is designed based on past experience and FE simulations, while process parameters are obtained in die tryouts based on trial and error. Among the process parameters, blank shape is critical as it influences other process parameters and significantly affects the material flow into the die cavity. Intelligent design of blank geometry and nesting of the blank in the coil could significantly improve the scrap rate and reduce the manufacturing cost. Also, post forming operation such as trimming, piercing, etc could also be avoided thereby reducing the manufacturing cost. Estimation of blank shape that reduces scrap rate and enhances the forming for new part geometries through trial and error die tryouts is time consuming even by a skilled engineer with many years of experience.

Several analytical and numerical methods have been proposed to predict the optimum blank shape. Among the methods, Slip-line methods [Chen et al. 1996], inverse method [Yang et al. 1998], backward tracing [Ku et al. 2000], constrained optimization method [Pegada et al. 2002], deformation path iteration method [Park et al. 1999], sensitivity method using FEA [Shim 2002] and Rollback method [Kim et al. 2000] are notable. Recently developed methods such as constrained optimization method, deformation path iteration method, sensitivity method and Roll back method, use FE analysis of the stamping process to obtain the optimal blank and therefore are more reliable compared 101 to other methods. However, the developed methods are limited to deformation of the sheet metal in the plane of blank holder and not on the side walls of the part. Also, the developed methods require several iterations of FE simulation as only the geometry of the formed part predicted by the FE simulation and not the straining of the material was used in calculation to estimate optimal blank shapes. In this study, new methodology that combines FE analysis with backward tracking using element shape functions was developed to consider both geometry and straining of the material to predict the blank shape involving fewer iterations and reducing the blank geometry estimation time.

5.2 Definition of optimal blank shape

Optimal blank shape is defined as the shape of the blank which after deformation conforms to the desired target boundary of the formed part, Figure 5.1. This includes the outer edge of the formed part and any features such as holes, slots in the formed part. Inherently, the stamping process requires minimum size of the blank to allow easy material flow into the die cavity and avoid failure, which is clearly evident in round cup deep drawing process as increase in drawing ratio (initial blank diameter/ punch diameter) increases thinning and reduces the draw ability of the sheet materials. Hence, process designers would like to have initial blank shape that leaves minimum flange to improve drawability and reduce the scrap by trimming. Thus, the target boundary of the formed part is known but the initial shape of the blank that would give the target shape is not known. Hence, this study focused on the estimation of the optimal blank shape through FE simulation to assist stamping process design engineers.

Final formed part Initial blank shape

Forming

Target part Target slot boundary boundary

Figure 5.1 Description of optimal initial blank shape

102

Start

Initial guess for FE Simulation of the blank shape forming process

End

Is the formed part boundary, hole Yes Optimum initial blank boundary matches shape obtained with target boundary?

FE Simulation of the No forming process Superpose target nodes on the formed part

Back track target nodes to the initial blank

Join target nodes location in the initial blank to get the new blank shape

Figure 5.2 Developed methodology to estimate optimal blank shape using FE simulation and element shape function based back tracking procedure.

5.3 Methodology to determine optimal blank shape

The developed methodology that couples FE simulation with element shape function based back tracking procedure is shown in Figure 5.2. The desired part boundary and/or the slot boundary in the formed part that need to be obtained after forming were discretized into points/ nodes called target points/nodes, Figure 5.3. The target nodes/points were assumed to lie in the midplane of the sheet metal along the thickness direction. At the start of the developed optimal blank estimation procedure, blank shape was assumed based on experience. FE simulation of the forming process was conducted using the assumed initial blank shape. The target points/nodes that describe 103 the desired target boundary were superimposed on the formed part geometry predicted by FE simulation to identify the elements in the formed part that contains the target nodes (Figure 5.3). The location of the superimposed target nodes/points on the final formed part were then backtracked to the initial blank using the FE results to obtain the superimposed target node location on the initial blank. The location of the back tracked target nodes in the initial blank were joined to get the new blank shape. This process was repeated until the part boundary and/or slot boundary in the formed part from FE simulation matches with the desired target boundary. The proposed methodology involves two steps, a) Identification of the target node location on the final formed part, b) Back tracking of the target nodes from the final formed part to the initial blank to obtain the blank shape.

Figure 5.3 Superimposition of target nodes on the deformed blank

5.3.1 Methodology for identification of the element in the final formed part that contains the target nodes

The formed part at the end of the FE simulation, which is discretized into four nodded quadrilateral elements and/or three nodded triangles, was superimposed on the target nodes. Two cases exist in identifying the element in the formed part that contains the target node. Case A: the target lies inside an element (Figure 5.3), Case B, target node does not lie inside on any element (Figure 5.3). The element that contains each target node was identified using the algorithm given in Figure 5.4. Initially, n closest element to 104 each target node was identified by calculating the distance from the center of each element to the target node. The center of each element in the final formed part can be calculated from the nodal coordinates using the element shape function (equation 3-6). Among the n closest element, the element that contains target node was identified by projecting the target node on each of the n closest elements along its normal and checking its location with respect to the element boundaries. The element normal was calculated using its nodal coordinates as all the coordinates of the element lie on the same plane, equation 5-1. To facilitate the nodal projection, the element along with target node was rotated such that the element normal is parallel to Z axis. Presence of the projected target node inside the element was checked by dividing the quadrilateral elements into triangles as shown in Figure 5.5. Presence of the target node in each triangle was checked using the Barycentric technique, Figure 5.5 [Hearn et al. 2000]. The barycentric technique uses the equation of the plane formed by three points of triangle to determine the location of the target point. if the projected target point does not lie inside any element, then the element that is closest to target node among the n closest elements was identified to contain the target node (case B).

X1, X2, X3 be the coordinate vector of point 1, 2, 3 Unit normal vector of plane that contains three points is given by

crossprodu ()()− ( − xxxxct 1312 ) nˆ = Equation 5-1 crossprodu ()()()−− xxxxct 1312

105

Start

Loop End For i = 1 to number of target nodes over

Formed part from Identify 10 closest FE simulation elements that contains target node

Loop No element For j = 1 to 10 over contains target node Calculate the element normal Pick the closest element for the Rotate the element and the target node target node until element normal parallel Z- axis

i = i +1 Divide the element to Go to next target node two triangles

Is the target Yes node inside element?

j = j +1

Figure 5.4 Methodology to identify the elements in the formed part that contains the target node

106

Element Normal

Target node (x3, y3) (x4, y4) Projected target node Triangle II (x1, y1)

(x1, y1) (x , y ) Quadrilateral Triangle I 2 2 Shell Element

1 ⎧ − xx 1 ⎫ ⎡ −− xxxx 1312 ⎤⎧α⎫ ⎨ 1 ⎬ = ⎢ ⎥⎨ ⎬ ⎩ − yy 1⎭ ⎣ −− yyyy 1312 ⎦⎩β⎭ if triangle inside Point inside triangle if

0 α 1, 0 β 1, βα ≤+≤≤≤≤ 1

Figure 5.5 Methodology to determine the presence of target node inside the element by Baycentric technique [Hearn et al. 2000]

5.3.2 Algorithm for back tracking the target node location to the initial blank

Back tracking was performed from final formed part to starting blank in the increments of time interval Δt used in the FE simulation. The procedure for back tracking from time tn+1

to tn is shown in Figure 5.6. This procedure was repeated from end to start of forming operation for all the target nodes to predict the initial blank. In the stamping simulation, the element undergoes deformation coupled with rigid body translation and rigid body

rotation from state tn to tn+1. Therefore, rigid body motions that do not contribute to strain of the element were compensated by translating the element from its position in the time

tn+1 along with the projected target node to the time tn such that node 1 of the element at

time tn+1 coincides with node 1 of the element at tn, Figure 5.7. Rigid body rotation was

compensated by rotating the element in time tn+1 along with the projected target node

about the node 1 such that element normal at tn+1 coincides with tn (Figure 5.7). For ease

of calculation the elements at time tn and tn+1 was rotated along with the target node such

107 that their normal is parallel to Z axis. The backward displacement at each node that would result in deformation of the element from tn+1 to tn was calculated by subtracting

the nodal coordinates of tn from tn+1for the corresponding node, Figure 5.7. The variation of the backward displacement within the element midplane from the nodal displacement is given by shape function (equation 4-10), which also interpolates the nodal coordinates within the element as isoparametric elements are used for modeling sheet materials in

FE simulation. The location of the projected target node in the element at tn+1 (value of local coordinates (ξ,η) in the shape function for the projected node) was estimated from the non linear equation 5-2 using Newton Raphson method. The estimated values (ξ,η) was used to calculate the backward displacement corresponding to the target node

location in the element tn+1 (equation 5-3). Thereby the strain in the element when it

deforms from tn to tn+1 was accounted while back tracking. The target node position in the element at time tn was calculated by adding calculated backward displacement to its coordinates at time tn+1. This procedure was repeated until t reached t=0 corresponding to start of forming to give the target node position in the initial blank for all the target nodes. The location of the target nodes on the initial blank are joined to give the initial blank shape.

⎧N1 εη ),( ⎫ * ⎪ ⎪ ⎧x ⎫ ⎧ xxxx '''' 4321 ⎫⎪N 2 εη ),( ⎪ F1()εη = 0, ⇒ ⎨ * ⎬ = ⎨ ⎬⎨ ⎬ ⎩y ⎭ ⎩ yyyy '''' 4321 ⎭⎪N3 εη ),( ⎪ F2 ()εη = 0, ⎪ ⎪ ⎩N 4 εη ),( ⎭ ()()11 −− ( )(11 +− ηςης ) N = ,N = , 1 4 2 4 Equation 5-2 ()()11 ++ ης ()()11 −+ ης N = ,N = 3 4 4 4

⎧ N 1 ξη ),( ⎫ 1 ⎪ ⎪ ⎧ u ⎫ ⎧ 1 2 3 uuuu 4 ⎫ ⎪ N ξη ),( ⎪ = 2 Equation 5-3 ⎨ 1 ⎬ ⎨ ⎬ ⎨ ⎬ ⎩ v ⎭ ⎩ 1 2 3 vvvv 4 ⎭ ⎪ N 3 ξη ),( ⎪ ⎪ ⎪ ⎩ N 4 ξη ),( ⎭

108

Start

Loop End over For t = tn to t1

For i = 1 to number of target nodes

FE result for step t Compensate for rigid body translation between time t and t-1 for element that FE result for step t-1 contains target node i

Compensate for rigid body rotation between time t and t- 1 for element that contains target node i

Calculate backward displacement

Calculation of target node location in the element at time t shape function coordinates (ξ,η)

Calculate backward displacement for target node

Calculation of target node location in the element at time t-1

Figure 5.6 Flow chart describing the backtracking methodology using element shape function

109

Element at time tn+1 Projected Target node

Node1 Element at time tn Node2 Node2 Node3 Node3 Node 4 Node1 After rigid body rotation Node 4 (x ,y ) 2 2 (x3,y3)

(x’2,y’2) (x’3,y’3) After rigid body translation (x4,y4) (x1,y1) (x’1,y’1) Node2 Node3 (x’4,y’4) Backward Deformation Node1 ⎡ui ⎤ ⎡ − xx 'ii ⎤ ⎢ ⎥ = ⎢ ⎥ Node4 v i − yy 'ii Node3 ⎣ ⎦ ⎣ ⎦ Node4

Figure 5.7 Schematic of the steps involved in the back tracking of target node

5.4 Numerical Implementation

The developed methodology was programmed in Matlab. The target nodes and the input blank mesh to the FE simulation in IDEAS UNV file format was provided as an input to the program. Also, the deformed sheet geometry at different time steps from FE simulations was provided as input to the program in IDEAS UNV file format. The program provides the location of the target nodes in the initial blank as an output in UNV file format. The output file can be opened in any CAD/CAE software that could read UNV file format, and the target node locations are joined by polyline to get the blank shape for the next iteration. This is repeated until the blank shape converges. This program is

110 referred as BLANKOPT in this thesis. The developed methodology was applied to estimate initial blank shape for torque converter turbine shell.

5.5 Application – Torque converter turbine shell

5.5.1 Background

An international automotive Tier-I supplier currently manufactures torque converter turbine shells [Figure 5.8] by transfer die forming followed by piercing slots [Figure 5.8] using Koppy machine. Koppy machine is similar to machining center where the punches are mounted at equal angle to balance the force and the part is indexed after each piercing until all holes are pierced in the part. The company is interested in avoiding piercing operation by the Koppy machine, as it is expensive and time consuming. Instead the company want to start with the blank that has a hole which after forming confirms to the desired shape and location in the final formed part. Hence, the developed methodology was applied for this cases study to estimate the optimal blank shape along with the hole that would result in desired formed part after forming.

Slots

Figure 5.8 Torque converter turbine shell used in this study to estimate blank shape

5.5.2 Process description The forming process of turbine shell involves two forming stages followed by piercing in koppy machine as shown in Figure 5.9. In forming stage – I, the inner curvature is

111 formed. While in stage –II, the inner curvature is held while the outer curvature is formed to get the desired shape. After forming, the part doesn’t require trimming and is ready for piercing in koppy machine.

After Ist stage (Stamping) Initial blank with center hole

After piercing

After IInd stage (Stamping)

Figure 5.9 Process sequence currently used to manufacture turbine shell

FE simulation of the existing process setup to form the turbine shell was conducted as a first step to validate the FE model for use with developed methodology to estimate the optimal initial blank shape.

5.5.3 FE Model

FE simulations of the process sequence currently used in production were conducted using commercial FE code PAMSTAMP. In the FE simulation, dies were modeled rigid while the mid surface of the sheet was modeled using shell elements. Figure 5.10 and Figure 5.11 show simulation model for stage 1 and stage 2 forming stage, respectively. The formed part from stage – I along with strain history was used as input blank/preform for stage – 2.The blank and die dimensions used in FE modeling were obtained from 2D CAD drawing supplied by the sponsor. Only 12.41° was modeled for both forming stages due to symmetric slot C location and boundary conditions. Table

112

5.1and Table 5.2 show various input parameters used in simulations for stage 1 and stage 2 respectively.

a) Isometric view b) Front view

Figure 5.10 First stage FE simulation model

a) Front view a) Isometric view

Figure 5.11 Second stage FE simulation model

113

Material MS-015 CR Sheet thickness 2 mm Young’s Modulus = 210 GPa, Poisson ratio = 0.3

0.58 Material property Flow stress :σ =+ε340 415 p MPa [ε p = Plastic strain] (Provided by Luk from tensile test) Friction coefficient 0.06 (Provided by Luk from pin and V block test - ASTM (μ) D-3222-92B) Nitrogen cylinder Blank holder 53.2 -88.8 kN (linearly varying with stroke) force

Table 5.1 Input parameters for first stage forming simulation

Material Preform from first stage

Young’s Modulus = 210 GPa, Poisson ratio = 0.3

0.58 Material property Flow stress :σ =+ε340 415 p MPa [ε p = Plastic strain] (Provided by Luk from tensile test)) 0.06 (Provided by Luk from pin and V block test - Friction coefficient (μ) ASTM D-3222-92B) Nitrogen cylinder Inner punch: 7-11 kN (linearly varying with stroke) force Binder: 34-53 kN (linearly varying with stroke)

Table 5.2 Input parameters for second stage forming simulation

5.5.4 FE results

Formed part dimensions obtained from FE simulations were compared with prototype part dimensions provided by sponsor. Figure 5.12 show the comparison of FEM results and prototype part dimensions for inner radius (R1), outer radius (R2) and part height (h). FE results agreed well the prototype part dimensions. Hence, the existing FE model was used in further analysis to predict the optimal blank shape with the slot.

114

R 1 h

R 2

Dimension Experiment (mm) Simulation Error (%) (mm) Inner radius (R1) 38.9 38.9 0 Outer radius (R2) 129.6 129.6 0 Height (h) 37.53 36.97 1.5

Figure 5.12 Comparison of formed part dimensions from FE simulation and prototype part

5.5.5 Estimation of optimal blank shape

5.5.5.1 Problem definition

The objective of this case study was to estimate the shape of the slot (Figure 5.8) in the blank such that, after forming using the developed forming process, Figure 5.9, the formed part conforms to the desired shape. The slots are close to the outer periphery of the formed part. Presence of the slots near the blank boundary would also influence the shape of the outer periphery of the part as well. Hence, both the slot shape in the blank and shape of the blank were estimated using the BLANKOPT such that after forming, the slots are of desired shape and are in desired location. Also, the outer periphery of the part is smooth without hills and valleys for assembly.

5.5.5.2 Blank shape prediction results

The developed methodology along with validated FE model was used to estimate the optimal blank shape with the hole. The target part boundary and the hole boundary was obtained from the CAD model of the turbine shell part provided by the sponsor. As FE model, part boundary corresponding to 12.41o that includes a slot were discretized to obtain the target nodes for the blank estimation, Figure 5.13. Initial blank of circular shape with inner hole radius of 30 mm and outer radius of 160 mm without the hole for 115 the slot was used as a starting blank for the optimum blank shape estimation, Figure 5.14. The results from the forming FE simulation using the blank without slot was provided as input to the BLANKOPT. Estimated blank shape by BLANKOPT based on the FE result for the blank without slot is shown in Figure 5.15. The estimated blank shape by BLANKOPT was used as an input to FE simulation. Figure 5.16 show the comparison of the predicted formed part geometry in iteration 1 using blank shape predicted by BLANKOPT, Figure 5.15 . It could be observed that the blank shape predicted by BLANKOPT in iteration 1 resulted in formed part very close to target shape. This procedure was repeated until the predicted blank shape converges and the formed part from FE simulation using the predicted blank shape from BLANKOPT matches with the target nodes. Five iterations were required for this case study to obtain the converged solution for the optimal blank shape. It should be noted that developed methods predicted blank shape close to optimal shape in first iteration. Remaining four iterations were required to fine tune the solution, Figure 5.17. Figure 5.18 show the comparison of the formed part with the target part using the optimal blank shape estimated by BLANKOPT. Slot and part boundary in the formed part using optimal initial blank matched well with desired target nodes indicating the slots in the formed part could be obtained using optimized blank shape instead of additional post forming operation such as piercing. The estimated optimum blank shape, Figure 5.19, was provided as input to the die tryout process to validate the FE predictions.

116

20 12.41o 15 Z Coordinate (mm) Z Coordinate 10

0 0 5 10 15 20 25 30 Y coordinate (mm)

Figure 5.13 Target nodes obtained from CAD data of the turbine shell part for use in estimation optimal blank shape

30 mm

160 mm

Figure 5.14 Schematic of initial blank used in FE simulation

117

15 12.41o

Y coordinate (mm) 10

0 90 100 110 120 130 140 150 160 X coordinate (mm)

Figure 5.15 Blank shape predicted by BLANKOPT for iteration 1 based on the FE results using initial blank without hole.

41 12.41o 39

37 Target nodes 35 Nodes of formed part from FE 33 31

Z Coordinate (mm) Z Coordinate 29 27 25 0 5 10 15 20 25 30 Y coordinate (mm)

Figure 5.16 Comparison of the formed part using iteration 1 blank predicted by BLANKOPT with desired target nodes. 118

25 12.41o 20

15 Iteration 1

Y coordinate (mm) Y coordinate 10 Iteration 2 Iteration 3 5 Iteration 4 Iteration 5 0 136 138 140 142 144 146 148 150 X coordinate (mm)

Figure 5.17 Comparison of the blank shape predicted by BLANKOPT for different iterations

40 12.41o 38 Target nodes 36 Nodes of formed part from FE 34

30 Z Coordinate (mm) Z Coordinate 28

26 0 5 10 15 20 25 30 Y coordinate (mm)

Figure 5.18 Comparison of the formed part using iteration 5 blank shape predicted by BLANKOPT with the desired target nodes. 119

Figure 5.19 Optimum blank geometry estimated by BLANKOPT and provided to sponsor for tryout

Figure 5.20 Formed part from die tryout using optimum blank geometry estimated using BLANKOPT and FE simulation.

5.5.6 Experimental validation

Figure 5.20 show the result from the tryout using the optimum blank, Figure 5.19, predicted using FE simulation and BLANKOPT. The part could be formed using the optimum blank without any failure. Several parts were formed using the optimum blank and the dimensions of the slot were measured by the sponsor and provided to ERC/NSM. The comparison of the desired slot dimensions with the measured dimensions in the formed part using the optimum blank is shown in Table 5.3. It could be observed the slot dimensions in the formed part were very close to the desired value.

120

However, the initial blank shape was latter fine tuned in the die tryout to meet the desired tolerance requirements.

Required Dimension Average Dimension obtained in tryout

Length 8.13 ±0.15 8.152±0.25

Width 1.61±0.15 1.440±0.20

Angle 55.5O±0.5 55.70±0.20

Position X 0.00±0.25 -0.272±0.15

Position Y 0.00±0.25 0.150±0.15

Table 5.3 Comparison of the CAD dimensions with tryout part dimensions formed using the blank predicted by BLANKOPT

5.6 Summary and conclusions

Among the process parameters, shape of the initial blank shape is critical as it influences other process parameters and affects the material flow into the die cavity. Also, intelligent design of blank shape could save material and reduce manufacturing cost. Conventionally, blank shape required to form the parts are estimated based on experience and based on blank geometry currently in use for similar parts. Hence, they are not optimum and estimation of blank shape by trial and error for new part geometries 121 in FE simulation requires significant amount computational time. Hence, in this study, a new method to estimate optimum blank shape using FE simulation was developed. The developed methodology was applied to industrial case study. Summary and conclusions drawn from this study are:

• Element shape function based backtracking methodology was developed to use with FE simulation results to predict the optimal blank shape.

• The developed methodology was applied to estimate optimal blank shape for the turbine shell part to eliminate piercing operation required for punching holes after forming.

• The developed methodology was able to predict the blank shape close to optimal in the first iteration. Four additional iterations were required to fine tune the estimate blank shape to get the optimum.

• The optimum blank shape resulted in formed part without failure in tryout. The slot in the formed part using the optimized blank was close to required dimensions. Also, the outer edge of the formed part was uniform without any scallops.

• The case study indicated that the developed methodology (BLANKOPT) could provide optimal blank shape that would result in formed part close to the target. The optimal blank shape was very good guess to fine tune in the die tryout.

• The developed methodology (BLANKOPT) need to be applied for different part geometries for further validation.

122

CHAPTER 6 OPTIMIZATION OF BLANK HOLDER FORCE TO PROGRAM MULTIPOINT CUSHION SYSTEM IN SHEET METAL FORMING

6.1 Introduction

In stamping as well as in sheet hydroforming with punch (SHF-P) and die (SHF-D), the quality of a formed part is determined by the amount of material drawn into the die cavity during the forming process. An excess material flow will cause wrinkling while insufficient material flow will cause tearing in the part, Figure 6.1. The blank holder plays a key role in regulating the material flow by exerting a predefined blank holder force (BHF). When designed correctly, this BHF can prevent wrinkling and delay tearing in the formed part. The required force for the blank holder to apply on the sheet as BHF is provided by the press cushion system or the nitrogen cylinders mounted on the tools. The BHF required to form the part are usually estimated during the process design by trial and error FE simulation and later refined during the die tryouts. Conventionally, the BHF is applied uniformly on the blank holder surface and held constant during the forming process. Therefore, it is relatively easy to estimate by trial and error in FE simulation and tryouts.

Modern presses are equipped with multipoint cushion systems that allow the BHF to vary with location and during the stroke [Pahl 1996, Siegert et al. 1993, Mueller Weingertan, 1999, Gloecker et al. 2004] Also, tools built with nitrogen cylinders to apply the required force allow the option to change the force between the cylinders/location and during the space [Hausermann 2000, Hengelhaupt et al. 2006, Metal forming controls, 2007]. Many investigations have shown that an optimized BHF profile, which varies in time/stroke and space/location, can improve the drawing process for parts with complex geometry by providing a better control of the metal flow [Manabe et al. 1987, Doege et al. 1987, Ahmetogulu et al. 1993, Siegert et al. 1993, Broek et al. 1995, Thomas et al. 1997, Neugebauer et al. 1997,Obermeyer et al. 1998, Mattiasson et al. 1998, Gunnarsson et al. 1998]. However, the multipoint cushion capability in the press is under utilized currently in production because a) it is difficult to estimate the BHF that 123 should be applied by each cushion and b) conventional steel sheets can be formed with existing method of constant blank holder force. However, increase in the complexity of the parts and emphasis to use lightweight materials such as aluminum alloys and Advanced High Strength Steels (AHSS) with low formability require the use of multi-point cushion capabilities to better control the material flow and expand the processing window. Successful application of a multi-point cushion system to form complex parts from the low formability materials requires a methodology to estimate the necessary blank holder force that will be applied by each cushion pin. This can be best done by FE simulation in the process design stage. Thus, little time will be necessary in tryout for production.

Hence, in this study, numerical optimization technique coupled with finite element analysis of the stamping/sheet hydroforming process was used to predict four possible modes for application of BHF, namely a) BHF constant in space/location and time/stroke, b) BHF variable in time/stroke and constant in space/location, c) BHF variable in space/location and constant in time/stroke and d) BHF variable in space/location and time/stroke for a multipoint cushion system as well as single point cushion system/nitrogen cylinders.

Maximum possible part height Wrinkles Fracture

Defect free part Part height (mm) (mm) height Part

Blank holder force (kN)

Figure 6.1 Potential failure modes in deep drawing influenced by the blank holder force 124

6.2 Methodology to estimate BHF

The estimation of BHF in sheet metal forming could be formulated as an optimization problem with the objective to minimize the risk of failure by tearing in the formed part and avoid wrinkling in the formed part as well. This optimization problem cannot be solved analytically as there is no explicit relationship between the blank holder force and tearing in the part, and blank holder force and wrinkles in the formed part except for the general trend. Also, such relationship varies significantly with the part geometry and sheet material. Therefore, numerical optimization techniques were used in this study to couple with FE simulation of the forming process to estimate the optimum blank holder force. The objective function and the constraint functions required for the optimization were calculated from the results of the FE simulation of the forming process. Figure 6.2 shows the strategy used to estimate the optimum BHF profiles required to program the mpultipoint cushion systems.

Cushion pin 1

480 ) 400 Inputs required 320 − Number of cushion cylinders Optimization program 240 developed by ERC/NSM to − Maximum and minimum force 160 estimate force required from limits for each cushion cylinder 80

− Quality parameters acceptable multipoint system Force (kN Holder Blank 0 wrinkling and thinning levels 0 102030405060 Punch Stroke (mm)

Commercial FE codes

PAMSTAMP Cushion pin n LSDYNA 480 ) 400 320 Inputs required 240 − Tool geometry 160 − Material properties 80 Blank Holder Force (kN Holder Blank − Process conditions 0 0 102030405060 Punch Stroke (mm)

Figure 6.2 Flowchart describing the overall approach to determine optimum BHF to using commercial FEA code PAMSTAMP 2000 and LSDYNA to program multipoint cushion system

125

6.3 Objective function

The objective of the optimization is to minimize the risk of failure by tearing in the formed part. Forming limit diagram and thinning in the part are commonly used to predict failure in the formed part. In this study thinning in the formed part was used as a indicator for failure by tearing because a) it can be easily measured in the formed part and b) it is independent of stress/strain path. Therefore the objective function (φ) was defined as the top 10 % of maximum thinning in the formed part and calculated with equation 6-1:

N ⎡ ' ⎤ ⎡ t i ⎤ φ ⎢−= ⎢ ⎥⎥ /ln N ∑ t i =1 ⎣⎢ ⎣ 0 ⎦⎦⎥ Equation 6-1 where t’I is the thickness in the formed part at node I , t0 is the initial sheet thickness and N is the number of nodes that are within top 10 % of the minimum thickness.

6.4 Constraint function

The constraints in this optimization problem are wrinkles that occur in the formed part. In sheet metal forming, due to complex shape of the tooling, wrinkles may be initiated at the start of the process and latter could get removed when the sheet is formed against the die at the end of the forming process. Hence, a geometry-based method could accurately guarantee the presence of wrinkles in the final formed part while other methods based on bifurcation theory [Baodeu et al. 1996, Chu et al. 1996, Kim et al. 2000, Tuguc et al. 2001, Correia et al. 2002] or energy method [Cao et al. 2000, Xie et al. 2002] could only predict the initiation of wrinkling but di not guarantee a presence of wrinkles. Therefore, wrinkles in the formed part were detected using a geometry based criterion which is described below. This criterion is mesh dependent therefore; refined mesh (element size ~ 2mm) need to be used in the FE simulations to model the sheet material.

126

6.5 Geometry based wrinkle criterion

6.5.1 Flange wrinkle

Flange wrinkle in the FE results is detected by calculating the normal distance between the die surface and the node at the mid plane of the sheet metal. This distance is compared with the current sheet thickness for each node of the sheet in the FE model that is between the die and blank holder. Flange wrinkle magnitude at a node (i) is as given in equation6-2. t −Δ i wrinkle magnitude at node (i) = i 2 Equation 6-2 where Δi is the normal distance between the die and the node i of the sheet metal

(Figure 6.3 )and ti is the sheet thickness at a given node i. Blank holder is modeled as elastic body in FEM ti Sheet Δ described by shell elements

Die is modeled as rigid body in FEM

Figure 6.3 Schematic illustrating the flange wrinkle magnitude calculation for elastic blank holder and rigid die

The flange wrinkle constraint in the optimization program is represented by

max(wrinklemagnitudenode ) wrinkleconstra pg ijflange )]([int = ≤− 00.1 Equation 6-3 δ f

where δf is the tolerable flange wrinkle magnitude.

The area in the flange over which the flange wrinkle calculation need to be performed was defined as an input to the optimization program. For example in the Hishida part, 127 the flange area is divided into 11 small convex areas/polygons of 4 or 5 sides, Figure 6.4. The coordinates for each area/polygon vertices are given in clockwise or anticlockwise order. The flange wrinkle in each polygon is calculated independently. Depending on the part geometry and/or the type of BHF profile being estimated, either maximum flange wrinkle calculated among all the area is used in optimization program or each area is used independently.

Area 2 Area 1 Area 3

Area 11 Area 4

Area 10 Area 5

Area 9 Area 6

Area 7 Area 8

Vertices of the area

Figure 6.4 Schematic illustrating the method to describe the flange area in the final part for flange wrinkle calculation.

Die

Sheet Punch Punch

Blank holder a) Closed die with bottom face b) Open die with no bottom face Figure 6.5 Schematic of the open die with no bottom face and closed die with bottom face 128

6.5.2 Side wall wrinkle

The methodology to evaluate the sidewall wrinkle from the FE results vary depending on the type of die geometry used in the FE simulation. Two possibilities namely; closed die with bottom face (Figure 6.5a), and open die with no bottom face (Figure 6.5b) were considered.

X Section plane

X Boundary plane Boundary plane Minimum (X limit) Maximum (Xlimit)

Figure 6.6 Strategy for sidewall wrinkle detection in closed die with bottom face

6.5.2.1 Closed die with bottom face

In this methodology for closed die with bottom face, the presence of wrinkle at the specific location in the FE simulation was detected by comparing the curvilinear length of

129 the sheet inside the die cavity to the curvilinear length of the tool inside the die cavity at the desired section as shown in Figure 6.6. The ratio of the curvilinear length of the sheet to the curvilinear length of the tool (punch+die) is used as an indicator for detecting sidewall wrinkle. In order to avoid any possibility of sidewall wrinkle at the

desired section, the wrinkle ratio (Wratio) should always be less than or equal to one at any step in the FE simulation. Wrinkle ratio higher than one indicates more material inside the die cavity than that cannot be accommodated at the end of stroke, resulting in wrinkling of the sheet metal. This geometry-based methodology is mesh dependent as the curvilinear length is calculated using the elements. Therefore, this wrinkle detection technique requires fine mesh in the corners of the punch, die and deforming sheet metal to accurately calculate the curvilinear length.

wrinklemagnitudeside = Wratio Equation 6-4 Once the side wall wrinkle amplitude is determined, the flange wrinkle constrain in the optimization program is represented by

wrinklemagnitude wrinklecon pgstra )]([int = side ≤− 00.1 Equation 6-5 ijside δ where δ is the tolerable flange wrinkle magnitude. In the simulations conducted δ was changed depending on the sides and part geometry. As an input to the optimization program, the origin and normal of the section plane and the origin of the boundary plane are required. It is assumed that limiting planes are parallel to the coordinate axis of the FE model.

6.5.2.2 Open die with no bottom face

In this methodology for open die with no bottom face, the deformed geometry of the sheet at the end of forming was not defined either by shape of the die or by the shape of the punch. Therefore, side wall wrinkle was detected by cutting sections in the formed part using the plane with the normal parallel to the stamping direction. The cut section was further subdivided into sides depending on the feature in the punch and die as straight sections and circular sections as shown in Figure 6.7 for IFU-Hishida die. Side 2, side 4, side 6, side 8 are the straight sections side1, side 3, side 5, side 7, are circular sections. Sidewall wrinkle magnitude along the straight and circular sections was estimated separately.

130

(Stamping direction) Z A A h

X Side 2 Side 1 Side 3

Y Side 4 Side 8 X

Side 5 Side 7 Side 6

Section A-A

Figure 6.7 Schematic to illustrate the detection of sidewall wrinkle for open die with no bottom face

Ylimit minimum Ylimit maximum

Xlimit maximum

Center of the circular section Xlimit minimum

Figure 6.8 Schematic illustrating the calculation of the sidewall wrinkle in the circular section.

6.5.2.2.1 Estimation of sidewall wrinkle magnitude along a circular section

Along the circular section, the radius from the center for various points in the sides within the limits specified along the X axis and Y axis are calculated as shown in the Figure 6.8. The sidewall wrinkle magnitude is the difference between the maximum radius and minimum radius as given by equation 6.6. 131

wrinklemagnitudeside _ circular = Maximum _ Radius − Minimum _ Radius Equation 6-6 Once the sidewall wrinkle amplitude is determined, the sidewall wrinkle constraint in the optimization program is represented by equation 6.7

wrinklemagnitude wrinkleconstra pg )]([int = side _ circular ≤− 00.1 Equation 6-7 side _ circular ij δ where δ is the tolerable side wall wrinkle magnitude. In the simulations conducted at δ was changed depending on the sides and part geometry

As an input to the optimization program, the center of the circular section, minimum and maximum limits on the X-axis and the Y-axis are required for each circular section. The center, maximum and minimum limits along X and Y axis can be defined as a function of Z coordinates.

6.5.2.2.2 Estimation of sidewall wrinkle magnitude along a straight section

In the estimation of sidewall wrinkle magnitude along the straight section, the deviation of the section from a straight line at various nodal points was calculated as shown in the Figure 6.9. The sidewall wrinkle magnitude is the difference between the maximum Y coordinate to the minimum Y coordinate in the section as given by equation 6-8.

Xlimit maximum

Ylimit maximum Maximum Y coordinate Y

X Minimum Y coordinate Ylimit minimum Xlimit minimum

Figure 6.9 Schematic illustrating the calculation of the sidewall wrinkle in the straight section.

wrinklemagnitudeside _ straight = Maximum Y __ Coordinate − Minimum Y __ Coordinate Equation 6-8 132

Once the sidewall wrinkle amplitude is determined, the sidewall wrinkle constraint in the optimization program is represented by equation 6.9.

wrinklemagnitude wrinkleconstra pg )]([int = side _ straight ≤− 00.1 Equation 6-9 side _ straight ij δ

where δ is the tolerable side wall wrinkle magnitude. In the simulations conducted δ was changed depending on the sides and part geometry.

As an input to the optimization program, minimum and maximum limits on the X-axis and the Y-axis and the angle made by the inclined line with respect to horizontal measured in anti clockwise direction are provided.

6.6 Design variables

The design variables depend on the type of the BHF that need to be estimated. In case the BHF is constant in space/location and time/stroke, the total BHF applied is the design variable therefore, the number of design variable is one. In case the BHF is variable in space/location and constant in space, the design variable is the BHF applied by each cushion pin. Therefore, the number of design variables is equal to the number cushion pins. In case the BHF is constant in space/location and variable in time/stroke, the BHF varying in stroke was defined as a B-Spline curve. Therefore the number of design variables is equal to the number points used to describe the B-Spline curve. In case the BHF is variable in space/location and in time/stroke, the design variables are the forces applied by each cushion pin described as B-Spline curve. Therefore, the number of design variables is equal to the number of cushion pins X number of points used to describe the B-Spline curve.

6.7 Optimization methodology

The optimization problem is to estimate the optimum blank holder force that minimize the thinning in part and avoid wrinkling (constraints).

Minimize: ϕ P)( Objective Function

133

j ,0)( =≤ ,1 mjPg Inequality sConstraint T Subject to: BHFMin < 21 n ],...,,[ < BHFPPP Max Equation 6-10 T D esign ],...,,[ where = 21 PPPP n ],...,,[ Design Variables Number of design variables depend on the type of the BHF to be estimated. According the solution methodology used to solve the optimization problem was different.

134

Start

Initial guess P0 i=0

Evaluate objective function, constraints, gradient of objective function and constraint function Forward difference method FE simulation ∂ϕ ∂g ϕ( 0 ), ( pgP 0 ), j = 1, ...mj j ∂P ∂P

Evaluate the increment in design variable using Newton Raphson method Max{}()Pg i P −=Δ j ∂()Max{}j ()Pg ∂P

Update the design variable i+1 i P =P +ΔP i FE simulation with BHF (P ) i=i+1 i i ϕ( ), j (PgP ) FE simulation with BHF (Pi) i i ϕ( ), j (PgP ) i= i+1 Root of second order i+1 i polynomial equation f(P) p If |Max {gj(p )}| ~= δ YES

NO Second order Polynomial l equation f(P) using Max{gj(p )}, pl, l= i,i-1,i-2

i NO If Max {gj(p )} <0 NO

YES i YES If |Max {gj(p )}| ~= δ Update the design variable Pi+1=Pi-1+0.5ΔP i=i+1 Optimum solution Pi

FE simulation with BHF (Pi) i i END ϕ( ), j (PgP )

Figure 6.10 Flow chart describing the methodology developed to estimate BHF constant in space/location and in time/stroke

135

6.7.1 Optimum BHF constant in space/location and time/stroke/pressure

In sheet metal forming, minimum BHF that prevents the formation of wrinkles would result in minimum thinning in the formed part. Hence, in this one dimensional optimization problem, the optimum BHF corresponds to the BHF that satisfies all the active constraints as equality constraints. Therefore, the defined optimization problem is reduced to the solution of a non linear equation involving active constraints, equation 6- 11. The solution procedure used to solve the problem is shown in Figure 6.10. This solution was obtained using Newton-Raphson (NR) method with line search (polynomial approximation). Line search method was used to avoid divergence of the solution and to avoid cycling between values near the solution. The gradients required for the NR method were estimated by finite difference method. Analytical calculations of gradients by direct differentiation method was not possible because commercial FE software’s (PAMSTAMP 2000, LSDYNA) used in this study restricts the access to the source code. It should also be noted that the initial guess for the optimization should start with lower BHF (outside design space) to avoid saturation of constraints.

{ j } ,0)( == ,1 mjPgMax Equation 6-11 B lank ][ = PP 1 ][ Blank forceholder

6.7.2 Optimum BHF variable in space/location and constant in time/stroke/pressure

In this case, the number of design variables is equal to number of independent cushion cylinders/pins available in the tooling/press to change force in the location/space of the blank holder. Hence, this is a nonlinear multi-dimensional constrained optimization problem with non-equality constraints. Direct method and indirect method are two common optimization techniques available to solve the problem. In indirect method, the constrained non-linear problem is transformed to unconstrained non-linear problem and the solution is obtained through optimization techniques developed for nonlinear unconstrained optimization problem. In the indirect methods, a merit function that comprises of the desired objective function and the constraints is used as objective function. The objective function is penalized every time a constraint is violated during the optimization. Thus in indirect method the search is mostly performed within the feasible region without coming out. Interior penalty method, Exterior penalty method, Lagrange

136 method and Augmented Lagrange method are common indirect methods and they differ mainly by the construction of merit function from the original objective function and constraints. Direct methods solve the optimization problem directly considering the constraints. Sequential Linear programming (SLP), Sequential Quadratic programming (SQP), Method of feasible directions, Generalized reduced gradient method are common direct methods. Among the direct methods, SQP method also known as constrained steepest descent method was used to solve this problem due to its robustness and faster convergence.

The SQP method is similar to gradient based method that involves two steps: a) determination of search direction and b) line search along the direction both uses the constraints. Figure 6.11 show the SQP methodology used to solve this problem. Quadratic programming method was used to obtain the search direction. Line search was performed by polynomial method [Arora 2000, Nocedal et al. 1999, Vanderplast 1984]. Augmented Lagrange function was used as descent function/ merit function to perform line search. Lagrange multipliers obtained from quadratic programming was used to build the merit function/ descent function. The solution is set be converged when the constraint functions are satisfied and magnitude of new direction is negligible indicating that further reduction in objective function (thinning) is not possible without causing wrinkling in the formed part. Hence, in this optimization, the optimum always lies on the boundary of the feasible region which is bounded.

The objective function gradients and constraint gradients required for the QP method were estimated by finite difference method. Analytical calculations of gradients by direct differentiation method was not possible because commercial FE software’s (PAMSTAMP 2000, LSDYNA) used in this study restricts the access to the source code. Thinning gradient for each element in the sheet metal were calculated based on FE results using FDM. The calculated gradients were stored and used in search direction calculation depending on the elements that constitutes the objective function during the optimization

137

Figure 6.11 Flow chart describing the methodology developed to estimate BHF variable in space/location and constant in time/stroke

138

6.7.2.1 Quadratic programming

Determination of search direction is a sub problem in the SQP optimization technique that can be formulated as given by equation 6-12. The original nonlinear objective function was reduced to quadratic function of search direction di. The original nonlinear constraint function was reduced to linear function of search direction di. This is an optimization problem with inequality constraints

2 ∂ϕ 1 T ∂ ϕ Minimize d i )( ϕϕ += d)(P ii + d i d i i = 1....n ∂Pi 2 ∂Pi PI

∂g j j d i )(g += d)(Pg iij =≤ ...1,0 mj Equation 6-12 ∂Pi

iiMin <+< BHFdPBHF Max The inequality constraints could be modified to equality constraints by adding slack variable but this further increased the number of unknowns to be estimated. In sheet metal forming, decrease in BHF (design variable) decreases the thinning (objective function) but increases the potential to form wrinkles (constraints). Hence, in this direction search, atleast one of the inequality constraints would become active (equality constraints) at the solution. Therefore, active set method was used to solve the quadratic programming sub problem [Nocedal et al. 1999]. Determination of search direction by quadratic programming with active set constraint strategy is shown in Figure 6.12. In active set method, the active constraint set, which is a subset of inequality constraints that are active are used to obtain the search direction. The obtained search direction is later checked for any inactive equality constraints for being potentially active. Any new active constraints are added to the subset and solved again for the solution. Projection method was used to consider the limits on the design variable.

The Lagrange function for search direction determination problem is given by equation 6-13.

2 ∂ϕ 1 ∂ ϕ ⎛ ∂g Sj ⎞ dL ,( ) ϕλ += d)(P + d T d + λ ⎜ + d)(Pg ⎟ Si ii i ⎜ iiSSi ⎟ Equation 6-13 ∂Pi 2 ∂Pi Pi ⎝ ∂pi ⎠ s = ofNumber active inequality ,sconstraint ≤ ms The solution to the search direction subproblem must satisfy the necessary first order conditions called the Kuhn-Tucker (KT) conditions given by equation 6-14 139

2 ∂L ∂ ∂ ϕϕ ∂g S = + d + λSi = 0 ∂di ∂Pi ∂Pi Pi ∂Pi ∂L ∂g += d)(Pg Sj = 0 iiS Equation 6-14 ∂λs ∂pi

λs ≥ 0 ,sconstrainty nonequalit active ofNumber s = ofNumber active nonequalit ,sconstrainty ≤ ms

Equation 6-14 reduced to set of simultaneous equations, equation 6-15.

2 ⎡ ∂ ϕ ∂g S ⎤ ⎢ ⎥ ⎡ ∂ϕ ⎤ ∂ P ∂PP ⎡di ⎤ − ⎢ i i i ⎥ = ⎢ ∂P ⎥ ∂g ⎢λ ⎥ ⎢ i ⎥ Equation 6-15 ⎢ S 0 ⎥⎣ s ⎦ )(Pg- ⎢ ⎥ ⎣ iS ⎦ ⎣ ∂pi ⎦

Solution to equation 6-15 is obtained by initially solving for λs followed by di

−1 −1 ⎡ 2 ⎤ 2 ∂g S ⎡ ∂ ϕ ⎤ ∂g S ∂g S ⎡ ∂ ⎤ ∂ϕϕ ⎢ ⎢ ⎥ ⎥[]λs −= ⎢ ⎥ + iS )(Pg ∂ ∂ P ∂pPp ∂ ∂ P ∂PPp ⎣⎢ i ⎣ i i ⎦ i ⎦⎥ i ⎣ i i ⎦ i −1 Equation 6-16 2 ⎡ ∂ ⎤ ⎡ ∂ϕϕ ⎡∂g S ⎤ ⎤ []di = ⎢ ⎥ ⎢− − ⎢ ⎥[]λs ⎥ ⎣∂Pi Pi ⎦ ⎣ ∂ i ⎣ ∂pP i ⎦ ⎦ ∂ϕ ∂g , S are gradients calculated from FDM method using FE simulation and were ∂P ∂pii ∂ 2ϕ known. However (Hessian) was not known. Its calculation by FDM is ∂∂ PP ii computationaly expensive. Hence, it was approximated in the calculations using BFGS method [Arora 2004, Nocedal et al. 1999, Vanderplast 1984]. The solution procedure is described in Figure 6.12. Estimated search direction was checked for any inactive constraints to be active. The search direction was estimated again using equation 6-16, by considering new inequality constraints that become active. The Lagrange multipliers obtained from solving equation 6-16 was also checked for negative values. The constraints corresponding to negative values were dropped from the active set as they were not active. This procedure was repeated until the active constraints were satisfied and the Lagrange multipliers for active constraints were positive thereby satisfying the K- T conditions to get optimum solution for the problem, equation 6-12.

140

Figure 6.12 Flow chart describing the quadratic programming methodology with active constraint set method developed to determine search direction for multidimensional optimization problem with inequality constraints.

141

6.7.3 Optimum BHF constant in space/location and variable in time/stroke/pressure

Two methods were considered to find the solution for this optimization problem. In Method-I, BHF variable in stroke was described using B-Spline curve, Figure 6.13. The control points of the B-Spline constitutes the design variable for the optimization problem. Control points more than one are required to describe the B-Spline curve other than a constant straight line. Therefore, the optimization problem for BHF variable in time/stroke/pressure is multidimensional with non equality constraints as given by equation 6-17.The optimization problem is similar to the BHF variable in space and constant in time with more than one design variable. Hence, the SQP method described in the section 6.7.2 was used to find optimum BHF variable in stroke/time/pressure and constant in space.

p0 p5

p1 p4 Blank Holder Force, BHF Force, Holder Blank p2 p3

Time, t

Figure 6.13 Schematic illustrating the Bspline and control points used to describe the B-Splines

Minimize: ϕ P)( Objective Function Subject to:

j ,0)( =≤ ,1 mjPg Inequality sConstraint T Equation 6-17 ],...,,[ BHFMin < 21 n ],...,,[ < BHFPPP Max T D esign ],...,,[ where = 21 PPPP n ],...,,[ Design Variables, Control points of Bspline curve

142

In method-II, the entire forming process was divided into n steps. The BHF was assumed to vary linearly within each step, Figure 6.14. The BHF required to minimize the thinning and satisfying the constraints were estimated at each step and continued until the stroke was complete to estimate the BHF variable in time and constant in space. In each step, the design variable that needs to be optimized is the blank holder force at the end of the step which is constant in space/location. Hence, a series of one dimensional optimization problem with inequality constraints were solved sequentially until the stroke/time reached the desired value, Figure 6.15. It is assumed that the blank holder force that minimizes thinning at each step would result in minimum thinning in the final part. The methodology described in section 6.7.1 was used to solve the optimization problem at each step.

Figure 6.14 Schematic illustrating the optimization performed in steps to obtain BHF varying in time time/stroke/pressure and constant in space/location

143

Start

Divide stroke/time to n steps

i= 1 to n

Estimate optimum BHF constant in space/ location at time ti / stroke si

i= i+1

End

Figure 6.15 Sequential optimization procedure to estimate BHF varying in time/stroke/pressure and constant in space/location

6.7.4 Optimum BHF variable in space/location and in time/stroke/pressure

Method – I and Method-II as described in section 6.7.3 could be used to obtain the optimum BHF variable in space and in time. In method-I, BHF for each cushion pin/cylinder would be described as B-Spline curve. The number of design variable is product of number of cushion pins/ cylinders and number of control points required to describe each cushion pin curve. This resulted in large number of design variables which significantly increased the computational time and cost. Therefore, only method –II was preferred to estimate the optimum BHF variable in space and time/stroke/pressure. In method-II, as described in section 6.7.3, Figure 6.14, the entire forming process is divided into n steps. The BHF in each cushion pin/cylinder was assumed to vary linearly within the step. The BHF at each cushion pin/cylinder required to minimize the thinning and satisfying the constraints were estimated at each step and continued until the stroke was complete to estimate the BHF variable in time/stroke/pressure and in space/location. In each step, the design variable that needs to be optimized was the blank holder force at the end of the step which is variable in space/location. Hence, a series of multi- 144 dimensional optimization problem with inequality constraints were solved sequentially until the stroke/time reached the desired value, Figure 6.16. It is assumed that the blank holder force that minimizes thinning at each step would result in minimum thinning in the final part. The methodology described in section 6.7.2 was used to solve the multidimensional optimization problem with inequality constraints at each step.

Start

Divide stroke/time/pressure to n steps

i= 1 to n

Estimate optimum BHF variable in space/ location at time ti / stroke si

i= i+1

End

Figure 6.16 Sequential optimization procedure to estimate BHF varying in time/stroke/pressure and in space/location

6.8 Numerical implementation

The optimization methodology to estimate optimum BHF for four cases a) BHF constant in space/location and in time/stroke/, b) BHF variable in space/location and constant in time/stroke, c) BHF variable in time/stroke and constant in space/location, and d) BHF variable in space/location and time/stroke were programmed separately in Language C. The optimization codes interacts seamlessly with commercial FE software’s LSDYNA and PAMSTAMP 2000 to evaluate the objective function, constraint functions, gradient of the objective function and constraint function required for every step in optimization for 145 a given design variable, Figure 6.17. The input to the optimization program consists of the FE simulation input file for PAMSTAMP or LSDYNA for the forming process to be optimized and the input file “CONTROL.FILE” that contains information to calculate objective function and constraints from the FE results for optimization. The execution of optimization program automatically starts FE simulation using the values for the design variables as estimated the optimization program. Once FE simulation is completed, the program calculates the results from FE results, processes the information and starts the next optimization run without any user interface. Thus, there is a seamless interaction between the optimization program and FE simulation software. The optimization program outputs several files. Among the output files, OUTPUT.DAT contains the result. More information about other input and output file can be obtained from the dedicated manuals prepared for the program [Palaniswamy et al. 2004, 2005a, 2005b, 2006]

Figure 6.17 Skeleton of the optimization program and its interaction with user and FEA code PAMSTAMP 2000 and LSDYNA

6.9 Summary and conclusions

In drawing, the blank holder plays a key role in regulating the material flow by exerting a predefined blank holder force (BHF). When designed correctly, this BHF can prevent

146 wrinkling and delay tearing in the formed part. Modern presses are equipped with multipoint cushion systems that allow the BHF to vary with location and during the stroke. Also, tools built with nitrogen cylinders to apply the required force allow the option to change the force between the cylinders/location and during the stroke. However this capability is under utilized in production as the force values needed to program the multipoint cushion systems are not known. Therefore, in this study a methodology to estimate the forces required for programming the multipoint cushion system was developed. Summary and conclusion drawn from this study are: − Four possible means of applying BHF by multipoint cushion system namely a) BHF constant in space/location and time/stroke/pressure, b) BHF variable in space/location and constant in time/stroke/pressure, c) BHF constant in space/location and variable in time/stroke/pressure, d) BHF variable in space/location and time/stroke/pressure, were considered. − Estimation of BHF was modeled as an optimization problem with the objective to minimize thinning in the formed part and avoid wrinkling. − FE simulation of the forming process was used with optimization to provide implicit relationship between the design variable and the objective function, constraint functions required for the optimization. − Newton Raphson method based technique was used for solving the one dimensional optimization problem with inequality constraints to estimate BHF constant in space/location and time/stroke/pressure. − Sequential Quadratic Programming (SQP) based technique was used for solving the multidimensional optimization problem with inequality constraints to estimate BHF variable in space/location and constant in time/stroke/pressure. − The problem to estimate BHF variable in stroke/time and constant in space/location was divided into n steps along the stroke/time of the forming process. Each step was solved sequentially to obtain the BHF variable in stroke/ time. In each step, the optimization problem is similar to BHF constant in space/location and stroke/time. Therefore, the optimization technique developed for BHF constant in space/location and stroke/time was used. − The problem to estimate BHF variable in stroke/time and in space/location was divided into n steps along the stroke/time of the forming process. Each step was solved sequentially to obtain the BHF variable in stroke/ time and location/space. In each step, the optimization problem is similar to BHF variable in space/location 147

and constant in stroke/time. Therefore, the optimization technique developed for BHF variable in space/location and constant in stroke/time was used. − The developed optimization technique for each case of BHF was programmed in C. The optimization program seamlessly interfaces with commercial FE codes PAMSTAMP 2000 and LSDYNA to estimate the required type of BHF profile.

148

CHAPTER 7

PROGRAMMING MULTIPOINT CUSHION SYSTEMS -APPLICATION –

STAMPING

7.1 Introduction

The developed methodology to estimate optimum blank holder force using FE simulation was applied to two parts a) IFU-Hishida die, and b) GM liftgate-inner part, Figure 7.1 formed by stamping process. This study was conducted as part of USCAR project “FLEXIBLE BINDER FORCE CONTROL FOR ROBUST STAMPING” funded by DoE, General Motors (GM), Ford, and Daimler Chrysler to study the advantages of multipoint cushion system in stamping and its effect on stamping die design, manufacturing and die tryout.

Figure 7.1 Schematic of the parts stamped by IFU-Hishida die and GM Liftgate- inner die used in this study.

149

7.2 IFU-Hishida part

7.2.1 Multipoint cushion system

IFU-Hishida die with multipoint cushion system was built by Institute for metal forming technology (IFU), University of Stuttgart, Germany as part of the USCAR project. Figure 7.2 show the schematic of the tooling with the multipoint cushion system. The cushion system consists of 10 hydraulic individual cylinders mounted on the lower die shoe and applies the blank holder force on the blank holder. Each cylinder can be independently controlled thereby all four possible means of changing the blank holder force namely: a) BHF constant in space/location and stroke/time, b) BHF variable in space/location and constant in stroke/time, c) BHF variable in stroke/time and constant in space/location, and d) BHF variable in space/location and time/stroke can be programmed in the cushion system. Each hydraulic cylinder has a maximum stroke of 150 mm and has maximum and minimum force capacity of 210 kN and 8 kN, respectively. The segmented elastic blank holder was used in the tooling to apply the blank holder force on the sheet. The segmented blank holder consists of 10 segments corresponding to each cushion pin, Figure 7.3. Thereby, the each cushion cylinder force can have a local effect on the pressure distribution at the blank holder-sheet interface, and the material flow. The segments were cast on top of thin plate that provides continuity of the blank holder surface. Also, it is flexible enough to deflect during the process depending on the applied blank holder force to ensure a uniform contact with deforming sheet. The die consists of honey comb structure, to provide high stiffness along the drawing direction,Figure 7.4. The punch, die and the blank holder are guided relative to each other and manufactured from grey cast iron GGG70L. The surfaces of the tools are not hardened to have less stiffness for more deflection. Hence the tooling is restricted to use with only aluminum alloys and mild steel. The tooling dimensions of importance for forming analysis is given in Figure 7.5.

150

Tooling Tank unit

Control unit Accumulator

Blank Hydraulic holder control unit Hydraulic cylinder

Figure 7.2 IFU-Hishida die with multipoint cushion system developed by IFU Stuttgart in cooperation with Moog and Hydac [Hengelhaupt et al. 2006].

151

Location of

Guides for cushion pins blank holder

Top view

Thin plate of Isometric View Front view 35 mm Pyramidal shape Figure 7.3 Schematic of the segmented elastic binder/ blank holder (IFU 2003)

Honeycomb

Top view structure with thin Thin plate structure of web (15 mm) 35 mm thickness Figure 7.4 Schematic of the prismatic draw ring (die) with honeycomb structure (IFU 2003)

152

642

R 78 R 78

Punch R 88 295 R 98 415

R 10 Die

100 Punch R 20 Blankholder

Figure 7.5 Dimensions of IFU Hishida die in mm (Hausermann 2000)

7.2.2 Objectives The objectives of this study were • To estimate optimum blank holder force profiles to form IFU hishida part of depth 75 mm from sheet material A5754-O of thickness 1.00 mm, and FU hishida part of depth 85 mm from sheet material BH210 steel of 0.815 mm using the developed optimization methodology. • To validate the estimated optimum BHF profiles through experiments.

7.2.3 FE model

The FE model used for simulating IFU Hishida die drawing process in FE codes PAMSTAMP 2000 and LSDYNA is shown in Figure 7.6. In conventional stamping simulations, the dies are modeled as rigid because of their high stiffness. In this analysis using segmented elastic blank holder and the prismatic draw ring (die), the die was modeled as a rigid object because of its high stiffness due to the honeycomb structure

153 as observed in Figure 7.4. The segmented elastic blank holder was modeled as an elastic object to account for its elastic deflection during forming process that enhances the application of force by multipoint cushion technology compared to conventional method of applying uniform blank holder force. The sheet metal was modeled as an elastic plastic object and the punch was modeled as rigid object as in conventional stamping simulations. The simulation input conditions and material properties used are given in Table 7.1. Sheet materials aluminum alloy A5182-O of thickness 1.15 mm and BH210 steel of thickness 0.815 mm were used in this study. Flow stress of the sheet material A5182 -O and BH210 obtained using bulge test was used in the FE simulation, Figure 7.7. Initial blank shape and dimensions is given in Figure 7.8. Part was drawn to a depth of 75 mm for both the sheet material. Coulomb friction coefficient of 0.08 was used at all the tool-sheet interfaces.

Die

Sheet

Segmented elastic Binder/ Blankholder

Guides for the binder

Punch

Figure 7.6 Schematic of the FE model used for IFU Hishida die drawing simulations

154

Sheet material A5182-O BH210 Initial sheet 1.00mm (0.0398’’) 0.815 mm (0.0383’’) thickness/dimensions Figure 7.8 Figure 7.8 Part Depth 75 mm 85 mm Flow stress (MPa) Refer Figure 7.7 Refer Figure 7.7 Direction (degrees) 0 45 90 0 45 90 Anisotropy 0.68 0.80 0.78 1.475 1.308 1.878 Friction coefficients Punch Blank holder Die with blank interface 0.08 0.08 0.08

Table 7.1 Process conditions and material properties used in FE simulation of IFU Hishida pan deep drawing

800 700 600 BH210 500 400 300 A5182-O

True stress (MPa) stress True 200 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True strain

Figure 7.7 Flow stress of BH210 steel and A5182-O sheet material obtained from bulge test

155

117 128 842

100 100 REF 117 128

Die radius

482 630

100 40 128 REF

175 50

Figure 7.8 Dimensions and shape of the initial blank geometry for the Hishida part

Modeling of the blank holder as an elastic object using the solid elements in the FE simulations increases the inertia of the system substantially compared to conventional methods of modeling blank holder as a rigid body with shell elements. Increase in the system inertia leads to unwanted oscillations in the FE results as the FE software PAMSTAMP and LSDYNA explicitly solves for dynamic equilibrium of the system (Equation 7-1) at every time step.

.. & =++ RKuuCuM Equation 7-1

Where M = Mass matrix of the system, C= Damping matrix of the system, K =Stiffness

.. . matrix of the system, R= Applied forces, u =Acceleration of nodes, u =Velocity of nodes,

u =displacement of nodes.

α += βKMC Equation 7-2

Where α = mass damping coefficient, β = stiffness damping coefficient In conventional stamping simulation using the shell elements, the mass of the system is negligible and the blankholder is considered as a rigid body therefore, only the stiffness- damping coefficient default value provided by software developers was included in the simulation. However, in the simulation with blankholder modeled using brick elements as an elastic object, nodal mass damping coefficient need to added to damp the oscillations. 156

7.2.4 Estimation of nodal damping coefficient

The nodal damping coefficient for the blank holder was estimated by conducting modal analysis/frequency analysis of the blank holder using FE software ABAQUS with similar boundary conditions as in stamping simulations. The boundary conditions used in modal analysis are shown in Figure 7.9. The applied boundary conditions reflect the constraints on the blank holder displacement during the stamping process. The critical nodal damping coefficient was calculated from the lowest frequency using the equation 7-3. Critical damping coefficient for a single degree freedom system is given by

4π qcritical == 2ϖ low Tmax Equation 7-3

max =< 9.0 qqq max

In case of multidegree freedom such as the flexible blankholder, Tmax corresponds to the lowest frequency of the system obtained from the modal analysis or from the response

due to applied load. The lowest frequency of the system (ϖ low ) that doesn’t correspond to rigid body displacement was obtained from modal analysis as 0.47 Hz. Nodal damping factor was chosen as 0.9 times the critical nodal damping coefficient Selection of higher frequencies for damping would result in over damping for low frequencies which would increase the time to converge to quasi static analysis thereby increases simulation time. Figure 7.10 shows the applied force obtained from FE simulation of forming process with and without nodal damping. It could be observed that nodal damping has significantly reduced the oscillations in the applied force at the contact of the sheet with blank holder.

157

Displacement normal to guide surface (Z axis) and rotation along X and Y axis were restricted on guide surface

Displacement along the Z axis and the rotation along X and Y axis were restricted on blankholder surface

Figure 7.9 Description of boundary condition on the blankholder for modal analysis

2500

Binder Contact Force - Undamped 2000 Binder Contact Force Damped

1500

Force (kN) Force 1000 Applied force 500

0 0246810121416 Time (Sec)

Figure 7.10 Comparison of blank holder forces obtained from FE simulation with and without mass damping.

7.2.5 Optimization problem definition

7.2.5.1 Objective:

Minimize thinning in the formed IFU-Hishida part 158

7.2.5.2 Constraints

The flange wrinkles and side wall wrinkles are constraints to this optimization problem. The flange wrinkles are calculated from FE results at the end forming using the methodology described in section 6.5.1. The locations for flange wrinkle calculations are shown in Figure 6.4. The entire flange is divided into 11 zones and is calculated in each zone. Only the maximum flange wrinkle value calculated from the 11 zones was provided as input to optimization program. The side wall wrinkles in the formed part were calculated from the FE results using the methodology described in section 6.5.2.2. In the side wall wrinkle calculation for IFU-Hishida part, a section of side wall was divided into four straight zones and four circular zones as shown in Figure 7.11. The zones were selected such that the constraint gradients are linearly independent. Optimization strategy requires prior knowledge on the tolerable wrinkle amplitude (critical wrinkle amplitude) at the flange and at the sidewall in the FE simulation at which no visible wrinkles would be observed in the experiment. The critical wrinkle amplitudes were estimated by conducting preliminary FE simulation with very high blank holder force.

Side 2 Side 1 Side 3

Side 4 Side 8

Side 5

Side 7

Side 6

Figure 7.11 Locations for side wall wrinkle calculation from FE results for IFU- Hishida part

159

7.2.5.3 Design variables

Design variables depend on the type of the blank holder force being estimated. In this study, the following cases of BHF were estimated: • BHF constant in space/location and time/stroke • BHF variable in space/location and constant in time/stroke • BHF constant in space/location and variable in time/stroke In BHF constant in space/location and time/stroke, all the cushion pins apply the same force. Hence number of design variable was one. In BHF constant in space/location and variable in time/stroke, all the cushion pins apply same force but the force varies in time/stroke. Bspline curve with four control points was used to model the variation of the BHF with stroke in each of the cushion pins. Hence, the design variables were the control points. The number of design variables was four. In BHF variable in space/location and constant in time/stroke, each cushion pin apply different force. Hence, the design variables were individual pin forces and the number design variable for IFU- Hishida tooling was 10. Pin 2 Pin 3 Pin 1 Pin 4

Pin 5 Pin 10

Pin 6

Pin 9 Pin 7 Pin 8 Figure 7.12 Location of cushion cylinders in IFU Hishida part.

7.2.6 Optimization results: Material- A5182-O

7.2.6.1 BHF constant in space and stroke

The methodology described in section 6.7.1 was used to estimate optimum BHF constant in space and time to form IFU-Hishida part of depth 75 mm from A5182-O sheet material. In the optimization, an initial guess of 60 kN was used for each of the 10 160 cylinders. Figure 7.13, Figure 7.14,and Figure 7.15show the evolution of the design variable, objective function, and constraints respectively during optimization. Initial force of 60kN in each cylinder violated all the constraints. Therefore, the force in the cylinders/piston was increased to reduce the constraint values to less than or equal to zero. At the end of fourth iteration, all the constraints were satisfied for the minimum constant force of 140 kN applied by each cylinder. Maximum thinning of ~23 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

160 140 120 100 80 60 40 by each pin (kN) 20

Blank holder force applied 0 01234 Iteration no

Figure 7.13 Evolution of the design variable during the optimization for the estimation of optimum BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material

10 Thinning (%) Thinning 5 Objective function - - function Objective 0 01234 Iteration no

Figure 7.14 Evolution of the objective function during the optimization for the estimation of BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material 161

9 8 Constraint - 1 7 Constraint - 2 6 5 4 3 2

Constraint function Constraint 1 0 -1 0 0.5 1 1.5 2 2.5 3 3.5 Iteration no

Figure 7.15 Evolution of the constraint function during the optimization for the estimation of BHF constant in all the cylinders and in stroke/time for A5182-0 sheet material

7.2.6.2 BHF constant in space and variable in stroke

The methodology described in section 6.7.3 was used to estimate optimum BHF constant in space and variable in stroke to form IFU-Hishida part of depth 75 mm from A5182-O sheet material. The BHF variable in stroke was modeled using Bspline curve of degree one with four control points equally spaced along the stroke. In the optimization, an initial guess of 60 kN was used for each of the 10 cylinders and for each control point. The predicted optimum force was constant at 20 kN in each of the 10 cylinder for initial stroke of 25 mm, Figure 7.16. The force increased to maximum value of 155 kN during stroke from 25 mm to 50 mm. During the last 25 mm, the predicted optimum force decreased to the value of 105 kN in each cylinder. In the optimization, FE simulations indicated that the force applied during initial 25 mm did not significantly influence the thinning or wrinkling in the final formed part. Hence, the force remained low at 20 kN. Figure 7.17, Figure 7.18, and Figure 7.19 show the evolution of the design variable, objective function, and constraints respectively during optimization. Initial force of 60kN in each cylinder and each control point violated all the constraints. Therefore, the force in all the cylinders was increased from stroke beyond 25 mm while the force during the initial stroke of 25 mm was reduced to minimize thinning and satisfy the constraints in 12

162 iterations. Maximum thinning of ~18 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

160 140 120 100 80 60 40 cushion pin (kN) pin cushion

Force applied by each each by applied Force 20 0 0 20406080 Stroke (mm)

Figure 7.16 Optimum BHF constant in all the cylinders and variable in stroke/time estimated by developed optimization methodology for A5182-0 sheet material

Bspline control point -1 Bspline control point - 2 160 Bspline control point - 3 Bsplinecontrol point - 4

140

120

100

pin (kN) pin 60

Blank holder force applied by each applied force Blank holder 0 024681012 Iteration no

Figure 7.17 Evolution of the design variables during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material

163

20 18 16 14 12 10 (%) 8 6 4 2

Objective function - Thinning Thinning - function Objective 0 024681012 Iteration no

Figure 7.18 Evolution of the objective function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material

2 Constraint function Constraint 0 024681012 -2 Iteration no

Figure 7.19 Evolution of the constraint function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for A5182-0 sheet material

164

250

205 200 160 140 150 133 127 122 96 100 70 Pin force (kN)

50 26 27

0 12345678910 Pin no

Pin 2 Pin 3 Pin 1 Pin 4

Pin 5 Pin 10

Pin 6

Pin 9 Pin 7 Pin 8 Figure 7.20 Optimum BHF variable in space/cylinders/pins and constant in stroke/time estimated by developed optimization methodology for A5182-0 sheet material

7.2.6.3 BHF variable in space/location and constant in stroke

The methodology described in section 6.7.2 was used to estimate optimum BHF variable in space and constant in stroke to form IFU-Hishida part of depth 75 mm from A5182-O sheet material. In the optimization, an initial guess of 60 kN was used for each of the 10 cylinders. Optimization predicted higher forces for the cylinders/pins 2, 3, 5, 7, 8, 10 that are located on the straight sides of the die to constraint material flow and avoid formation

165 of wrinkles, Figure 7.20. Lower force were predicted for the pins at the corners, pin 1, pin 4, pin 6, and pin 9 to allow material flow and reduce thinning in the formed part, Figure 7.20. Figure 7.21, Figure 7.22, Figure 7.23 and Figure 7.24 show the evolution of the design variables, objective function, and constraints, respectively during optimization. In the optimization, the initial force of 60kN in each cylinder resulted in violation of all the constraints. Therefore, the force in the cylinders corresponding to wrinkle locations was increased to minimize thinning and satisfy the constraints. 17 iterations were required to obtain the optimum solution. Maximum force was applied by pin 10 to minimize thinning in the adjacent corners and avoid winkling in constraint 8 of the formed part. Maximum thinning of ~17 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

Pin 1 Pin 2 Pin 3 Pin 4 Pin 5 180

160 140 120 100 80 60

40 Blank holder force (kN) 20 0 0 5 10 15 20 Iteration no

Figure 7.21 Evolution of the design variables (pin 1 – 5) during the optimization for the estimation of optimum BHF variable in pins/cylinders/space and constant in stroke/time for A5182-0 sheet material

166

Pin 6 Pin 7 Pin 8 Pin 9 Pin 10 250

200

150

100

Blank holder force (kN) force Blank holder 50

0 0 5 10 15 20 Iteration no

Figure 7.22 Evolution of the design variables (pin 6-10) during the optimization for the estimation of optimum BHF variable in pins/cylinders/space and in constant in stroke/time for A5182-0 sheet material 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Objective function - Thinning (%) Thinning - function Objective 0 0 5 10 15 20 Iteration no

Figure 7.23 Evolution of the objective function during the optimization for the estimation of optimum BHF variable in all the pins/cylinders/space and constant in time/pressure for A5182-0 sheet material

167

14 Constraint - 1 12 Constraint -2 10 Constraint - 3 Constraint - 4 8 Constraint - 5 6 Constraint - 6 Constraint - 7 4 Constraint - 8

Constraint function Constraint 2 Constraint - 9

-2 0 5 10 15 20 Iteration no

Constraint -2 Constraint -1 Side 2 Constraint -3 Side 1 Side 3

SideConstraint 4 -4 Side 8 Constraint -8

Constraint -5 Side 5

Side 7

Constraint -7 Side 6 Constraint -6

Figure 7.24 Evolution of the constraint functions during the optimization for the estimation of optimum BHF variable in all the cylinders/space/pins and constant in stroke/time for A5182-0 sheet material

7.2.6.4 Comparison of thinning distribution

Figure 7.25, and Figure 7.26 show the comparison of the thinning distribution along section X-X and section Y-Y, respectively, predicted by FE simulation for the estimated optimum blank holder forces. Maximum reduction in thinning of 3%, and 6% was observed along section A-A and section B-B , respectively for the BHF varying in space 168 compared to BHF constant in space and stroke. Lower force at corners for BHF variable in space/pins/cylinders allowed better material flow and reduced thinning in the formed part compared to BHF constant in space/cylinder/pins.

Side 2 x Side 1 Side 3

Pin1 Pin2 Pin3 Pin4

Side 4 Side 8 Pin5 Pin10 Pin6 Pin7 Pin9 Pin8 Side 5

Side 7

Side 6 x BHF CONSTANT IN SPACE AND TIME BHF VARIABLE IN TIME AND COSNTANT IN SPACE BHF VARIABLE IN SPACE AND CONSTANT IN TIME 22 20 18 16 4% 14 12 10 8 6

Thinning (%) Thinning 4 2 0 -2 -4 0 200 400 600 800 1000 Curvilinear length (mm)

Figure 7.25 Comparison of thinning distribution predicted by FE simulation along section X-X in the 75 mm deep IFU Hishida pan for the estimated optimum forces.

169

Y Side 2 Side 1 Side 3

Pin1 Pin2 Pin3 Pin4

Side 4 Side 8 Pin5 Pin10

Pin6 Pin7 Pin9 Pin8 Side 5 Side 7 Y Side 6

BHF CONSTANT IN SPACE AND TIME BHF VARIABLE IN TIME AND COSNTANT IN SPACE BHF VARIABLE IN SPACE AND CONSTANT IN TIME 20

2% 15 6%

5 Thinning (%) Thinning

-5 0 200 400 600 800 1000 Curvilinear length (mm)

Figure 7.26 Comparison of thinning distribution predicted by FE simulation along section Y-Y in the 75 mm deep IFU Hishida pan for the estimated optimum forces.

7.2.7 Optimization results: Material - BH210

7.2.7.1 BHF constant in space and stroke

The methodology described in section 6.7.1 was used to estimate optimum BHF constant in space and time to form IFU-Hishida part of depth 85 mm from BH210 sheet 170 material. In the optimization, an initial guess of 100 kN was used for each of the 10 cylinders. Evolution of the design variable, objective function, and constraints during optimization are provided in Appendix I. Initial force of 100kN in each cylinder violated all the constraints. Therefore, the force in the cylinders/piston was increased to reduce the constraint values to less than or equal to zero. At the end of fifth iteration, all the constraints were satisfied for the minimum constant force of 175 kN applied by each cylinder, Figure 7.27. Maximum thinning of ~35 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

200 180 160 140 120 100 80 60

by each pin (kN) pin each by 40 20

Blank holder force applied applied force holder Blank 0 0 20406080100 Iteration no

Figure 7.27 Optimum BHF constant in all the cylinders and in stroke/time estimated by developed optimization methodology for BH210 sheet material

7.2.7.2 BHF constant in space and variable in stroke

The methodology described in section 6.7.3 was used to estimate optimum BHF constant in space and variable in stroke to form IFU-Hishida part of depth 85 mm from BH210 sheet material. The BHF variable in stroke was modeled using Bspline curve of degree one with four control points equally spaced along the stroke. In the optimization, an initial guess of 100 kN was used for each of the 10 cylinders and for each control point. The predicted optimum force in each cylinder increased from 20 kN to 210 KN until the stroke of 57 mm and remained constant at 210 KN until stroke of 85 mm, Figure 7.28. It should be noted that 210 KN is the maximum force each cylinder. Evolution of

171 the design variable, objective function, and constraints respectively during optimization are provided in Appendix I. Initial force of 100kN in each cylinder and each control points violated all the constraints. Therefore, the force in all the cylinders was increased to minimize thinning and satisfy the constraints in 9 iterations. Maximum thinning of ~28 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

250

200

150

100

cushion pin (kN) pin cushion 50 Force applied by each each by applied Force 0 0 20406080100 Stroke (mm)

Figure 7.28 Optimum BHF constant in all the cylinders and variable in stroke/time estimated by developed optimization methodology for BH210 sheet material

7.2.7.3 BHF variable in space/location and constant in stroke

The methodology described in section 6.7.2 was used to estimate optimum BHF variable in space and constant in stroke to form IFU-Hishida part of depth 85 mm from BH210 sheet material. In the optimization, an initial guess of 100 kN was used for each of the 10 cylinders. Optimization predicted higher forces for the cylinders/pins 2, 3, 5, 7, 8, 10 that are located on the straight sides of the die to constraint material flow and avoid formation of wrinkles, Figure 7.29. Lower force were predicted for the pins at the corners, pin 1, pin 4, pin 6, and pin 9 to allow material flow and reduce thinning in the formed part, Figure 7.29. Evolution of the design variables, objective function, and constraints during optimization are provided in Appendix -I. In the optimization, the initial force of 100kN in each cylinder resulted in violation of all the constraints. Therefore, the force in the

172 cylinders corresponding to wrinkle locations was increased to minimize thinning and satisfy the constraints. 17 iterations were required to obtain the optimum solution. Maximum thinning of ~25 % and no flange and side wall wrinkles were observed in the formed part for the optimum BHF.

250 208 205 210 200 182 165 152 146 150 140 120

100

Pin force (kN) 51 50

0 12345678910 Pin no

Pin 2 Pin 3 Pin 1 Pin 4

Pin 5 Pin 10

Pin 6

Pin 9 Pin 7 Pin 8

Figure 7.29 Optimum BHF variable in all the cylinders/space/pins and constant in stroke/time estimated by developed optimization methodology for BH210 sheet material

173

7.2.8 Comparison of thinning distribution

Figure 7.30 and Figure 7.31 show the comparison of the thinning distribution along section X-X and section Y-Y, respectively, predicted by FE simulation for the estimated optimum blank holder forces. Maximum reduction in thinning of 10%, was observed along section A-A and section B-B for the BHF varying in space compared to BHF constant in space and stroke. Lower force at corners for BHF variable in space/pins/cylinders allowed better material flow and reduced thinning in the formed part compared to BHF constant in space/cylinder/pins.

Side 2 x Side 1 Side 3

Pin1 Pin2 Pin3 Pin4

Side 4 Side 8 Pin5 Pin10 Pin6 Pin7 Pin9 Pin8 Side 5

Side 7

Side 6 x

BHF VARIABLE IN STROKE AND CONSTANT IN SPACE 35 BHF CONSTANT IN SPACE AND STROKE BHF VARIABLE IN SPACE AND CONSTANT IN STROKE 30 10% 25

Thinning (%) Thinning 10

-5 0 200 400 600 800 1000 Curvilinear length (mm)

Figure 7.30 Comparison of thinning distribution predicted by FE simulation along section X-X in the 85 mm deep IFU Hishida pan for the estimated optimum forces. 174

Y Side 2 Side 1 Side 3

Pin1 Pin2 Pin3 Pin4

Side 4 Side 8 Pin5 Pin10

Pin6 Pin7 Pin9 Pin8 Side 5 Side 7 Y Side 6

40 BHF CONSTANT IN SPACE AND STROKE 35 BHF VARIABLE IN STROKE AND CONSTANT IN SPACE 30 10% BHF VARIABLE IN SPACE AND CONSTANT IN STROKE 25

Thinning (%) Thinning 10

-5 0 200 400 600 800 1000 Curvilinear length (mm)

Figure 7.31 Comparison of thinning distribution predicted by FE simulation along section Y-Y in the 85 mm deep IFU Hishida pan for the estimated optimum forces.

7.2.9 Experimental validation

7.2.9.1 Material: A5182_O

Figure 7.32 show the formed part using the predicted optimum BHF constant in space/pins and stroke, Figure 7.13 using the tooling, Figure 7.2. BHF constant in space/constant and time resulted in failure at the locations (location 1 and location 2) were maximum thinning was predicted in the FE simulation for estimated optimum BHF

175 constant in space and time/stroke, Figure 7.25,and Figure 7.26. Also, it should be noted that the corners at which fracture was observed in experiment correspond to smallest radius in the punch, Figure 7.5.

Location 1 Location 2 Fracture Fracture

Figure 7.32 Formed IFU-Hishida part of depth 75 mm from sheet material A5182- O sheet material of thickness 1.00 mm using the estimated optimum BHF constant in space/pins/cylinders and in stroke/time

Figure 7.33 show the formed part using the predicted optimum BHF constant in space/pins and variable in stroke, Figure 7.17 using the tooling, Figure 7.2. BHF constant in space/constant and variable in time avoided failure at location 1 but resulted in failure at the location 2. It should be noted that the at location 1, FE simulations predicted that the optimum BHF variable in stroke/time reduced thinning by 2% compared to optimum BHF constant in space/pins and stroke/time, Figure 7.25.

Figure 7.34 show the formed part using the predicted optimum BHF variable in space/pins and constant in stroke, Figure 7.20 using the tooling, Figure 7.2. BHF variable in space/location and constant in time avoided failure at location 1 location 2 but minor wrinkles were observed in the side wall. It should be noted that at both location 1 and location 2, FE simulations predicted that the optimum BHF variable in space/location reduced thinning by 4% compared to optimum BHF constant in space/pins and stroke/time, Figure 7.25, and Figure 7.26, thereby avoiding failure in the experiment/tryout. The wrinkles were eliminated in the tryout by increasing the force in pin 1 and pin 9.

176

Location 2 Fracture

Figure 7.33 Formed IFU-Hishida part of depth 75 mm from sheet material A5182- O sheet material of thickness 1.00 mm using the estimated optimum BHF constant in space/pins/cylinders and variable in stroke/time

Side wall wrinkles

Figure 7.34 Formed IFU-Hishida part of depth 75 mm from sheet material A5182- O sheet material of thickness 1.00 mm using the estimated optimum BHF variable in space/pins/cylinders and constant in stroke/time

7.2.9.2 Material : BH210

Figure 7.35 show the formed part using the predicted optimum BHF constant in space/pins and stroke, Figure 7.27 using the tooling, Figure 7.2. BHF constant in space/constant and time resulted in failure at the locations (location 1 and location 2) were maximum thinning was predicted in the FE simulation for estimated optimum BHF constant in space and time/stroke, Figure 7.30,and Figure 7.31. Also, it should be noted

177 that the corners at which fracture was observed in experiment correspond to smallest radius in the punch, Figure 7.5.

Location 1 Location 2 Fracture Fracture

Figure 7.35 Formed IFU-Hishida part of depth 85 mm from sheet material BH210 sheet material of thickness 0.813 mm using the estimated optimum BHF constant in space/pins/cylinders and in stroke/time

Figure 7.36 Formed IFU-Hishida part of depth 85 mm from sheet material BH210 sheet material of thickness 0.813 mm using the estimated optimum BHF variable in space/pins/cylinders and constant in stroke/time

Figure 7.36 show the formed part using the predicted optimum BHF variable in space/pins and constant in stroke, Figure 7.29 using the tooling, Figure 7.2. BHF constant in space/constant and time resulted in failure at the locations (location 2) towards very end of the stroke were maximum thinning was predicted in the FE 178 simulation for estimated optimum BHF constant in space and time/stroke, Figure 7.31. The fracture was avoided in by reducing the force in pin 1 in tryout to form part without failure and wrinkles.

7.3 Liftgate –inner part

7.3.1 Multipoint cushion system

Figure 7.37 show the schematic of the GM Lift gate tooling used in this study. The tooling consists of upper die, lower punch, inner blankholder, outer blankholder and guide for outer blankholder. The upper die, lower punch and the inner blankholder are made from Zn alloy (Kirksite material) commonly used in making prototype dies for stamping. The outer blankholder is made of cast iron using the segmented elastic blankholder concept developed by IFUM, University of Stuttgart [Hausermann et al. 2002]. The inner and the outer blankholder are attached to the multipoint cushion system [Figure 7.38] that has 26 cylinders. Force applied by each cylinder can be independently adjusted and controlled during the stamping process. Figure 7.39 show the schematic of the whole unit mounted on a mechanical press. During the forming process, the required blankholder force is provided by the multipoint cushion system rather than the cushion in the press.

Upper Die

Lower die / Punch

Outer blankholder /binder

Guide for the outer blankholder/binder

Figure 7.37 Schematic of the GM lift gate die

179

Figure 7.38 Schematic of the 26 point cushion system used for the GM lift gate tooling

Upper Die

Outer blankholder /binder Guide for the outer blankholder/binder

26-pin cushion system

Figure 7.39 Schematic of the press with the 26 point cushion system and the GM lift gate tooling

The die and the punch geometry was designed using the conventional methods for rigid blankholder to form the part initially from A6111-T4 sheet material 1 mm using single point cushions/nitrogen cylinders. However, the lift gate part could not be formed from A6111-T4 material with rigid blankholder despite several days of tryouts by spotting the

180 dies and adjusting the blankholder force in all the cushions. The die geometry for serial production was modified to form the part using 5xxx series aluminum alloy in production with a constant blankholder force in a single point cushion system. Therefore, considering the complexity and difficulty in manufacturing the part from A6111-T4 alloy of 1mm thickness, lift gate die geometry was selected for this study to enhance the formability/drawability and form the part through use of multipoint cushion system. The conventional outer blankholder in the lift gate die was replaced by the segmented elastic blankholder based on the IFU design with multipoint cushion system for this study. The blankholder force required at each of the pins to form the part without any defects need to be estimated. Therefore, in this study, FE simulations coupled with optimization techniques was used to estimate the optimum BHF profiles.

7.3.2 Objectives

The objectives of this study were • To estimate optimum blank holder force profiles to form GM liftgate-inner part from sheet materials A6111-T4 of thickness 1.00 mm and BH210 steel of 0.815 mm using the developed methodology. • To validate the estimated BHF profiles through experiments.

7.3.3 FE simulation of the liftgate-inner stamping process

7.3.3.1 CAD model of the lift gate die

The die geometry was initially manufactured as per the design (CAD model) and repeatedly modified in die tryouts in earlier attempts to form the part from A6111-T4 material of thickness 1 mm with rigid blankholder and single point control cushion system. Therefore, the CAD model for the tooling needs to be updated to reflect the latest modifications in the die. Hence, as part of this study, the tooling (upper die, lower die and the blankholder) surfaces were scanned for the updated geometry and provided as surface patches in *.stl file format to ERC/NSM. Also, the scanned surface was divided into number of sections and the sections were provided as *.igs file to ERC/NSM for accurate construction of FE model (Figure 7.40). The CAD software available at ERC/NSM namely Unigraphics, Ideas, Solid edge and Pro-Engineer could not convert 181 the patches in *.stl file format into surfaces for meshing and to use in the FE simulations. Therefore, the information from the sections of the scanned geometry was used to update the CAD model available to ERC/NSM. The CAD model from sponsor did not have the geometry of the drawbeads in the die. Therefore, the drawbead geometries and its locations were extracted from the sections of the scanned die geometry to update the CAD model for use in the FE simulation.

Upper die

Punch

Blankholder Inner Blankholder outer

Y X

Figure 7.40 Sections of the upper die, punch and the blankholder obtained from the scanned surface of the lift gate tooling for FE simulation 182

7.3.3.2 Extraction of the drawbead geometry and location from the scanned die geometry

Bead 8 Bead 9 Bead 7

Bead 14 Bead 15 Bead 13

Bead 6 Bead 12

Bead 11

Bead 3,4,5 Bead 10 Bead location in inner blankholder Punch boundary Bead 2

Y Bead location in Bead 1 X outer blankholder

Figure 7.41 Schematic illustration of the different types of bead geometry and its location in the lift gate tooling

The geometry (cross section) and the location of the drawbead significantly influence the restraining force on the sheet material during the stamping process. The scanned sections of the die geometry at different locations were analyzed for the bead geometry (cross section). The drawbead geometry in the lift gate die changed with locations. Fifteen different types of draw beads depending on its geometry as shown in Figure 7.41 was observed. The cross section of the drawbead 1 is shown in Figure 7.42. The cross section of the other drawbeads obtained from the scanned sections is provided in Appendix J. These scanned sections for each drawbead were used in the FE

183 simulations to estimate the drawbead restraining force and the drawbead normal force. The estimated drawbead normal force and the restraining force were used in the FE model for the forming simulation of the lift gate part.

Bead in the die Z

Bead in the blankholder

Figure 7.42 Schematic of the drawbead 1 cross section extracted from scanned data of the lift gate tooling die and the blank holder

7.3.3.3 Blank geometry and the material

Figure 7.43 show the schematic of the blank geometry used in the FE simulation. The blank geometry was estimated in tryouts conducted previously at GM for this die geometry using the conventional blankholder and single point cushion system. The blank has a hole/cut out at the center where it is in contact with the inner blankholder. This cutout allows material flow from the hole outwards under the influence of the drawbead and the blankholder force during the stamping process. Two materials 1) Bake hardened steel (BH210) of thickness 0.813 mm, and 2) Aluminum alloy A6111-T4 of thickness 1.0 mm were considered for this study. Blanks of different material and thickness were selected to demonstrate that a) adjusting the individual pin forces in the multipoint cushion system with the segmented blankholder could accommodate inherent variation in the blank thickness and properties in the incoming coil to the stamping plant which currently lead to large scrap rate, and b) draw development of a part for a bake hardened steel commonly used in the past could be used with minor modifications for stamping from new light weight materials such as aluminum alloys (5xxx series and 6xxx series) and AHSS steels (DP steels and TRIP steels) of different sheet thickness thereby saving draw development time. Material properties of the selected sheet materials were obtained from tensile test conducted at U.S.Steel facility and VPB test at ERC/NSM, Palaniswamy et al. 2006.

184

Figure 7.43 Schematic of the blank geometry used in the FE simulation (All dimensions are in inches).

7.3.3.4 Estimation of the drawbead forces using FE simulation.

The drawbeads are characterized by small features and smaller corner radius compared to other features in the stamping tooling. Modeling of the drawbeads in forming simulation with exact geometry in the tooling requires the blank to be discretized with small element size < 1mm to capture the bending strains when the sheet bends and unbends on the smaller radius in the drawbeads. Smaller element size of the sheet significantly increases computational time as the time increment FE calculation advances at each step is proportional to the element size. Therefore, in the stamping simulation, the drawbeads are modeled approximately by replacing the exact geometry 185 in the tooling with a simplified drawbead model represented by a line. The simplified model applies the equivalent forces exerted by the drawbead on the sheet when it passes through the line while straining of the material when it bends and unbends in drawbead is neglected. The simplified draw model requires the drawbead restraining force and the drawbead normal force as an input. Drawbead restraining force is the force acting on the sheet in direction opposite to the direction of material flow to restrict the material flow into the die cavity during stamping. Drawbead normal force is the force acting normal to the bead trying to open the bead when the sheet material flows over the drawbead. The drawbead forces are function of the drawbead geometry and the interface friction conditions. Therefore drawbead forces for all the 15 types of beads in the GM liftgate-inner was estimated by drawbead simulation for use in forming FE simulation with simplified drawbead model.

7.3.3.5 Drawbead simulation-FE model

Figure 7.44 shows the FE model for the drawbead simulation. In the drawbead simulation, the physical shape of the bead obtained from the scanned sections of the upper die and blankholder of the GM liftgate-inner die were modeled. Initially the sheet of width 10 mm was clamped between the upper bead and the lower bead such that the gap between the upper and lower bead at the end of clamping is equal to the initial sheet thickness. The material flow over the bead was simulated by pulling one end of the sheet as shown in Figure 7.44. During the pulling simulation, the force required for pulling and the normal force acting on the upper and lower bead were recorded. The edges of the sheet were fixed in the Y direction to emulate plane strain conditions in the Y direction. The input parameters to the FE simulations are shown in Table 7.2. Pulling FE simulation was conducted with the gap between the upper bead and lower bead equal to a) sheet thickness (1mm) and b) sheet thickness + 1 mm (2 mm) after clamping. The length of the narrow strip of sheet metal for drawbead FE simulation was decided such that the sheet material is not completely pulled out of the drawbead at end of the pulling simulation.

186

Upper bead

Pulling Clearance / direction Sheet Lower bead Gap X Y Initial setup At the end of clamping Pulling Figure 7.44 Schematic of the FE model for the drawbead simulation to obtain the drawbead normal force and drawbead restraining force.

800 700 600 BH210 500 400 300 A5182-O

True stress (MPa) stress True 200 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 True strain

Figure 7.45 Flow stress of the sheet materials A6111-T4 and BH210 steel obtained from bulge test

Sheet Material BH210 A6111-T4 Sheet Thickness 0.815 mm 1 mm Flow stress obtained from bulge test , Material Figure 7.45 Properties: r =1.475, r =1.308, r 0 45 90 r =0.677, r =0.534, r =0.562 =1.878 0 45 90

Table 7.2 Input parameters for liftgate-inner drawbead FE simulation

187

7.3.3.6 Drawbead simulation-FE Results

Figure 7.46 & Figure 7.47 show the drawbead restraining force and the drawbead normal force, respectively obtained from drawbead FE simulation for an example bead geometry for sheet material A6111-T4. The bead force initially increase and remains approximately constant reflecting the steady state deformation of the sheet metal as it bends and unbends over the bead radius during the pulling simulation. The fluctuations observed in the force curves are due to the contact and release of the nodes as the sheet metal slides over the drawbeads. Figure 7.48 shows the comparison of the drawbead restraining force for two different gaps/clearances between the upper and lower bead. Increase in the gaps/clearances decreased the drawbead restraining force due to increase in the effective bending radius of the sheet metal when it passes through the drawbead. Table 7.3 shows the drawbead restraining force and the drawbead normal force for different drawbead geometries of the liftgate-inner die for two different gaps/clearances when sheet material A6111-T4 passes through it. It could be observed that the bead forces vary significantly with the bead geometry. Drawbead forces for sheet material BH210 were estimated by similar procedure and provided in Appendix K.

0.8 0.7 0.6 0.5 0.4 0.3 kN/ 10mm kN/ 0.2

Drawbead normal force normal Drawbead 0.1 0 0 102030405060 Pulling length (mm)

Figure 7.46 Drawbead restraining force obtained from FE simulation for drawbead 1

188

1 0.9 0.8 0.7 0.6 0.5 0.4 (kN/10mm) 0.3 0.2 0.1 Drawbead restraining force 0 0 102030405060 Pulling length (mm)

Figure 7.47 Drawbead normal force obtained from FE simulation for drawbead 1

1 Clearance 1 mm 0.9 (Sheet thickness) 0.8 0.7 0.6 0.5 0.4 (kN/mm) 0.3 Clearance 2 mm 0.2 (Sheet thickness+1 mm) 0.1 Drawbead restraining force force restraining Drawbead 0 0 102030405060 Pulling length (mm)

Figure 7.48 Drawbead restraining force obtained for different clearance from FE simulation for drawbead 1

189

Clearance 1mm (sheet Clearance 2mm (sheet Draw bead thickness) thickness + 1mm) Information Pulling Force Normal Force Pulling Force Normal Force kN/10 mm kN/ 10 mm kN/ 10 mm kN/ 10 mm Bead 1 0.7 0.55 0.55 0.45 Bead 2 0.8 1.1 0.6 0.5 Bead 3 0.5 0.8 0.35 0.6 Bead 4 0.5 0.8 0.4 0.65 Bead 5 0.5 0.8 0.4 0.65 Bead 6 1.1 1.2 1 0.9 Bead 7 0.9 0.7 0.75 0.6 Bead 8 10.850.90.6 Bead 9 0.45 0.45 0.35 0.35 Bead 10 0.7 0.8 0.55 0.5 Bead 11 0.5 0.6 0.4 0.35 Bead 12 0.45 0.4 0.3 0.3 Bead 13 0.45 0.45 0.4 0.4 Bead 14 0.5 0.5 0.4 0.35 Bead 15 0.4 0.4 0.26 0.3

Table 7.3 Drawbead forces for the liftgate-inner drawbeads obtained from drawbead simulation for A6111-T4 sheet material using two different clearances

7.3.3.7 Forming simulation of the lift gate stamping process

The liftgate-inner die was designed for use in a single action press. The sheet is initially placed on the blankholder. The upper die attached to the top ram moves down and initially clamps the sheet against the blankholder then the clamped sheet is formed against the stationary punch during the forming process. The sheet initially placed on the blankholder sags due to its own weight. Therefore, the deformation of the sheet due to gravity needs to be considered to accurately simulate the stamping of the part. Therefore, stamping simulation for the lift gate part was conducted in three stages. Stage I was the gravity simulation where the blank deforms due to gravity after being placed on the blankholder. Stage II was the binder wrap simulation where the blank is clamped between the blankholder and the upper die. Stage III was the forming simulation where the blank is formed against the stationary punch by the motion of the upper die attached to the press ram. The blank holder force plays role only in forming simulation, Therefore, only forming simulation is described here. Gravity stage and holding stage simulation results can be obtained from Palaniswamy et al. 2005c. 190

In the stamping process with the single action tooling, the die attached to the top ram moves down and clamps the sheet against the blankholder followed by forming against a stationary punch. Therefore in the stamping process the die and the blankholder move down while the punch remains stationary. In the FE simulation of the forming process the die and the blankholder are kept stationary while the punch is moved up against the die cavity to form the part thereby reproducing the same relative motion between the tools in a stamping process. In the FE simulation, movement of the die and blankholder was avoided to minimize the effect of inertia force of the solid blankholder on the deformation of the sheet. During the forming process the material is being drawn from the flange into the die cavity. Therefore the drawbeads needs to be considered in this FE model.

The drawbeads are modeled using bar elements in the conventional stamping simulation. The drawbeads are attached to the blankholder to apply the drawbead restraining force on the sheet material when it passes over the bead. Also it applies drawbead normal force on the attached object (blankholder) in the direction opposite to the blankholder force. However, strains in the sheet material due to bending and unbending when it slides over the drawbead are assumed negligible compared to strains in forming and therefore neglected. Commercial FE codes (LS-DYNA and PAMSTAMP) used for stamping simulations only allow the drawbeads (bar elements) to be attached to the blankholder modeled using shell elements. In the current FE model for lift gate, the blankholder was modeled as an elastic object to account for the elastic deflection. Therefore, the simplified drawbead represented by bar elements could not be attached to the solid blankholder. Different FE models were considered in this study to account for the drawbead forces and the elastic deflection of the blankholder. FE model that could model the drawbeads with elastic blank holder was used in the analysis.

Figure 7.49 show the schematic of the final FE model. The model constitutes of the inner blankholder, outer blankholder, sheet, punch, upper die and the drawbeads. The drawbeads modeled using shell elements are placed in between the solid blankholder and the deforming sheet material. In the FE model, contact interface was defined between the drawbead and the sheet to apply the drawbead restraining force on the sheet and drawbead normal force on the drawbead. Also contact interface was defined between the drawbeads and the solid blankholder to transmit the drawbead normal force

191

on the blankholder and to hold the drawbead from displacement due to the drawbead normal force during the stamping simulation. Thus, the effect of the drawbead normal force along with the elastic deflection could be accounted in the FE model. Only a half of the lift gate part was modeled due to symmetry of the die, punch and blankholder geometry. The deformed sheet from the stage II holding simulation with stress and strain history was used as input to the forming FE simulation. Input parameters to the process simulation are provided in Table 7.4. The shell drawbeads were defined with negligible thickness and unit width. Interface friction condition of 0.08 was used between the sheet - blankholder and sheet – die and sheet – punch in the forming simulation. The FE model was damped using the procedure explained in section 7.2.4 to avoid unnecessary oscillations due to high kinetic energy

Die Z Deformed sheet X after holding simulation Y Shell drawbead

Drawbead modeled using shell elements

Inner solid blank holder

Punch Outer solid blank holder

Figure 7.49 Schematic of the FE model for the forming simulation of the lift gate part in PAMSTAMP

192

Sheet Material BH210 A6111-T4 Sheet Thickness 0.815 mm 1 mm Flow stress obtained from bulge test ,Figure 7.45 Material Properties: r0=1.475, r45=1.308, r90 r0=0.677, r45=0.534, r90 =1.878 =0.562 Friction coefficient (μ) 0.08 Inner blankholder Zinc alloy cast (Kirksite) material Elastic material Young’s Modulus=83 GPa, Poisson ratio = 0.32 properties Sheer Modulus = 250 MPa Outer blankholder Grey Cast Iron (SAE G4000) material Elastic material Young’s Modulus =150 GPa, Poisson ratio = 0.29 properties Shear Modulus = 48 GPa

Table 7.4 Input parameters for the liftgate-inner forming FE simulation

Preliminary FE simulation using force of 45 kN in each cushion pin was conducted to check the model. Displacement of drawbeads in the direction of drawbead normal force was not observed indicating that the drawbead forces were accounted during the FE simulation and transmitted to the blankholder. It could be observed that the effective blankholder force acting on the sheet is less than actual blankholder force due to the drawbead force generated opposite to the blankholder force when the sheet is drawn over the drawbead into the die cavity, Figure 7.50. The sum of the drawbead normal force and the effective blankholder force acting on the sheet was equal to the applied force, Figure 7.51 indicating that the final FE model with the elastic blankholder could successfully account for the drawbead normal force.

193

800

700 Force applied by the cushion pin on the outer blankholder 600 Effective blankholder force applied on the sheet by the outer blankholder 500

400

Force (kN) Force 300

200 Drawbead normal force applied on 100 the outer blankholder by the drawbead 0 0 20406080100120 Stroke (mm)

Figure 7.50 Comparison of the force applied by the cushion pin on the blank holder, force applied by the drawbead on the outer blankholder and the force applied by the blankholder on the sheet.

800 Force applied by the cushion pin 700 on the outer blankholder 600

500

400 Effective blankholder force applied on

Force (kN) Force 300 the sheet by the outer blankholder 200 + Drawbead normal force applied on the 100 outer blankholder by the drawbead

0 0 20406080100120 Stroke (mm)

Figure 7.51 Comparison of the force applied by the cushion pin, and sum of force applied by the drawbead on the outer blankholder and the force applied by the blankholder on the sheet.

194

7.3.4 Optimization problem

7.3.4.1 Objective

Minimize thinning in the formed liftgate-inner part

7.3.4.2 Constraints

The flange wrinkles and side wall wrinkles are constraints to this optimization problem. The flange wrinkles are calculated from FE results at the end forming using the methodology described in section 6.5.1. The locations for flange wrinkle calculations are shown in Figure 6.4. The entire flange is divided into 15 zones and is calculated in each zone. The side wall wrinkles in the formed part were calculated from the FE results using the methodology described in section 6.5.2.2. The location for side wall wrinkle calculation is shown in Figure 7.11. The zones were selected such that the constraint gradients are linearly independent. Optimization strategy requires prior knowledge on the tolerable wrinkle amplitude (critical wrinkle amplitude) at the flange and at the sidewall in the FE simulation at which no visible wrinkles would be observed in the experiment. The critical wrinkle amplitudes were estimated by conducting preliminary FE simulation with very high blank holder force.

Location 11 Location 10 Location 12 Location 13

Location 9

Location 8 Location 7

Location 6

Location 5 Location 2 Location 4 Location 3 Location 1

Figure 7.52 Locations for flange wall wrinkle calculation from FE results for liftgate-inner part 195

A B

Y Y

X Z Figure 7.53 Locations for side wall wrinkle calculation from FE results for liftgate-inner part

7.3.4.3 Design variables

Design variables depend on the type of the blank holder force being estimated. In this study, the following cases of BHF were estimated: • BHF constant in space/location and time/stroke • BHF variable in space/location and constant in time/stroke • BHF constant in space/location and variable in time/stroke • BHF variable in space/location and time/stroke In BHF constant in space/location and time/stroke, all the cushion pins apply the same force. Hence number of design variable was one. In BHF constant in space/location and variable in time/stroke, all the cushion pins apply same force but the force varies in time/stroke. Bspline curve with four control points was used to model the variation of the BHF with stroke in each of the cushion pins. Hence, the design variables were the control points. The number of design variables was four. In BHF variable in space/location and constant in time/stroke, each cushion pin apply different force. Hence,

196 the design variables were individual pin forces and the number design variable for liftgate-inner tooling was 15 due to symmetric conditions.

Pin 9 Pin 10 Pin 11 Pin 8 Pin 13 Pin 14 Pin 7

Pin 6

Y Pin 12 Pin 15 Pin 5 X

Pin 4

Pin 3 Pin 2 Pin 1

Figure 7.54 Location of cushion cylinders/pins and their numbering in liftgate- inner part.

7.3.5 Optimization results – Material: A6111-T4

7.3.5.1 BHF constant in space/location and stroke/time

The methodology described in section 6.7.1 was used to estimate optimum BHF constant in space and time to form liftgate-inner part from A6111-T4 sheet material. In the optimization, an initial guess of 30 kN was used for each of the 15 cylinders/pins. Initial force of 30kN in each cylinder violated all the constraints. Therefore, the force in the cylinders/piston was increased to reduce the constraint values to less than or equal to zero. Maximum thinning of ~27 %, Figure 7.56 and no flange and side wall wrinkles were predicted by FE simulation in the formed part for the optimum BHF.

197

60 50

40 (kN) 30

20 10 Force applied at each cushion cushion each at applied Force 0 0 20 40 60 80 100 120 Stroke (mm)

Figure 7.55 Optimum BHF constant in space/location and stroke/time estimated by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm Location B 27 % Thinning

Location A 27 %

X Location C 20 % Y

Figure 7.56 Thinning distribution predicted by FE simulation for the optimum constant blankholder force of 72 kN in each pin

198

7.3.5.2 BHF variable in space/location and constant in time/stroke

The methodology described in section 6.7.2 was used to estimate optimum BHF variable in space and constant in stroke to form liftgate-inner part from A6111-T4 sheet material. Figure 7.57 show the BHF variable in space and constant in time predicted by optimization coupled with FE simulation. Forces in pin1, pin11, pin 14 and pin 15 needs to be multiplied by two before being used in tryout/experiments because only half of the pin was modeled due to symmetric boundary conditions. The force at pin 1, pin 11, pin 14 and pin 15 were less compared to other pins because the material flow near to these pins was close to plain strain conditions and there were less compressive stresses in the flange due to the geometry of the punch/die. Therefore, minimum force required to avoid lifting of blankholder due to the drawbead normal force was sufficient to avoid flange wrinkles as well. Force at pin 2, pin 3 and pin 4 were in the range of 45 kN – 50 kN to allow easy material flow and avoid formation of wrinkles in the two corners close to pin 2, pin 3 and pin4. Among the pins in the sides (pin 5, 6, 7, 8) pin 6 applied higher BHF because the flange region close to pin 6 is subjected to compressive stress and resulted in thickening of the sheet due to the inherent shape of the die compared to flange regions in pin 5, pin 7, pin8. In the inner blankholder pins, pin 12 applied less BHF to reduce thinning in location b (Figure 7.56). However pin 13 applied high BHF to reduce material flow and avoid thinning in the location A (Figure 7.56). Higher BHF at pin 13 resulted in more material flow from the outer blankholder into the die cavity near the region of pin 13. Therefore higher BHF was required in the top pins (pin8, pin 9 and pin 10) compared to bottom pins (pin2, pin 3 and pin 4) to avoid flange wrinkle formation.

Figure 7.58 shows the thinning distribution in the formed part predicted by FE simulation for the estimated optimum BHF. Maximum thinning of ~23 % was predicted for the optimum BHF variable in space/location and stroke/time at location A and location B (Figure 7.58), compared to 27 % (Figure 7.56) for BHF constant in space and time.

199

120

100

Blank holder force (kN) 20

0 123456789101112131415 Pin numbers Pin 9 Pin 10 Pin 11 Pin 8 Pin 13 Pin 14 Pin 7

Pin 6

Y Pin 12 Pin 15 Pin 5 X

Pin 4

Pin 3 Pin 2 Pin 1

Figure 7.57 Optimum BHF variable in space and constant in time predicted by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm

200

Location B 23 %

Thinning Location A 23 % Z Y Y

Z X

Location C 17 % Y

Figure 7.58 Thinning distribution predicted by FE simulation for the optimum blankholder force variable in space and constant in time

7.3.5.3 BHF constant in space/location/pins and variable in stroke/time

The methodology described in section 6.7.3 was used to estimate optimum BHF constant in space and variable in stroke to form liftgate-inner part from A6111-T4 sheet material. The BHF variable in stroke was modeled using Bspline curve of degree one with four control points spaced at stroke of 0mm, 40mm, 65mm, and 100mm. In the optimization, an initial guess of 30 kN was used for each of the 15 cylinders and for each control point. After 6 iterations of optimization, the blank holder force variable in time/stroke and constant in space/location that resulted in minimum thinning satisfying the constraints was obtained (Figure 7.59). Minimum blank holder force of 20 KN was obtained at the punch stroke of ~40 mm. Higher blank holder force of 118 kN was observed at the final stroke of 100 mm. Comparison of the optimum BHF constant in space/location and time/stroke with optimum BHF variable in time/stroke and constant in space/location (Figure 7.59) indicates that lower BHF is required during the initial stroke of 60 mm and higher BHF beyond the stroke of 60mm. Figure 7.60 show the thinning distribution in the formed part for the optimum blank holder force constant in space/location and variable in time/stroke. Maximum thinning of ~25 % was predicted at location A and location B for the optimum BHF constant in space/location and variable in stroke/time compared to 27 % for BHF constant in space/location and in stroke/time. 201

120

100 Optimized BHF constant in 80 space and variable in time

60 Optimized BHF constant (kN) 40 in space and time

20 Initial guess of optimization

Blank holder force in each pin 0 0 20 40 60 80 100 120 Stroke (mm)

Figure 7.59 Comparison of optimum BHF constant in space and variable in time with optimum BHF constant in space and in time estimated by optimization for forming the lift gate part from aluminum alloy A6111-T4 of thickness 1 mm Location B 25 % Thinning

Location A 25 %

Y Z Y

X Z Location C Y 21 %

Figure 7.60 Thinning distribution predicted by FE simulation for the optimum BHF constant in space and variable in time/stroke in each pin

202

7.3.5.4 BHF variable in space/location and in time/stroke

The sequential optimization methodology described in section 6.7.4 was used to estimate optimum BHF constant in space and variable in stroke to form liftgate-inner part from A6111-T4 sheet material. The entire forming sequence was divided into steps as shown in Figure 7.61. Initial step covered a stroke of 40 mm, where no significant change in thinning was observed at locations were maximum thinning was observed in the final formed part, Figure 7.61. Beyond stroke of 40 mm, step size of 5 mm was used. Blank holder force variable in space and stroke predicted by sequential optimization is shown in the Figure 7.62 and Figure 7.63. Beyond the stroke of ~40 mm, the blank holder force increased significantly to avoid flange wrinkle caused due to material flow. In the cushion pins for the inner blank holder, pin 13 applied higher blank holder force to avoid excessive material flow from the inside blank holder thereby avoiding potential risk of failure by tearing at the cutout edge in the blank. Among the pins in the outer blank holder, pin 6 applied higher force because in the flange location near the pin 6, compressive stresses are induced during the material flow due to inherent shape of the die. Lower blank holder force was applied by the pin1, pin 11, pin 14 and pin 15 as the material flow near these pins are close to plain strain and is controlled by drawbeads. Therefore minimum force to oppose the drawbead normal force in this area to keep the die closed was sufficient. Figure 7.64 shows the thinning distribution in the formed part for the optimum BHF variable in space/location and time/stroke estimated by sequential optimization. Maximum thinning of ~23 % was predicted at location A and location B for the optimum BHF variable in space/location and in stroke/time compared to 27 % for BHF constant in space/location and stroke/time.

203

25 Location C Location A Location C 20 Location A 15 Location B Location B 10 Thinning % 5 Step 1 Step n 0 0 20 40 60 80 100 120 Punch stroke (mm)

Figure 7.61 Step size used in sequential optimization to predict the blank holder force variable in space/location and time/stroke for lift gate-inner part

120 Pin 6 100 Pin 8 Pin 7 11 80 10 9 8 13 Pin 3 7 60 15 6 40 14 12 5 2 4

Cushion pin force (kN) Cushion pin force Pin 1 3 20 Pin 4 Pin 1 Pin 2 0 Pin 5 0 20406080100120 Punch stroke (mm)

Figure 7.62 Schematic of the blank holder force variable in space and in time predicted by sequential optimization technique for pin 1 to pin 8

204

120 Pin 13

100 Pin 10

Pin 9 11 80 10 9 8 Pin 11 13 60 15 7 6 40 Pin 12 14 12 5 4 Cushion pin force (kN) 20 Pin 15, Pin 14 Pin 1 2 3

0 0 20 40 60 80 100 120 Punch stroke (mm)

Figure 7.63 Schematic of the blank holder force variable in space and in time predicted by sequential optimization technique for pin 9 to pin 15

Location B 23 % Thinning

Location A 23 %

X Location C Y 17 % Figure 7.64 Thinning distribution predicted by FE simulation for the optimum BHF constant in space and variable in time/stroke in each pin

7.3.6 Optimization results – Material: BH210

Optimum BHF profiles for A6111-T4 material indicated that BHF variable in stroke/time did not significantly improve thinning in the formed part. Also, preliminary experiments indicated by sponsor indicated that the part could not be formed with BHF constant in 205 space and time. Therefore, only BHF variable in space/location and constant in stroke/time was estimated by the developed optimization methodology for sheet material BH210 steel. Figure 7.65 show the BHF variable in space and constant in time predicted by optimization coupled with FE simulation for material BH210. The estimated optimum BHF variation with respect to cushion pins for sheet material BH210 show similar trend as A6111-T4 sheet material except the estimated force are higher compared to A6111- T4 sheet material.

Figure 7.66 shows the thinning distribution in the formed part from BH210 steel predicted by FE simulation for the estimated optimum BHF. Maximum thinning of ~25 % was predicted for the optimum blankholder force at location B. At critical location A & D, maximum of thinning of 22% and 21 % was observed Further decrease in the blankholder force at pin 12 (Figure 7.65) resulted in more material flow to reduce thinning at location B. However, easy material flow from the inner blankholder to the die cavity enlarges the cutout in the blank which induces tensile stresses at the circumference of the cutout and increase the thinning at location A (Figure 7.66). Similarly, decrease in the blankholder force at pin 13 (Figure 7.65) results in more material flow to reduce thinning at location D. However, easy material flow from the inner blankholder to the die cavity enlarges the cutout in the blank which induces tensile stresses at the circumference of the cutout and increase the thinning at location A (Figure 7.66). Hence the predicted blank holder force was optimum that resulted in maximum thinning of ~22.5± 2 % at three different critical locations.

206

140 125 120

100 97 82 78 80 74 76 75 76 70 60 60 60

Force (kN) Force 50 43 40 40 40

0 123456789101112131415 Pin numbers

Pin 9 Pin 10 Pin 11 Pin 8 Pin 13 Pin 14 Pin 7

Pin 6

Y Pin 12 Pin 15 Pin 5 X

Pin 4

Pin 3 Pin 2 Pin 1

Figure 7.65 Optimum BHF variable in space and constant in time predicted by optimization coupled with FE simulation for forming the lift gate part from BH210 steel of thickness 0.81 mm

207

Location B ~25 %

Thinning Location A ~20 %

X Location D ~22 % Location C ~16 % Y Figure 7.66 Thinning distribution predicted by FE simulation for the optimum blankholder force variable in space and constant in time for BH210 sheet material

7.3.7 Experimental validation

Experiments/tryouts to form the liftgate inner part were conducted at Troy Design and Manufacturing (TDM) facility at Detroit using the multipoint cushion system and the flexible blank holder in a mechanical press, Figure 7.39. Optimum BHF profiles estimated by FE simulation coupled with optimization techniques were used in the tryout to form the liftgate-inner part and validate the FE predictions. Experiments conducted with BHF constant in space resulted in failure of the part indicating the liftgate-inner part could not formed with use of BHF variable in space. Hence, experimental results for BHF profiles variable in space are presented in this section.

7.3.7.1 Material : A6111-T4

7.3.7.1.1 BHF variable in space /location and constant in stroke/time

Figure 7.67 show the schematic of the formed part with the optimum BHF variable in space and constant in stroke predicted by FE simulation coupled with optimization. The

208 part was formed without splits. However minor wrinkles were observed in the flange geometry beyond the drawbeads on top and bottom and on one of the corners as shown in Figure 7.67.

Minor wrinkles

Figure 7.67 A6111-T4 liftgate-inner part formed in the tryout using the optimum BHF variable in space and constant in time predicted by FE simulation coupled with optimization

7.3.7.1.2 BHF variable in space /location and in stroke/time

Figure 7.68 shows the schematic of the formed part with the optimum BHF variable in space and variable in stroke predicted by FE simulation coupled with optimization. The part was formed with split towards the end of the forming process at the location were maximum thinning was observed in the FE simulation, Figure 7.66. Excessive material flow from the inner blank holder force resulted in opening of the cut out in the blank, which causes tensile stress at the cutout edges leading to failure. Experiment/tryout was repeated by increasing the pin force from 110 kN to 125 kN at the end of the forming process (Figure 7.69). Modified force at pin 13 resulted in good part with minor wrinkles observed in the flange geometry beyond the draw beads at top and bottom and on one of the corners as shown in Figure 7.70.

209

Figure 7.68 A6111-T4 liftgate-inner part formed in the tryout using the optimum BHF variable in space and in time predicted by FE simulation coupled with optimization

140 Predicted force at Pin 13 120 Modification to predicted force at 100 Pin 13 80

60 Force (kN) Force 40

0 0 20 40 60 80 100 120 Stroke (mm)

Figure 7.69 Comparison of the predicted force from optimization coupled with FE simulation and the modified force in tryout for pin 13 to form the part from A6111-T4 210

Minor wrinkles Figure 7.70 A6111-T4 part formed in tryout using the modified optimum BHF variable in space and in time at pin 13

7.3.7.2 Material : BH210 steel

Figure 7.71 show the formed part with the optimum BHF variable in space and constant in stroke predicted by FE simulation coupled with optimization. The part was formed without splits and wrinkles without any modifications to predicted forces.

7.3.7.3 Comparison of FE results with the experiments

Parts formed in the tryouts were cut along three sections (section A, section B and section C) as shown in Figure 7.72. Thinning in the part along the cut sections were measured using micrometer. The measured thinning was compared with the FE predictions for sheet material A6111-T4, and BH210 steel.

211

Figure 7.71 The formed part in tryout using the optimum blankholder force variable in space and constant in time predicted by FE simulation coupled with optimization for BH210 steel

212

A B

A Figure 7.72 Location of the sections in the formed part for comparison of the thinning distribution from experiment with FE predictions

7.3.7.3.1 Material : A6111-T4

Thinning from the FE simulations for the predicted optimum blankholder force variable in space and stroke were compared with a) experiment that used the same force and resulted in split panel (Figure 7.68) and b) experiment with modified force at pin 13 that resulted in split free panel (Figure 7.70). Figure 7.73, Figure 7.74, and Figure 7.75 shows the comparison of the FE predictions with the experiments for the split and split free panel along section A-A, Section B-B and section C-C, respectively. It could be observed that the thinning distribution along section A-A were identical for both split and split free panel (Figure 7.73) as the only difference in forming these two panels was the change in the blankholder force at pin 13 which is far away from section A-A. Thinning predicted by FE simulation along section A-A agrees with the experiments (Figure 7.73). Maximum thinning along the section A-A in the experiments was less (12 %) compared to the maximum thinning of 15% predicted by FE simulation. The difference in thinning could be due to the difference in geometry (radius) between the actual physical die,

213 which was spotted before to form the part without multipoint cushion system and the FE model along the section A-A.

20 Experiment-Split_free 15 Experiment-Split FE Simulation 10

Thinning % Thinning 0 0 100 200 300 400 500 600 700 800 900 -5

-10 Curvilinear length [mm]

Figure 7.73 Comparison of the thinning distribution along section A-A from FE simulation with experiments using FE predicted BHF that resulted in split and modified BHF that resulted in split free panel for A6111-T4 sheet material

Along the section B-B, the thinning distribution in split panel and panel without panel was same (Figure 7.74) as the increase in the blankholder force at pin 13 does not affect the material flow in the section B-B. Thinning predicted by FE simulation along section B-B agrees with the experiments. Maximum difference between FE simulation and experiment was observed at curvilinear length of 50 mm to 100 mm (Figure 7.74) where the material flows over the draw bead during the forming process. In the FE simulation, the material flow over the draw bead is approximately simulated by equivalent forces while the strain/thinning is not considered which could have resulted in lesser thinning in FE simulation compared to the experiment.

214

20 Experiment-Split free 15 Experiment-Split FE Simulation 10

Thinning % Thinning 0 0 100 200 300 400 500 -5

-10 Curvilinear length (mm)

Figure 7.74 Comparison of the thinning distribution along section B-B from FE simulation with experiments using predicted optimum BHF that resulted in split panel and modified BHF that resulted in split free panel for A6111-T4 sheet material

Along the section C-C, the thinning distribution in split free panel was slightly higher compared to split panel (Figure 7.75) as the increase in the blankholder force at pin 13 reduced the material flow and increased thinning. Thinning predicted by FE simulation along section C-C agrees with the experiments. Maximum difference between FE simulation and experiment was observed at curvilinear length of 25 mm to 75 mm and 375 mm to 425 mm (Figure 7.75) where the material flows over the draw bead during the forming process. In the FE simulation, the material flow over the draw bead is approximately simulated by equivalent forces while the strain/thinning is not considered which could have resulted in lesser thinning in FE simulation compared to the experiment. Maximum thinning of 18 % was predicted by FE simulation compared to

215

14 % in the experiments at the curvilinear length of 300 mm. This could be due to the difference between the geometry between the physical die and FE model at the die corner at curvilinear length of 300 mm.

20 Experiment_Split Experiment_Nosplit 15 Simulation

Thinning % Thinning 0 0 100 200 300 400 500 600 -5

-10 Curvilinear length (mm)

Figure 7.75 Comparison of the thinning distribution along section C-C from FE simulation with experiments using predicted optimum BHF that resulted in split panel and modified BHF that resulted in split free panel for A6111-T4 sheet material

7.3.7.3.2 Material : BH210

Thinning from the FE simulations for the predicted optimum BHF variable in space and constant stroke were compared with experiments that resulted in split free panel with no

216 wrinkles (Figure 7.71). Figure 7.76, Figure 7.77, and Figure 7.78 shows the comparison of the FE predictions with the experiments along section A-A, Section B-B and section C-C, respectively. Thinning predicted by FE simulation along section A-A , B-B and C-C agrees with the experiments (Figure 7.76). The thinning distribution in BH210 liftgate- inner part showed similar trend as liftgate-inner part formed from A6111-T4.

16 FE Simulation 14 Experiment/Tryout 12 10 8 6 4

Thinning % Thinning 2 0 -2 0 200 400 600 800 1000 -4 Curvilinear length (mm)

Figure 7.76 Comparison of the thinning distribution along section A-A from FE simulation with experiments using predicted optimum blankholder force for BH210 sheet material

217

14 12 FE Simulation 10 Experiment 8 6 4

Thinning % Thinning 2 0 -2 0 100 200 300 400 500 600 -4 Curvilinear length (mm)

Figure 7.77 Comparison of the thinning distribution along section B-B from FE simulation with experiments using predicted optimum BHF for BH210 sheet material

218

18 16 Experiment 14 FE Simulation 12 10 8 6

Thinning % Thinning 4 2 0 -2 0 100 200 300 400 500 600 -4 Curvilinear length (mm)

Figure 7.78 Comparison of the thinning distribution along section C-C from FE simulation with experiments using FE predicted optimum BHF for BH210 sheet material

7.4 Summary and conclusions

In stamping, the applied BHF control material flow and significantly influence the characteristics of the formed part. When designed correctly, this BHF can prevent wrinkling and delay tearing in the formed part. Introduction of multipoint cushion system allows changing the BHF in space/location and during the forming process, thereby, enhancing the drawability of the part. However, this capability is underutilized in production as the force required to program the multipoint cushion system is not known 219 and difficult to estimate by trial and error. Therefore, in this study, the developed optimization methodology was used to estimate optimum BHF required to program multipoint cushion system for forming a) IFU-Hishida part and b) GM liftgate-inner part by minimizing thinning in the formed part without wrinkles. The estimated blank holder force profiles for IFU-Hishida die and GM liftgate-inner part were later validated using experiments. The summary and conclusions drawn from this study are

7.4.1 IFU-Hishida die • Developed optimization routine was used to estimate a) BHF constant in space/location and time/stroke, b) BHF variable in space/location and constant in time/stroke, and c) BHF constant in space/location and variable in time/stroke to form IFU- Hishida part of depth 75 mm from A5182-O sheet material of thickness 1.00 mm, and IFU- Hishida part of depth 85 mm from BH210 sheet material of thickness 0.815 mm. • Optimum BHF variable in space/location resulted in significant (6% - 10%) improvement in maximum thinning of the formed part compared to BHF constant in space /location and stroke/time for both the materials. • BHF variable in space applied less force at the corners resulting in more material flow and reduction in thinning in the formed part. Also, higher BHF was applied at straight sides to avoid wrinkle in the formed part. • Optimum BHF constant in space/location and variable in time/stroke resulted in modest (3% - 5%) improvement in maximum thinning of the formed part compared to BHF constant in space /location and stroke/time for both the materials. • Estimated optimum BHF profiles were used in the experiments to validate the optimization methodology and FE predictions. • Optimum BHF constant in space/location and stroke/time resulted in failure at two locations in the formed part for both A5182-O sheet material and BH210 sheet material indicating that it is not possible to form the part with BHF constant in space and stroke as applied conventionally. • Optimum BHF constant in space/location and variable in stroke/time resulted in failure in one location in the formed part for A5182-O sheet material indicating modest improvement.

220

• Optimum BHF variable in space/location and constant in stroke/time resulted in no failure in the formed part but with minor wrinkles for A5182-O sheet material indicating the significance of multipoint cushion system in improving the drawability of the formed part. • Optimum BHF variable in space/location and constant in stroke/time resulted in failure towards very end of the stroke in the formed part from BH210 sheet material. Force applied by one of the pin near the fracture location was modified from the predicted value in tryout to form a good part without failure. This indicated the significance of multipoint cushion system in improving the drawability of the formed part.

7.4.2 GM liftgate-inner part

• Developed optimization routine was used to estimate a) BHF constant in space/location and time/stroke, b) BHF variable in space/location and constant in time/stroke, c) BHF constant in space/location and variable in time/stroke, and d) BHF variable in space/location and time/stroke to form GM liftgate-inner part from A6111-T4 sheet material of thickness 1.00 mm, and BH210 sheet material of thickness 0.815 mm. • Optimum BHF variable in space/location resulted in significant (4% - 5%) improvement in maximum thinning of the formed part compared to BHF constant in space /location for A6111-T4 sheet material. • BHF variable in stroke/time did not result in significant improvement in maximum thinning of the formed part from A6111-T4 sheet material. • Estimated optimum BHF profiles were used in the experiments to validate the optimization methodology and FE predictions. • Optimum BHF constant in space/location and stroke/time resulted in failure in the formed part for both A6111-T4 sheet material and BH210 sheet material indicating that it is not possible to form the part with BHF constant in space and stroke as applied conventionally. • Optimum BHF constant in space/location and variable in stroke/time resulted in failure in the formed part for A6111-T4 sheet material. • Optimum BHF variable in space/location and constant in stroke/time resulted in no failure in the formed part and with minor wrinkles for A6111-T4 and BH210

221

sheet material indicating the significance of multipoint cushion system in improving the drawability of the formed part. • Optimum BHF variable in space/location and in stroke/time resulted in failure towards very end of the stroke in the formed part for A6111-T4 sheet material. Estimated profile was modified in one of pins tryout to form a good part without failure. This indicated that the significance of multipoint cushion system in improving the drawability of the formed part. • Thinning distribution in the formed part was measured along three sections and compared with FE predictions. FE predictions agreed well with experiments.

222

CHAPTER 8

PROGRAMMING SINGLEPOINT CUSHION SYSTEM - APPLICATION –

SHEET HYDROFORMING WITH PUNCH (SHF-P)

8.1 Introduction

In the SHF-P process, the sheet is deep drawn against a counter pressure in the pot, as shown in Figure 8.1, rather than a female die as in a regular stamping operation. The medium in the pressure pot can be either “passive” (pressure generated due to the incompressibility of the medium during forward stroke of the punch) or “active” (pressure generated by external pump). The fluid pressure acting on the sheet metal results in a high friction force at the sheet – punch interface that prevents stretching of the sheet material after it comes in contact with the punch. Therefore, the sheet metal in the cup wall is not stretched as much in the SHF-P process, compared to stamping, resulting in a more uniform wall thickness and higher LDR. Elimination of the female die results in lower tool cost and lower die development time. Furthermore, elimination of sidewall wrinkles during the forming process, due to external fluid pressure, allows more freedom in designing auto body panels. The ability to form complicated shapes and features results in fewer forming operations compared to conventional stamping, which reduces manufacturing costs. SHF-P processes can also be combined with regular stamping operations to reduce forming stages [Siegert et al. 1999, Maki, 2003].

223

Fbh Fp Fbh Punch Blank holder

Sheet P Pressure pot

Fbh + Fp

a) Process description [Aust 2001] b) Example parts [Maki 2003], [Aust 2001]

Figure 8.1 Schematic of the SHF-P process and example automotive parts manufactured using SHF-P

8.2 Process limits, defects and process window in SHF-P process

The process window is an area where the SHF process can be performed without any defects in the formed part and is dependent on the part geometry, Figure 8.2. The boundaries of the process window are the process limits. The two process parameters that are used to define the process limits are the pressure versus stroke and the blank holder force versus stroke curve. Fracture and wrinkling in the part are the common defects in the part manufactured by SHF-P process, similar to conventional deep drawing. Flange wrinkling occurs when the blank holder force is too small. While side wall wrinkle can be attributed to insufficient pressure, nature of punch geometry (tapered walls) and due to excessive flange wrinkle. Fracture occurs after excessive thinning due to high blank holder force and/or insufficient pressure in the pot (see point A in Figure 2- 6). Fracture can be avoided/postponed by increasing the pot pressure (moving from A to A” in Figure 2-6) or by decreasing the blank holder force (moving from A to A’ in Figure 2-6). Leaking of the pressurizing medium and bursting against the drawing directions are unique possible modes of failure in the SHF-P process. Leaking occurs when the blank holder force was not sufficient resulting in wrinkling in the flange followed by leaking of the medium (see point B in Figure 2-6). Also leaking results in reduction of fluid pressure in the pot, which could result in fracture of the part. Leaking can be avoided by increasing the blank holder force (moving from B to B” in Figure 2-6) or by reducing the pressure in the pot (moving from B to B’ in Figure 2-6). Also, sealing is used to reduce the possibility of leaking during the SHF-P process. Bulging against the drawing direction

224 occurs due to excessive fluid pressure. This is usually observed in parts with tapered walls. Bulging can be avoided by decreasing the pressure (moving from C to C’ in Figure 2-6). Thus, in order to obtain the loading path (relationship between the pressure versus punch stroke and blank holder force versus punch stroke) to form a part, several trial and error experiments or simulations have to be conducted. This is a time consuming process and might not result in a feasible loading path to form a part. Any technique, which reduces the number of iterations in the trial and error process, would considerably save time and money. In this study, the developed optimization techniques coupled with FE simulation was used to estimate the optimum loading path for forming round cup part geometry. This study was done in cooperation with Schnupp Hydraulic, Germany

Fracture due to Fracture due to Low pressure+ Fracture due to High BHF 4 High BHF High pressure + 8 High BHF D

2 Bulge against D’ A drawing A” C’ C direction

A’ 7 5

Blank Holder Force [BHF] Force Holder Blank B” 6 B’

B Leaking and Wrinkling

Pressure in the Pot

Figure 8.2 Process window for the SHF-P process

225

8.3 Objectives

The specific objectives of this study are • Estimate the optimum blank holder force and the pot pressure versus stroke of the punch to successfully form an axisymmetric part – 90 mm diameter round cup of depth 105 mm.

• Verify the predicted optimum process parameter profiles using experiments at Schnupp Hydraulic, Germany.

8.4 Experimental setup

Maximum punch force 800 kN

Maximum blank holder force 4 x 100 kN

Maximum water pressure 400 bar

Maximum punch stroke 500 mm

Maximum blank holder stroke 250 mm

Shut height 395 mm

Area of press bed 580 mm X 450 mm

Table 8.1 Technical specifications of the SHF-P press at Schnupp Hydraulik

8.4.1 Press

Figure 8.3 show the schematic of the double action press at the Schnupp Hydraulic GmbH, Germany facility used for SHF-P experiments. Technical specifications of the press are given in Table 8.1. In the double action press, punch is attached to the inside ram and the blank holder is attached to the outer ram. A hydraulic cylinder actuates inside ram independently while four hydraulic cylinders actuates the outer ram. The four hydraulic cylinders in outside ram are independently controlled thereby allowing the user to change blank holder force in space and stroke during the SHF-P process. The pressure pot is attached to the press bed. Separate hydraulic circuit controls the pressure inside the pot during the process. The closed loop control system for the blank holder force and the pot pressure helps the press to trace the user specified pressure and blank holder force profile as a function of stroke during the SHF-P process. The

226 pressure and the blank holder force profiles can be programmed as a step function and/or as a linear function of stroke.

Punch cylinder

Blankholder cylinders

Blankholder

Pressure Pot

Press control unit

Hydraulic control unit

Figure 8.3 Schematic of the 80 ton hydraulic press used for experiments at Schnupp Hydraulic GmbH, Germany

Draw-ring

Pressure Pot

Figure 8.4 Schematic of the die ring and pressure pot for forming 90mm diameter cup

227

8.4.2 Tooling

The tooling consists of upper die that constitutes the punch and the blank holder and the lower die includes the pressure pot and the draw ring. The draw ring is attached mechanically to the pressure pot as shown in Figure 8.4. Sealing is placed between the pressure pot and the die ring to avoid leakage of the medium at the interface. It should be noted that there is no additional sealing on the draw ring surface that comes in contact with the blank during the forming process. The tools were made from non- hardened tool steel material to form 90 mm diameter round cups.

8.5 Optimization problem

8.5.1 Objective

Minimize thinning in the formed round cup

8.5.2 Constraints

The flange wrinkles are constraints to this optimization problem. The flange wrinkles are calculated from FE results at the end forming using the methodology described in section 6.5.1. The flange wrinkle is calculated over entire area of the flange. Side wall wrinkles were not considered as the fluid pressure outside the sheet prevents occurrence of side wall wrinkles during forming.

8.5.3 Design variables

Design variables depend on the type of the process parameter being estimated. In this study, BHF and fluid pressure constant in space/location and variable in time/stroke was estimated. Therefore, the number of design variable is two.

8.6 FE model

Figure 8.5 shows the schematic of the FE model. The die, blank holder and pressure pot were considered rigid while the blank was modeled as anisotropic elastic plastic material. Only a quarter model was used in the FE simulation due to the axisymmetric 228 deformation and boundary conditions, and the symmetric material properties of the sheet metal. Table 8.2 shows the various input parameters used in the process simulation. The flow stress of the material obtained from VPB test conducted at ERC/NSM was used in the simulation ,Figure 8.6. The anisotropy constant for the sheet material st14 was obtained from Technical university of Munich. The Coulomb friction coefficient between sheet - blank holder and sheet –die was set to 0.06 in the FE simulation, as thin elastomer foil was used as a dry film lubricant in experiment. In the FE simulation, interface friction coefficient of 0.12 was used at the punch - sheet interface, as there was no lubricant used in the process. It should be noted that higher friction at the punch sheet interface enhances the SHF-P process. In the FE model, the hydraulic pressure acting on the sheet during the process was modeled using the Aquadraw interface available in the FE code PAMSTAMP 2000.

Punch

Blank holder

Shoulder

Sheet

Pressure pot

Figure 8.5 Schematic of the FE-Model for the SHF-P process

229

Initial blank geometry Ø 230 mm, thickness 1.00 mm

Blank material St14 steel

Flow stress: Bulge test (Figure 8.6) Material properties r0 = 912.1 ,83r45 = .1 ,14r90 = .2

Punch/sheet = 0.12

Friction coefficient Die/sheet =0.06 Blank holder/sheet = 0.06

Table 8.2 Input parameters used in the FE simulation of round cup hydroforming process

700 Bulge test data 600

500

400

300 Bulge test data compensated for anisotropy

True Stress [MPa] 200

100

0 0 0.2 0.4 0.6 0.8 True Strain

Figure 8.6 Flow stress of st14 steel obtained from the VPB test.

8.7 Optimization results

The optimum pressure versus stroke and blank holder versus stroke curve required to form the round cup was estimated using the sequential optimization procedure described in section 6.7.4. During the initial stroke of ~ 10 mm the sheet metal bends and warps around die corner radius. Any increase pot pressure only stretches the material and 230 further increases the thinning. Hence, the pot pressure is nearly zero at the start of forming process. During the stroke of 10 mm – 60 mm, forming pressure increases to wrap the sheet against the punch. Beyond the stroke of 60 mm, the pot pressure is constant at 400bar due to limitation of the press and tooling design. During the initial stroke of 60 mm, the BHF was maintained constant as 5tons is the minimum force the blank holder can apply. BHF increased towards the end of the forming process to restrain the material flow and avoid excessive thickening, which would otherwise result in flange wrinkles and leakage of the pressurizing/forming medium.

450 450 400 Pressure 400 350 350 300 300 250 Blank Holder Force 250 200 200 150 150 Pressure [bar] Pressure 100 100 Blank holder force [kN] force holder Blank 50 50 0 0 0306090120 Stroke [mm]

Figure 8.7 BHF and forming pressure varying with punch stroke estimated by optimization coupled with FEM.

8.8 Experimental validation

The predicted BHF and forming pressure varying with punch stroke were used in the experiments conducted at Schnupp Hydraulik, Germany. The part was formed using the predicted optimum blank holder force with negligible amount of leakage of the medium, no wrinkling and no split (Figure 8.8).

231

Figure 8.8 Picture of the deep drawn samples using the optimized pressure and BHF curve from FE simulations

Blank holder force Blank holder force cylinder 3 cylinder 4

Blank holder force Less material flow (draw-in) Blank holder force cylinder 1 cylinder 2

Figure 8.9 Schematic of the non uniform material flow (drawin) observed in the formed cup

The material flow was not symmetric about the axis of anisotropy. Less material flow (drawin) was observed close to the blank holder force cylinder 1 compared to other

232 locations in the flange. In the experiments, it was observed that the blank holder force cylinder 1 always applied slightly higher blank holder force compared to other cylinders as measured by the press measurement system, Figure 8.9. Also, the force from the blank holder force cylinder 1 was fluctuating about the desired value. This could be the possible reason for the less material flow (drawin) observed near the blank holder force cylinder 1.

8.8.1 Comparison of thinning distribution

Thinning distribution along section A-A and section B-B, Figure 8.10 were measured from the experiment and compared with FE predictions. FE simulation predictions agree well with the experimental measurements, Figure 8.11, Figure 8.12.

Blank holder force Blank holder force cylinder 3 B cylinder 4

Blank holder force cylinder 2 Blank holder force cylinder 1

Figure 8.10 Location of sections for comparison of FE predictions with experiments

233

155mm 170mm 15 Experiment 10 140mm 5

-5 FE simulation Thinning [%] -10 50mm

-15 0mm 40mm

-20 0 50 100 150 Curvlinear Length [mm]

Figure 8.11 Comparison of thinning distribution from experiment and FE simulation along section A-A for optimized pressure and BHF profile

20 155mm 170mm Experiment 15 10 140mm 5 0 -5 -10 FE simulation 50mm

Thinning [mm] Thinning -15 -20 0mm 40mm -25 -30 050100150 Curvelinear Length [mm]

Figure 8.12 Comparison of thinning distribution from experiment and FE simulation along section B-B for optimized pressure and BHF profile.

8.9 Summary and conclusions

In sheet hydroforming process with punch (SHF-P), the relationship between the blank holder force and the fluid forming pressure with respect to punch stroke significantly influences the characteristics of the formed part. Conventionally, fluid pressure and

234 blank holder force is maintained constant during the forming process. However, variable blank holder force and fluid pressure allows better material flow during the forming process, thereby enhancing the drawability of the part. Therefore, in this study the developed optimization methodology was used to estimate optimum process parameters to form a round cup with minimum thinning and no wrinkling. The estimated blank holder force and fluid pressure profiles were later validated using experiments at Schnupp Hydraulic, Germany. The summary and conclusions drawn from this study are

• FE simulation coupled with developed optimization routine to estimate optimum BHF and forming pressure constant in location/space and variable in stroke/ time to form 90 mm diameter round cup of depth 105mm with minimum thinning and no wrinkling from st14 sheet material.

• Optimum process parameter resulted in maximum thinning of 15% in the formed part.

• SHF-P experiments for round cup were conducted using the estimated optimum piston profiles for at Schnupp Hydraulik, Germany

• Estimated process parameter profiles resulted in good part without fracture and minor wrinkles in the flange.

• The thinning distribution in the experiments agreed well with FE predictions for the estimated optimum process parameters.

235

CHAPTER 9

PROGRAMMING MULTIPOINT CUSHION SYSTEM - APPLICATION –

SHEET HYDROFORMING WITH DIE (SHF-D)

9.1 Introduction

In SHF-D, the sheet metal is formed against a female die by the hydraulic pressure of the fluid as shown in Figure 9.1.The forming operation in SHF-D process can be divided into two phases. Phase I involves the free forming where the sheet bulges freely in the die cavity until it contacts the die. This introduces uniform strain distribution throughout the sheet and thereby a) the formability of the material is used effectively compared to the conventional stamping process where deformation is localized in the sheet at the punch and die corner radii, and b) the dent resistance of the hydroformed part is improved compared to the stamped part. Phase II involves calibrating the sheet against the die cavity to obtain the desired shape. High fluid pressure is required in phase II and its value depends on the material, sheet thickness, and the smallest corner radius in the die geometry.

Initial state Phase I Phase II

Figure 9.1 Schematic of the sheet hydroforming process [Kleiner 1999] 236

9.2 Process limits, defects and process window in SHF-D process

The quality of the formed part in a SHF-D process depends on the part geometry, properties of the sheet material, sheet/die contact interface conditions (friction) and process parameters, which consist of the relationship between the forming pressure and the applied blank holder force. Common defects in SHF-D are excessive thinning leading to fracture, wrinkling and the leaking of the pressurizing medium during forming.

a Fluid/Forming Pressure Figure 9.2 Process window in the SHF-D process

The process window is an area where the SHF D process can be performed without any defects in the formed part and is dependent on the part geometry [Figure 9.2]. The boundaries of the process window are the process limits. The two process parameters that are used to define the process window are the forming pressure and the blank holder force (BHF). The effect of the blank holder force (BHF) and forming pressure on

237 the common defects in SHF-D process (process window) was experimentally studied by Novotny et. al, 2001,Shulkin et al. 2000, Kleiner et al. 2004.

Fracture and wrinkling in the part are the common defects in the part manufactured by SHF-D process, similar to conventional deep drawing. Flange wrinkling and leaking occurs when the blank holder force is too small. Therefore, a minimum amount of blank holder force as a function of forming pressure (line a-b, Figure 9.2) is required to form the part without leakage. Further increase in the blank holder force beyond the minimum value would result in either deep drawing or stretching of the sheet as shown in the Figure 9.2. Therefore, depending on the required characteristics of the part, the blank holder force (BHF) can be changed to have either stretching or deep drawing. In parts that require high dent resistance, stretching is performed followed by drawing. The stretching zone(c,d,e) decreases with increase in the forming pressure as higher pressure and high blank holder result in failure of the part due to excessive thinning. The stretching zone in most parts is limited to low forming pressure corresponding to stage I of the process before the sheet material touches the die cavity. In deep drawing zone (a,b,c,d), increase in the blank holder force at the low forming pressure would result in stretching. At higher forming pressure, higher blank holder force would result in failure due to excessive thinning. Depending on the complexity of the part geometry and the formability of the sheet material, the operating window could be very narrow and would require several trial and error runs to estimate the process parameters to form the part. The presence of multipoint cushion system in the tooling further complicates the determination of process window experimentally. Therefore, in this study the developed methodology to estimate optimum BHF versus forming pressure was used to estimate process parameters required to form a rectangular part using SHF-D process with multipoint cushion system. A successfully formed part would be characterized by minimum thinning and no wrinkles, and will be formed without leaking/ minor leakage. This study was done in cooperation with IUL, University of Dortmund, Germany.

9.3 Objectives The specific objectives of this study are

238

• Estimate the optimum blank holder force versus the forming pressure to successfully form a rectangular part by SHF-D process using multipoint cushion system.

• Verify estimated optimum BHF profiles through experiments at IUL, University of Dortmund, Germany to validate optimization methodology and FE predictions.

9.4 Experimental setup

9.4.1 High pressure sheet hydroforming press

High pressure sheet hydroforming presses and tools are designed and manufactured based on the technology developed and available in tube hydroforming. However, in sheet hydroforming, higher clamping force due to large area of the sheet and blank holding mechanism need to be considered. University of Dortmund, (IUL), Germany in cooperation with Siempelkamp Pressen Systeme (SPS), Germany had built a 10,000 ton press for high pressure sheet hydroforming of large automotive parts (Figure 9.3). The press is designed to have horizontal mounting for inexpensive compact design, easy handling of work piece, short stroke for cylinder to reduce cycle time. The press frame is cast and prestressed by wire winding to withstand cyclic loading and unloading during forming. During hydroforming, large volume of fluid at relatively low pressure is required during the phase I (free bulging) and a small amount of fluid at high pressure is required in phase II for calibration. Therefore, to save cost, hydraulic systems in the press were designed for two different pumping conditions a) 100 l/min at max pressure of 315 bar and b) 5 l/min at max pressure of 2000 bar [Kleiner et al. 2001].

9.4.2 Tooling

The tooling of the SHF-D process consists of the die and the intermediate plate. The blank is placed between the intermediate plate and the die as shown in Figure 9.4. Pressurizing fluid is introduced between the sheet and the intermediate plate to form the sheet against the die cavity. During the forming process, the intermediate plate is held against the die and the sheet by the press ram. Thus, the intermediate plate acts as a blankholder as well as seal to avoid leakage of the pressurizing medium.

239

9.4.3 Multipoint blankholder system

In conventional high-pressure sheet hydroforming tooling, intermediate plate applies the required blankholder force to seal the pressurizing medium and to control the material flow during hydroforming. Hence it is difficult to precisely apply the required blankholder force on the sheet. IUL University of Dortmund, designed a new tooling as shown in Figure 9.4, where the sealing force is applied by the intermediate plate while the blankholder that is mounted in the flange portion of the die applies the blankholder force. Thus, the material flow from the flange is precisely controlled. In forming asymmetric parts, thickening in the flange is not uniform resulting in a gap that causes the leakage. IUL developed a modular tooling design with multipoint blankholder as shown in Figure 9.4, where the pressure at each piston can be varied independently. The multipoint blankholder system consists of 10 pieces of blankholder plate of thickness 15 mm arranged around the die as shown in Figure 9.4. The blankholder plates are supported by 10 piston/cylinders which provide the necessary blankholder force [Kleiner et al. 2001, Kleiner et al. 2003,].

Figure 9.3 Schematic of horizontal high pressure sheet hydroforming press at IUL [Kleiner et al. 2001]

240

Die Seal HFA- Supply plate

Sheet

Blank holder Piston for applying blank holder force

BH10 BH1 Piston 1

BH2 Piston 10 Piston 2 BH9 BH3 Piston 3 Piston 9 BH8

Piston BH4 Piston 4 Piston 8 BH7 Piston 7 Piston 5 BH6 BH5 Piston 6 Blankholde r

Front view of the blankholder layout Back view of the blankholder layout

Figure 9.4 Schematic of tool design by IUL with multipoint cushion system in the die for blankholder force application [Kleiner et al. 2003]

~ 480 mm

~960 mm

Figure 9.5 Schematic of the die for the rectangular part formed by SHF-D process at IUL,University of Dortmund

241

9.4.4 Die geometry

Figure 9.5 shows the schematic of the die for rectangular part formed by SHF-D process at IUL, University at Dortmund.

9.5 FE model

The geometry of the die I is symmetric along two planes therefore, only a quarter model was considered for the FE simulation, Figure 9.6. The die, pistons, and intermediate plate were modeled as rigid bodies. The blank holders were modeled with brick elements as an elastic object to account for elastic deflection that significantly influences the interaction of the blank holder with the sheet during the forming process. The blank and the spacer blank were modeled as elastic plastic objects using shell elements. Input parameters to the process simulation are shown Table 9.1. The flow stress and anisotropy of the sheet material DP600, obtained from bulge test was used in FE simulation. In the FE model, the hydraulic pressure acting on the sheet during the process was modeled using the Aquadraw interface available in the FE code PAMSTAMP 2000. The input instantaneous pressure curve is shown in Figure 9.7. The maximum value of the pressure was fixed at 300 bar while the shape of the curve was adjusted such that the kinetic energy of the deforming sheet is negligible compared to the internal energy in the FE simulation.

Sheet material DP600, thickness =1mm Sheet material properties Flows stress from bulge test(Figure 4.42) Anisotropy constants from bulge test

DP600 - ( r0 = 38.1 , r45 = 2.1 , r90 = 42.1 ) Piston pressure To be determined Forming pressure (Figure 9.7) Interface friction conditions Coulomb friction µ = 0.10

Table 9.1 Input parameters for FE simulation of rectangular part hydroforming process

242

Intermediate plate and Pressure Pot

Spacer blank Blank

Cylinder/Piston2 Blank holder Cylinder/Piston1

Cylinder/Piston3

Die

Figure 9.6 Schematic of the FE model of SHF-D process for rectangular part geometry used to estimate blank holder force by optimization

350

300

250

200

150 ` 100

Forming pressure (Bar) pressure Forming 50

0 010203040 Simulation time (mSec)

Figure 9.7 Instantaneous maximum forming pressure versus time curve used as an input to the aquadraw model in the FE simulation. 243

9.6 Optimization problem

9.6.1 Objective

Minimize thinning in the formed rectangular part

9.6.2 Constraints

The flange wrinkles were constraints to this optimization problem. The flange wrinkles were calculated from FE results at the end forming using the methodology described in section 6.5.1. The entire flange was divided into four regions, Figure 9.8 and flange wrinkle in each region was calculated from FE results. Side wall wrinkles were not considered as the fluid pressure inside the sheet forces it against the die and prevents occurrence of side wall wrinkles during forming.

Region 3 Region 4 Region 2 Region 1

Figure 9.8 Locations in the FE model monitored for the flange wrinkle

9.6.3 Design variables

The design variables were the pressure applied by each piston on the blank holder. In this study, piston pressure constant in space/location and in time/forming pressure, and piston pressure variable in space/location and constant in time/forming pressure were estimated. The number of design variable was one for the former case. In the latter case, since, only a quarter of the die was modeled in FE simulation therefore, the number of design variable was 3.

244

9.7 Optimization results

9.7.1 Piston pressure constant in space/location and in time/forming pressure

Piston pressure constant in space/location and time/stroke was estimated using the optimization methodology described in section 6.7.1. In the optimization, an initial guess of 2 bar was used for the pressure in all the piston. Figure 9.9,Figure 9.10, and Figure 9.11show the evolution of the objective function, constraints and design variable, respectively during optimization. At the initial piston pressure value of 2 bar, all the constraints were violated. Therefore, the pressure in the cylinders/piston was increased to reduce the constraint values to less than or equal to zero. At the end of fourth iteration, all the constraints were satisfied for the minimum constant piston pressure of 37.0 bar. Maximum thinning of 22.4 % and no flange wrinkling was observed in the formed part.

21.8 21.6 21.4 21.2 21 20.8 20.6 20.4

Maximum thinning (%) thinning Maximum 20.2 20 012345 Simulation number

Figure 9.9 Evolution of the objective function during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material

245

5 Constraint 1 Constraint 2 4 Constraint 3 3 Constraint 4 2 Constraint not satisfied 1 0 -1

Blankholder force (kN) force Blankholder Constraint satisfied -2 012345 Simulation number

Figure 9.10 Evolution of the constraint functions during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material

50 45 40 35 30 25 20 15 10

Blankholder force (kN) force Blankholder 5 0 012345 Simulation number

Figure 9.11 Evolution of the design variable during the optimization for the estimation of piston pressure constant in all the cylinders and in time/pressure for DP600 sheet material

246

9.7.2 Piston pressure variable in space/location and constant in time/forming pressure

Piston pressure variable in space/location and constant in time/forming pressure was estimated using the developed methodology described in section 6.7.2. In the optimization, an initial guess of 2 bar was used for the pressure in all the blank holder force cylinder/piston. Figure 9.12, Figure 9.13, and Figure 9.14 show the progress of the objective function, constraints and design variable, respectively, during optimization. Flange constraints in the flange area 1, 3, 4, Figure 9.8, were considered for the optimization. Flange constraint in flange area 2 was eliminated due to linear dependency in constraint gradients between constraint in flange area 2 and flange area 3, Figure 9.8. At the initial piston pressure value of 2 bar constraints 2 and 3 were violated. Therefore, the pressure was increased to reduce the constraint values to less than or equal to zero and minimize thinning. At the end of seventh iteration, all the constraints were satisfied for the minimum constant blank holder cylinder /piston pressure of 19 bar, 5 bar and 35.5 bar in cylinder/piston1, 2, and 3, respectively. It could be observed that all the constraint function values are close to the critical value for the optimized BHF [Figure 9.13]. Maximum thinning of 20.9 % with no flange wrinkling was observed in the formed part for the optimized piston pressure. The piston 2 at the corner applied the minimum force compared to the other cylinders thereby allowing easy material flow and reduce stretching of the sheet metal at the corner while the piston 1 and 3 applied higher force to avoid any flange wrinkling.

21 20.9 20.8 20.7 20.6 20.5 20.4 20.3

Maximum thinning (%) Maximum 20.2 20.1 012345678 Iteration number

Figure 9.12 Evolution of the objective function during the optimization for the estimation of piston pressure variable in all the cylinders and constant in time/ forming pressure for DP600 sheet material

247

10 Constraint 1 8 Constraint 2 6 Constraint 3

2 Constraint not satisfied

Constraint values Constraint 0

-2 Constraint satisfied 012345678 Iteration number

Figure 9.13 Evaluation of the constraint functions during the optimization for the estimation of blank holder cylinder pressure variable in all the cylinders and constant in time/forming pressure for DP600 sheet material

40 Piston 1 35 Piston 2 Piston 3 30 Piston 1 25 Piston 2 20 Piston 3 Piston 15 10

Piston pressure (Bar) pressure Piston 5 0 012345678 Iteration number

Figure 9.14 Evolution of the design variable during the optimization for the estimation of piston pressure variable in all the cylinders and constant in time/forming pressure for DP600 sheet material

248

9.7.3 Comparison of thinning distribution

Figure 9.15 show the comparison of the thinning distribution predicted by FE simulation along section C-C for the optimum piston pressure variable in each cylinder and constant in time/forming pressure and piston pressure constant in each cylinder/space and time/forming pressure for DP600 sheet material. Piston pressure variable in space didn’t result in significant improvement in the maximum thinning distribution of the formed part. In SHF-D process, in addition to applied blank holder force, forming pressure and the interface conditions between the sheet and the die influences the material flow. At higher forming pressures, the friction force between the sheet and die are higher and restrains the material flow as observed in Figure 9.16. Hence, towards to end of forming the corners (maximum thinning locations) are stretched formed at higher forming pressure with negligible influence from applied blank holder force.

25 B Constant piston pressure 20 Variable piston pressure 15

10 B

5 Thinning % Thinning 0 410 -5 130

-10 60 0 0 100 200 300 400 500 Curvilinear length (mm)

Figure 9.15 Comparison of thinning distribution along section B-B predicted by FE simulation for the optimum piston pressure variable in all the cylinders and constant in time/pressure and optimum piston pressure constant in all the cylinders and time/pressure for DP600 material.

249

4.2 Bar 6.8 Bar

75 Bar 47 Bar 15 Bar 265 Bar 58 Bar 8.5 Bar

Figure 9.16 Schematic of the formed sheet along section B-B predicted by FE simulation at different forming pressure for friction condition of μ = 0.10

9.8 Experimental validation

Experiments were conducted using the predicted piston pressure for sheet material DP600 at the horizontal sheet hydroforming press in IUL, Dortmund, Germany. The sheet was laser cut to the desired dimension and coated with commonly used lubricants. The piston pressures, a) constant in space and time, and b) piston pressure variable in space and constant in time, predicted using FE simulation coupled with optimization was used in the experiments. Three parts were formed for each case of the predicted piston pressures. The parts could be formed without any fracture and minor wrinkle in the flange, Figure 9.17, for both cases of predicted piston pressures.

Piston pressure constant in space Piston pressure variable in space

Figure 9.17 Picture of the formed part using the predicted piston pressure from FE simulations coupled with optimization

250

9.9 Comparison of FE results with experiments

9.9.1 Draw-in

Figure 9.18 and Figure 9.19 show comparison of the flanges in the drawn parts obtained from experiments and FE simulations. The boundary of the drawn part in FE simulations agrees well with experiment for both the cases except at location A where the material flow is restrained more in the experiment and not symmetric compared to symmetric material flow in the FE simulation. This could be due to difference in the interface conditions in the experimental setup at location A (due to friction or elastic deflection of the tooling). Also, it could observed that there was no significant difference in drawn part boundary between the two cases of blank holder force in the experiment further validating FE predictions, Figure 9.20.

9.9.2 Thinning distribution

The draw-in of the formed part in experiments for both the constant piston pressure and variable piston pressure were not symmetrical, Figure 9.19, and Figure 9.18. Therefore, thinning distributions along sections A-A and B-B, from experiment were compared with FE predictions for the optimized piston pressures. Figure 9.21 and Figure 9.22 show the comparison of the thinning distributions along section A-A and section B-B respectively for uniform pressure in all the pistons. Along section A-A, the material flow in the experiment was restrained leading to less draw-in in experiments compared to FE predictions, Figure 9.18. Hence higher thinning was measured in the formed part in experiments compared to FE predictions. Along section B-B Figure 9.22, the thinning predicted by FE simulation agreed well with experiments as draw-in agreed with FE predictions Figure 9.18. Similar results were also observed for the optimum piston pressure varying with space / location, Figure 9.23 and Figure 9.24. As predicted in FE simulations, piston pressure variable in space/location resulted in marginal improvement in thinning in the final formed part compared to uniform optimum piston pressure in all the pistons, Figure 9.25.

251

600 Experiment 500

400 FE result

300

200 Location A

100

0 -400 -300 -200 -100 0 100 200 300 400 -100 Y Coordinate (mm) Y Coordinate Initial sheet -200

-300

Die cavity -400

-500

-600 X Coordinate (mm)

Figure 9.18 Comparison of drawn part boundary in FE simulation with experiments for optimum piston pressure uniform in all the pistons for DP600 sheet material

252

600

500 Experiment

400 FE result 300

200 Location A 100

0 -400 -300 -200 -100 0 100 200 300 400 -100 Y Coordinate (mm) Y Coordinate -200 Initial sheet -300

-400 Die cavity -500

-600 X Coordinate (mm)

Figure 9.19 Comparison of drawn part boundary in FE simulation with experiments for optimum piston pressure varying with space/location for DP600 sheet material

253

Experiment - Uniform piston pressure 600

500

400 Experiment – Non- 300 Uniform piston pressure

200

100

0 -400 -300 -200 -100 0 100 200 300 400 -100 Y Coordinate (mm) Y Coordinate -200

Initial sheet -300

-400

Die cavity -500

-600 X Coordinate (mm)

Figure 9.20 Comparison of drawn part boundary in the experiments for optimum piston pressure varying with space/location with optimum piston pressure constant in space/location for DP600 sheet material

254

30 A Experiment - Sample 1 25 Experiment - Sample 2 20 FE Simulation 15 10 A 5

Thinning (%) Thinning 0 -5 0 50 100 150410 200 250 -10 130

-15 60 0 Curvilinear length (mm)

Figure 9.21 Comparison of thinning distribution in the experiment with FE predictions along section A-A for uniform piston pressure in all the pistons (material DP600)

Experiment - Sample 1 25 Experiment - Sample 2 20 FE Simulation 15 B 10

Thinning % Thinning 0 B -5 0 50 100 150410 200 250 130 -10 Curvilinear length (mm) 60 0

Figure 9.22 Comparison of thinning distribution in the experiment with FE predictions along section B-B for uniform piston pressure in all the pistons (material DP600)

255

30 A Experiment - Sample 1 25 Experiment - Sample 2 20 FE Simulation 15

10 A 5

Thinning (%) Thinning 0 -5 0 50 100 150 200 250 410 -10 130 -15 60 0 Curvilinear length (mm)

Figure 9.23 Comparison of thinning distribution in the experiment with FE predictions along section A-A for optimum piston pressure variable in space/ location (material DP600)

Experiment - Sample 1 25 Experiment - Sample 2 20 FE Simulation 15 B 10 5 0 B Thinning % Thinning -5 0 50 100 150 200 250 410 -10 130 -15 60 Curvilinear length (mm) 0

Figure 9.24 Comparison of thinning distribution in the experiment with FE predictions along section B-B for optimum piston pressure variable in space/ location (material DP600)

256

15 B

5 B

Thinning % Thinning 0 410 0 50 100 150130 200 -5 60 0 -10 Experiment - Optimum variable piston pressure

-15 Experiment - Optimum constant piston pressure Curvilinear length (mm)

Figure 9.25 Comparison of thinning distribution along section B-B in the experiment for optimum piston pressure constant in space and optimum piston pressure variable in space (material ;DP600)

9.10 Summary and conclusions

In sheet hydroforming process with die (SHF-D), the relationship between the blank holder force and the fluid forming pressure significantly influences the characteristics of the formed part. Conventionally, blank holder force is applied uniformly around the circumference of the flange and maintained constant during the forming process. However, development of multipoint cushion system allows changing the blank holder force in space/location and during the forming process, enhancing the drawability of the part. However, this capability is underutilized in production as the force required to program the cushion system is not known and difficult to estimate by trial and error. Therefore, in this study the developed optimization methodology was used to estimate optimum process parameters to form a rectangular part using a multipoint cushion system with minimum thinning and no wrinkling by SHF-D process. The estimated blank holder force profiles were later validated using experiments at IUL, University of Dortmund. The Summary and conclusions drawn from this study are

257

• FE simulation coupled with developed optimization routine to estimate a) optimum piston pressure constant in location/space and time, and b) optimum piston pressure variable in space/location and constant in time required to form the rectangular part with minimum thinning and no wrinkling from and DP600 sheet material.

• Optimum piston pressure variable in space/location resulted in marginal improvement in thinning (~ 1% - 2%) in the formed part compared to optimum piston pressure constant in space/location in the FE simulation.

• In SHF-D process, in addition to applied blank holder force, forming pressure and the interface conditions between the sheet and die influence the material flow. At higher forming pressures, the friction force between the sheet and die are higher and restrains the material flow. Hence, towards to end of forming, the corners (maximum thinning locations) are stretched formed at higher forming pressure with no significant influence from applied blank holder force/ piston pressure.

• SHF-D experiments for rectangular part were conducted using the estimated optimum piston profiles for DP600 sheet material at IUL, University of Dortmund, Germany

• Estimated piston pressure profiles resulted in good part without fracture and minor wrinkles in the flange.

• The draw-in and the thinning distribution in the experiments agreed well with FE predictions except at one of the corners in the rectangular part where the material flow was restrained in experiment.

• Piston pressure variable in location/space resulted in marginal improvement in thinning distribution (~ 1%-2%) in the formed part in experiment compared to uniform pressure in all the pistons as predicted by FE simulation.

258

CHAPTER 10

CONCLUSIONS AND FUTURE WORK

10.1 Conclusions In the stamping and sheet hydroforming process, the quality of a formed part is determined by the amount of material drawn into the die cavity. An excess material flow will cause wrinkling while insufficient material flow will cause tearing in the part. The material flow into the die cavity is influenced by the Blank Holder Force (BHF). Conventionally, the BHF is applied uniformly on the blank holder surface and held constant during the forming process. However, increase in the complexity of the parts and emphasis on the low formability lightweight materials require the use of multipoint cushion system in modern presses. This system allows the blank holder/binder/cushion force to change in space/location and during press stroke/time. Thus, better control material flow can be achieved for enhancing the formability / drawbility. However, the force required to program the multipoint cushion system is is costly and difficult to estimate by trial and error. Hence, it is hardly used in production. Estimation of BHF to program the multipoint cushion system could be best done through FE simulation in process design stage so that its advantgaes can be incorporated in process design. Also, the estimated force can be used in die tryout to prove them before used in production. Another significant process parameter is the initial blank shape which influences the material flow in stamping that need to be estimated through FE simulation to avoid trial and error in die tryouts to reduce tryout time and save material cost by reducing scrap.

Successful application of FE simulations in process design depends on the accuracy of the input parameters. Conventionally, material properties of sheet materials obtained from tensile test are used in FE simulation. This data is insufficient for the analysis because a) maximum strain obtained in uniaxial tensile test is small compared to strains encountered in stamping operations b) the stress state in tensile test is uniaxial while in regular stamping it is biaxial. Therefore, there is a need for biaxial test to accurately

259 determine the material properties (flow stress and anisotropy constants) over larger strain range for forming FE simulations.

The issue of appropriate test method to determine material properties for process simulation, optimal blank shape determination, and estimation of optimum force to program multipoint cushion system were addressed in this study. The summary and conclusion drawn from this study are:

10.1.1 Material properties determination

• Elliptical bulge test was developed to use along with circular bulge test to estimate flow stress and anisotropy of sheet materials in a biaxial stress state over a large strain range nearly twice of tensile test. • Developed elliptical test was used to estimate the flow stress and anisotropy of AKDQ steel, DP600 steel, DDS steel and A5754-O sheet materials. • Flowstress obtained from bulge test were higher compared to tensile test for same strain. Ansiotropy values obtained from bulge test were lower compared to tensile test for the tested sheet materials. • Difference in flow stress between tensile test and bulge test is attributed to inability of the Hill’s yield criteria and corresponding equivalent stress and strain relation ship to describe a) plastic yielding in multiaxial stress state for tested sheet materials, and b)relate multiaxial stress state to corresponding equivalent uniaxial stress state.

• Anisotropy value r45 could not be estimated using the developed test as the test measurements (pressure and doem height) are not very sensitive to anisotropy

value r45. • Deep drawing and bulge test simulations conducted using the material properties from the developed bulge test better correlated with experiments, compared to simulation results using tensile test data indicating that material properties from buge test is more appropriate for process simulation compared to tensile test.

260

10.1.2 Optimal blank shape determination

• Element shape function based back tracking method was developed to couple with FE results for estimation of the optimum blank shape. • The developed methodology was programmed in Matlab to work with commercial FE codes PAMSTAMP 2000 and LSDYNA • The developed methodology was used to estimate the optimal blank shape with holes for industrial part – Torque converter turbine shell. • Estimated optimal blank shape resulted in formed part without any failures in tryout. The dimensions of the formed part using the estimated optimum blank shape agreed well with desired CAD dimensions of the part. • Estimated optimal blank eliminated piercing and trimming, thereby reducing the manufacturing cost.

10.1.3 Programming multipoint cushion system

• Determination of force required to program multipoint cushion system in stamping and sheet hydroforming was formulated as an optimization problem with the objective to minimize thinning in the formed part and avoiding wrinkles. Wrinkles in the formed part were treated as constraints to the optimization problem. • FE simulation of the forming process was used to determine the implicit relationship between the design variables and the objective function, and design variables and constraint function required for optimization as explict relationships are not available for a given part geometry and sheet material. • Newton-Raphson based numerical optimization scheme was used to solve one dimensional optimization problem with constraints (BHF constant in space/location and stroke/time). • Sequential quadratic programming method was used to solve multidimensional optimization problem with constraints (BHF variable in space/location and constant in stroke/time, BHF constant in space/location and variable in stroke/time).

261

• Estimation of optimum BHF variable in stroke/time was divided into series of steps. Each step was solved sequentially to obtain BHF variable in stroke/time (sequential optimization). • The developed optimization strategy was programmed in C language and interfaced seamlessly with commercial FE codes for stamping; PAMSTAMP 2000 and LSDYNA to estimate optimum BHF profiles.

10.1.3.1 Stamping

• The developed technique was applied to estimate the blank holder force required to form the IFU Hishida part from sheet material A5182-O and BH210, and GM liftgate-inner part from sheet material A6111-T4 and BH210 using multipoint cushion system. • Estimated optimum BHF profiles variable in space/location and constant in stroke/time resulted in good part without failure and minor wrinkles validating the optimization and FE predictions. In other BHF cases, fracture was obaserved indicating the parts could not be formed without use of the multipoint cushion system i.e. BHF variable in space/location. • Programming multipoint cushion system for liftgate-inner part by trial and error in die tryout required 3 days while 3 hours were required to form the part without any defects using the estimated optimum BHF profiles. • In the investigated part geometries, BHF variable in space/location significantly improved thinning in the formed part and avoided failure by tearing compared conventional method of BHF constant in space/location and stroke/time that resulted in failure. This illustrated the enchancement to the drawbility of the material by using multipoint cushion system in drawing process.

10.1.3.2 Sheet hydoforming with punch (SHF-P)

• The developed technique was applied to estimate the optimum blank holder force and forming fluid pressure variable in stroke/time required to form 90 mm round cup upto depth of 100 mm using single point cushion system.

262

• Estimated optimum BHF and forming fluid pressure profiles resulted in good part without failure and wrinkles in the experiments validating the optimization and FE predictions. • Optimum BHF and forming fluid pressure resulted in less thinning and no wrinkling compared to profiles estimated through trial and error by press manufacturer.

10.1.3.3 Sheet hydroforming with die (SHF-D)

• The developed technique was applied to estimate the blank holder force constant in space/location and stroke/time, and BHF variable in space/location and constant in stroke/time to form rectangular pan from DP600 sheet material by multipoint cushion system. • Estimated optimum BHF profiles resulted in good part without failure and wrinkles for both predicted BHF profiles validating optimization and FE predictions. • BHF variable in space/location did not significantly improve thinning in the formed part as higher forming pressure and the interface conditions between the sheet and die restrained the material flow towards end of forming in SHF-D. Hence, the corners (maximum thinning locations) are stretched formed at higher forming pressure with no significant influence from applied blank holder force.

10.2 Future work

10.2.1 Material properties determination

• The developed methodology needs to be applied to several steel alloys of different thickness to further validate this concept of biaxial test for determination of anisotropy and flow stress. • The developed methodology needs to be extended for aluminum alloys using Barlat 1996 yield criterion

263

10.2.2 Optimal blank shape determination

• The developed methodology needs to be applied to different part geometries to further validate this concept.

10.2.3 Programming multipoint cushion system

• Currently gradients of the objective function and constraints are calculated by finite difference method in the developed optimization methodology. Direct differentiation method which is currently under development (Yang et al. 2002) to evaluate the gradients of objective function and constraint function could be implemented to reduce the computational time by maximum of 50 %. • Modeling the blankholder as an elastic object significantly increase the time required for each FE simulation run. Alternative method to consider elastic blankholder that under goes very little deformation compared to the sheet metal in the FE simulation such as condensing the stiffness of the blankholder to the degree freedom of the nodes/elements that are in contact with the sheet and nodes/elements on which the boundary conditions such as force, displacement is applied would significantly reduce the computational time. • In this project it was assumed that the locations of the cylinders of the multipoint cushion system around the blank holder were known. However, when Nitrogen cylinders are used in constructing dies, the location of each cylinder is selected based on experience and trial and error. Future work should include the prediction of optimum a) number of cylinders, b) location of cylinders around the blank holder, and c) the magnitude of the force exerted by each cylinder

264

REFERENCES

Ahmetoglu et al. 1993 Ahmetoglu, M., Coremans, A., Kinzel, G., Altan, T., 1993, “Improving drawability by using variable blank holder force and pressure in deep drawing of round cups”, SAE paper no 930287

Arora 2004 Arora, J.S., 2004,”Introduction to optimum Design”, Elsevier academic press, U.S.A.

Arrieux 1995 Arrieux, R., 1995, “Determination of the forming limit stress diagrams in sheet metal forming”, Journal of material processing technology, Vol 53, pp, 47-56

Aust 2001 Aust, M., 2001, “Modified hydromechanical deep-drawing”, Hydroforming of Tubes, Extrusions and Sheet Metals, Edited by K Siegert, Vol 2, pp 215 – 234

Aue-u-lan 2004 Aue-u-lan, Y., Ngaile, G., Altan, T., 2004, “Optimization of tube hydroforming process using process simulation and experiments”, Journal of material processing technology, Vol 146, pp, 137-143

A/SP 2000 A/SP, 2000, “High strength steel stamping design manual “, Auto Steel Partnership, www.a-sp.org.

Banabic et al. 2000 Banabic, D., Bunge, H. J., Pohlandt, K., Tekkaya, A.E., 2000, ”Formability of metallic materials”, Springer Verlag,Newyork

265

Banabic et al. 2001 Banabic, D., Bala, T., Comsa, D.S., 2001, “Closed form solution for bulging through elliptical dies”, Journal of Material Processing Technology, Vol 115, pp 83-86.

Barlat et al. 1991 Barlat F., Lege, D.J., Brem, J.C., 1991 “A six component yield function for anisotropic materials”, International Journal of Plasticity, Vol 7, pp 693-712.

Barlat et al. 1997 Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997, “Yield function development for aluminum alloy sheets”, Journal of Mechanics Physics Solids”, Vol 45, pp 1727-1763.

Barlat et al. 2003 Barlat, F., Brem , J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., 2003, “Plane stress yield function for aluminum alloy sheets – Part 1: Theory”, International Journal of Plasticity, Vol 19, pp 1297 – 1319

Bathe 1997 Bathe, K.J.,, 1997, “Finite Element Procedures”, Prentice Hall Inc, Englewood Cliffs, N.J., U. S. A.

Bleck et al. 2005 Bleck, W., Blumbach, M.,2005, “Laser aided flow curve determination in hydraulic bulging”, Steel research international, Vol 76, pp 125-130.

Boger et al. 2005 Boger, R.K.,Wagoner, R.H., Barlat, F., Lee, M.G., Chung, K., 2005, “Continuous, large strain, tension/compression testing of sheet material”, International journal of plasticity, Vol 21, pp 2319-2343.

266

Boudeau et al. 2000 Boudeau, N., Lejeune, A, Gelin, J.C., 2000, “Necking, bursting and wrinkling in stamping and tube hydroforming”, Proceedings of Metal forming 2000, pp391-396.

Bouvier et al. 2006 Bouvier, S.; Gardey, B.; Haddadi, H.; Teodosiu, C, 2006, “Characterization of the strain-induced plastic anisotropy of rolled sheets by using sequences of simple shear and uniaxial tensile tests”, Journal of Materials Processing Technology , Vol 174 , pp 115–126.

Broek et al. 1995 Broek, T., Muderrisoglu, A., Ahmetoglu,M., Chandorkar, K., Kinzel, G., Altan, T., “Improving drawability of aluminum alloy 2008-T4 sheet by optimizing the blank geometry and controlling the blankholder force”. Report No: ERC/NSM –S- 95-05, The Ohio State University, Columbus, 1995.

Cao et al. 1997 Cao, J., Boyce, M.C., 1997, “ A predictive tool for delaying wrinkling and tearing failures in sheet metal forming”, Transactions of the ASME, Vol 119, pp 354-366.

Cao 1999 Cao, J., 1999,”Prediction of the plastic wrinkling using the energy method”, Transactions of ASME, Vol 66, pp646-652.

Cao et al. 2000 Cao, J., Wang, X., 2000, “On the prediction of side wall wrinkling in sheet metal forming processes”, International journal of mechanical sciences, Vol 42, pp 2369-2394.

Cao et al. 2002 Cao, J., Wang, X., Mills, F.J., 2002, “Characterization of sheet buckling subjected to controlled boundary conditions”, Journal of manufacturing science and engineering, Vol 124, pp493-501

267

Cao et al. 2003 Cao, J., Krishnan, N., 2003, “Estimation of optimal blank holder force trajectories in segmented binders using ARMA model”, Journal of manufacturing science and engineering, Vol 125, pp763-770.

Cazacu et al. 2003 Cazacu, O., Barlat, F., 2003, “Application of the theory of representation to describe yielding of anisotropic aluminum alloys”, International Journal of Engineering Science, Vol 41, pp 1367-1385.

Chen et al. 1996 Chen, X., Sowerby, R.,1996, “Blank Development and the Prediction of Earing in Cup Drawing”, International Journal of Mechanical Sciences, vol 8, pp 509-516.

Chu et al. 1996 Chu, E., Li, M., 1996, “The onset and growth of elastoplastic wrinkling”, proceedings of NUMISHEET 1996, pp228-238.

Chung et al. 1992 Chung, K., Richmond, O., 1992, “Sheet forming process design based on ideal deformation theory”, Proceedings of NUMIFORM 1992, pp455-462.

Cook et al. 2000 Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J., 2000, “Concepts and applications of finite element analysis”, John wiley, Newyork.

Correia et al. 2002 Correia, J.P.M., Ferron, G., 2002, “Wrinkling predictions in the deep drawing process of anisotropic metal sheets”, Journal of material processing technology, Vol 128, pp, 178- 190.

Danckert et al. 1998 Danckert, J., Nielsen, K.B., “Determination of the plastic anisotropy ratio in the sheet metal using automatic tensile test equipment”, Journal of Material Processing Technology, Vol 73, pp 276-280.

268

Dingle et al. 2001 Dingle, M.E., Hodgson, P.D., Cardew-Hill, M.J.,2001, “Analysis of elastic behavior of the press system in draw die forming”, Innovations in processing and manufacturing of sheet materials, The Minerals, Metals & Society, pp 405-413.

Doege et al. 1987 Doege, E., Sommer, N., 1987, “Blank holder pressure and blank holder layout in deep drawing of thin sheet metal,” Advanced Technology of plasticity, Vol 2, pp 1305-1314.

Doege et al. 2001 Doege, E., Elend, E.L., 2001, “Design of pliable blank holder system for the optimization of process conditions in sheet metal forming”, Journal of material processing technology, Vol 111, pp, 182-187.

Gantar et al. 2002 Gantar, G., Kuzman, K., 2002, “Sensitivity and stability evaluation of the deep drawing process”, Journal of material processing technology, Vol 125, pp, 302-308.

Gelin et al. 2002 Gelin, J.C., Boudeau, N., Lejeune, A., 2002, “Wrinkling predictions in metal forming”, Proceedings of NUMISHEET 2002, pp 210-215.

Ghouati et al. 1999 Ghouati, O., Lenoir,H., Gelin, J.C., 1999, “Optimization techniques for the drawing of sheet metal parts”, Proceedings of NUMIFORM 1999, pp 293-298.

Ghosh 2002 Ghosh, S., 2002, “Lecture notes of ME 839 – Advanced finite element methods in engineering”, Department of mechanical engineering, The Ohio State University.

269

Gloecker et al. 2004 Glocker, M., Schmolz, K., Luginger, F., 2004, ‘Standard control for presses and automation mechanisms”, New developments in sheet metal forming, edited by K.Siegert, 2004, pp 297 – 315.

Groche et al. 2005 Groche, P., Metz, C., 2005, “Hydroforming of unwelded metal sheets using active elastic tools”, Journal of material processing technology, Vol 168, pp, 192-201.

Gunnarsson et al. 2001 Gunnarsson, L., Schedin, E., 2001, “Improving the properties of exterior body panels in automobiles using variable blank holder force”, Journal of material processing technology, Vol 114, pp, 168-173.

Gutscher et al. 2000 Gutscher,G., Wu, H.C., Ngaile, G., Altan, T., 2000, “Evaluation of formability and determination of flow stress curve of sheet metals with hydraulic bulge test”, Report no : S-ERC/NSM-00-R-15, The Ohio State University.

Halloquist 1998 Halloquist, J.O, 1998, “LS-DYNA Theoretical manual”, Livermore software technologies corporation, www.lstc.com.

Hausermann 2000 Hausermann, “Multipoint – Cushion – Technology Advances and Die Design”, Advances in sheet metal forming, edited by K.Siegert, 2000, WGP, pp 341 – 363.

Hearn et al. 2000 Hearn, D., Baker, M.P., 2000, “Computer graphics”, Prentice-Hall.

270

Hengalhaupt et al. 2006 Hengelhaupt, J., Vulcan, M., Darm, F., Ganz, P., Schweizer, R., 2006, “Robust deep drawing process of extensive car body panels”, New developments in sheet metal forming, edited by M. Liewald, pp 276-304.

Hill 1956 Hill, R., 1956, Mathematical theory of plasticity, Oxford press Hoffmann et al. 2006 Hoffmann,H., Hong,S., 2006, “Determination of flow stress for very thin sheet metal considering scale effects”, Annals of CIRP, pp. 263-266.

Hughes 2000 Hughes, TJ.R., 2000,” The Finite Element Method : Linear static and dynamic finite element analysis”, Prentice-Hall Inc, NewJersy.

Jirathearanat 2004 Jirathearanat, S., 2004, “Advanced methods for finite element simulation for process design in tube hydroforming”, Ph.D dissertation, The Ohio State University.

Kojic et al. 2005 Kojic, M., Bathe, K.J., 2005, “Inelastic analysis of solids and structures”, Springer Verlag, Newyork.

Kim et al. 2000 Kim, J.Y., Kim, N., Huh, M.S., 2000, “Optimum blank design of an automobile sub-frame”, Journal of material processing technology, Vol 101, pp, 31-43.

Kim et al. 2001 Kim, J.B., Yoon, J.W., Yang, D.J., Barlat, F., 2001.”Investigation into wrinkling behavior in the elliptical deep drawing process by finite element analysis bifurcation theory”, Journal of material processing technology, Vol 111, pp, 170-174.

Kim et al. 2002 Kim,S.H., Huh, H., 2002, “Design sensitivity analysis of the sheet metal forming process with direct differentiation”, ”,

271

Journal of material processing technology, Vol 130, pp, 504- 510.

Kleiner et al. 1999 Kleiner, M., Hellinger, V., Homberg, W., Klimmek, Ch., 1999, “Trends in sheet metal hydroforming”, Hydroforming of Tubes , Extrusions and Sheet Metals, Edited by K Siegert, Vol 1, pp 249 – 260.

Kleiner et al. 2001 Kleiner, M., Homberg, W., 2001, “New 100,000kN press for sheet metal hydroforming”, Hydroforming of Tubes , Extrusions and Sheet Metals, Edited by K Siegert, Vol 2, pp 351 – 362.

Kleiner et al. 2003 Kleiner, M., Homberg, W., Trompeter, M., 2003, “High – pressure sheet Metal forming of large –area sheat metal parts”, Hydroforming of Tubes , Extrusions and Sheet Metals, Edited by K Siegert, Vol 3, pp 435 – 445.

Kleiner et al. 2004 Kleiner, M., Kopp, R., Homberg, W., Krahn, M., Krux, R., Putten, K.V., 2004, “High-pressure sheet metal forming of tailor rolled blanks”, submitted for publication.

Ku et al. 2001 Ku, T., Lim, H., Choi, H., Hwang, S., Kang, B., 2001, “Implementation of backward tracing scheme of the FEM to blank design in sheet metal forming”, Journal of Materials Processing Technology, vol 111, pp 90-97.

Lee et al. 1998 Lee, C.H., Huh, H., 1998, “Blank design and strain estimates for sheet metal forming process by finite element inverse approach with initial guess of linear deformation”, Journal of material processing technology, Vol 82, pp, 145-155

272

Maki 2003 Maki, T., 2003, “Current status of fluid forming in the automotive industry”, Hydroforming of Tubes, Extrusions and Sheet Metals, Edited by K Siegert, Vol 3, pp 25 – 44.

Malo et al. 1998 Malo, K.A., Hopperstad, O.S., Lademo, O.G., 1998, “Calibration of anisotropic yield criteria using uniaxial tension tests and bending tests”, Journal of Material Processing Technology, Vol 80-81, pp 538-544.

Manabe et al. 1987 Manabe, K., Nishimura, H., 1987, “An improvement in deep drawbility of steel/plastic laminate sheets by control of blank holding force”, Advanced Technology of plasticity, Vol 2, pp 1297-1304.

Marciniak et al. 2002 Marciniak, Z., Duncan, J.L., Hu, S.J., 2002, “Mechanics of sheet metal forming”, Butterworth-Heinmann, Oxford.

Mattiasson et al. 1998 Mattiasson, K., Bernspang, L., 1998, “On the use of variable blank holder force in sheet metal stamping”, Simulation of materials processing: Theory, Methods and Applications, pp 831-838.

Metal forming controls ,2007, www.metalformingcontrols.com

Muller Weingarten, 1999 “Basic principles of Drawing Technology”, Technology – Journal, 1999, Muller Weingarten. http://www.mueller- weingerten.de/English/Technology

Naceur et al. 2001 Naceur, H., Guo, Y.Q., Batoz, J.L., Lenoir, C.K., 2001, “optimization of drawbead restraining forces and drawbead

273

design in sheet metal forming process”, International journal of mechanical sciences, Vol 43, pp 2407-2434.

Neugebauer et al. 1997 Neugebauer, R., Leib, U., Braun, H., 1997, “Influence of material flow in deep drawing using individual controllable draw pin and smooth blank holder design” SAE paper no 9709.

Nocedel et al. 1999 Nocedal, J., Wright, S., “Numerical optimization”, Springer Verlag, Newyork.

Novotny et al. 2001 Novotny, S., Hein, P., 2001, “Hydroforming of sheet metal pairs from aluminum alloys” Journal of Material Processing Technology, Vol 2001, pp65 – 69.

Obermeyer et al. 1998 Obermeyer, E.J., Majilessi, S.A., 1998, “A review of recent advances in the application of blankholder force towards improving the forming limits of sheet metal parts”, Journal of materials processing technology, Vol 75, pp 222-234.

Ohata et al. 1999 Ohata, T., Katayama, T., Nakamachi, E., Nakamura, Y., Kawahara, H., 1999, “Improvement of optimum process design system by numerical simulation – Discretized optimization method”, Proceedings of NUMIFORM 1999, pp 299-304.

Pegada et al. 2002 Pegada, V., Chun, Y., Santhanam, S., 2002, “An algorithm for determining the optimal blank shape for the deep drawing of aluminum cups”, Journal of material processing technology, Vol 125, pp, 743-750.

Palaniswamy et al. 2004 Palaniswamy, H., Thandapani, A., Altan, T., 2004, “Manual for optimization – BHF constant in space and time”, Report 274

no: ERC/NSM -04-R-46, The ohio state University, Columbus, Ohio.

Palaniswamy et al. 2005a Palaniswamy, H., Vavilikolane, V., Altan, T., 2005, “Manual for optimization – BHF variable in space and constant in time”, Report no: ERC/NSM-05-R-05, The ohio state University, Columbus, Ohio.

Palaniswamy et al. 2005b Palaniswamy, H., Vavilikolane, V., Altan, T., 2005, “Manual for optimization – BHF constant in space and variable in time”, Report no: ERC/NSM-05-R-33, The ohio state University, Columbus, Ohio.

Palaniswamy et al. 2005c Palaniswamy, H., Vavilikolane, V., Altan, T., 2005, “Prediction of blank holder force for liftgate-inner part - 1”, Report no: ERC/NSM-05-R-22, The ohio state University, Columbus, Ohio.

Palaniswamy et al. 2006 Palaniswamy, H., Altan, T., 2006, “Manual for optimization – BHF variable in space and in time”, Report no: ERC/NSM- 06-R-10, The ohio state University, Columbus, Ohio.

Pahl 1997 Pahl, K.J., 1997, “New developments in multipoint die cushion technology”, Journal of material processing technology, Vol 71, pp, 168-171.

Park et al. 1999 Park, S.H., Yoon, J.W., Yang, D.Y., Kim, Y.H., 1999, “Optimal blank design in sheet metal forming by the deformation path iteration method”, International journal of mechanical sciences, Vol 41, pp1217-1232.

275

Rauch 1998 Rauch, E.F., 1998, “Plastic anisotropy of sheet metals determined by simple shear tests”, Material science and engineering A. Vol 241, pp 179 – 183.

Rees 1995 Rees, D.W.A., 1995, “Plastic flow in elliptical bulge test”, International journal of mechanical sciences, vol 37, No 4, pp 373-389.

Roy et al. 2002 Roy, S., Kunju, R., Kirby, D.,2002, “An automated and flexible approach to optimal design of shape and process variables for stamped parts”, Proceedings of the NUMISHEET 2002, pp 573-578.

Schuler, 1998 “Metal Forming Hand Book - Schuler”, Springer Verlag, 1998.

Shim 2002 Shim, H., 2002, “Determination of optimal shapes for the stampings of arbitrary shapes”, Journal of material processing technology, Vol 121, pp, 116-122.

Shulkin et al. 1995 Shulkin, L.B.,Jansen, S.W., Ahemtoglu, M.A., Kinzel, G.L., Altan, T., 1995, “Elastic deflections of the blank holder in deep drawing of sheet metal”, Journal of material processing technology, Vol 59, pp, 34-40.

Shulkin et al. 2000 Shulkin, L.B.,Posteraro,R., Ahemtoglu, M.A., Kinzel, G.L., Altan, T., 1995, “Blank holder force control in viscous pressure forming of sheet material”, Journal of material processing technology, Vol 98, pp, 7-16.

Sheng et al. 2004 Sheng, Z.Q., Jirathearanat, S., Altan, T., 2004, “Adaptive FEM simulation for prediction of variable blank holder force 276

in conical cup drawing”, International Journal of Machine Tools and Manufacture Vol 44. pp. 487-494.

Siegert et al. 1993 Siegert, K., Doege, E., 1993, “CNC hydraulic multipoint blank holder system for hydraulic presses”, Annals of CIRP,Vol 1, pp 319- 322.

Siegert et al. 1999 Siegert, K., Losch, B., 1999, Sheet metal hydroforming, Hydroforming of Tubes, Extrusions and Sheet Metals, Edited by K Siegert, Vol 1, pp 221 – 248.

Siegert et al. 2000 Siegert, K., Haussermann, M., Haller, D., Wagner, S., Ziegler, M., 2000, “Tendencies in presses and dies for sheet metal forming process”, Journal of materials processing technology, vol 98, pp 259-264.

Siegert et al. 2001 Siegert, K., Beck, S., 2001, “Deep drawing with segmented elastic binders”, Production Engineering, Vol VIII/2, pp 35-40 Simo et al. 1997 Simo, J.C., Huhes, T.J.R., 1997,“Computational Inelasticity”, Springer Verlag, Newyork.

Strano et al. 2004 Strano, M., Carrino, L., 2004, “Adaptive selection of loads during FEM simulation of sheet forming processes”, Proceedings of NUMIFORM 2004, pp 802-807.

Thomas et al. 1997 Thomas, W., Ahmetoglu, M., Kinzel, G., Altan, T., “Improving the deep drawability of 2008-T4 aluminum and 1008 AKDQ EG steel sheet with location variable blankholder force control”; Report No: ERC/NSM –S-97-08, The Ohio State University, Columbus, 1997.

Thomas 1999 Thomas, W., 1999, “Product, Tool and Process design methodology for deep drawing and stamping of sheet materials”, Ph.D dissertation, The Ohio State University

277

Tuguc et al. 2001 Tuguc, P., Neale, K.W., Wu, P.D., MacEwen, S.R., 2001,”Effect of planar anisotropy on the wrinkle formation tendencies in curved sheets”, International journal of mechanical sciences, Vol 43, pp 2883-2897.

Vanderplaats 1984 Vanderplaats, G.N., 1984, “Numerical optimization techniques for engineering design: with applications”, McGraw Hill, Newyork.

Yang et al. 1998 Yang, S., Nezu, K., 1998, “Application of an inverse FE approach in the concurrent design of sheet stamping”, Journal of Materials Processing Technology, vol 79, pp 86- 93.

Yang et al. 2001 Yang, J.B., Jeon, B.H., Oh,S.I., 2001, “Design sensitivity analysis and optimization of the hydroforming process”, Journal of material processing technology, Vol 113, pp, 666- 672.

Xie et al. 2002 Xie, L.S., Xiong, Z.Q., 2002, “Study on the theory and application of the energy method and application of energy method for analyzing compressive instability in sheet forming”, Journal of material processing technology, Vol 129, pp, 255-260.

Zienkwicz et al. 2000 Zienkwicz, O.C., Taylor, R.L., 2000, “The Finite Element Method”, Vol 2, Butterworth-Heinmann, Oxford

278

APPENDIX A

ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 1

The bulging of the sheet in the elliptical was analyzed by assuming the sheet metal as a thin membrane that do not account for the bending stress and deformation. Figure A-1 show the schematic of the assumed geometry of the bulged sheet in the elliptical die.

Figure A.1 Schematic of the shape of the bulged specimen in the elliptical die assumed for analytical calculations A.1 Strain calculations at the apex of the dome

Let δt be the change in the thickness at the apex of the dome, then the strain increment along the thickness of the sheet is given by δt δε = t t Equation A-1 Where t is the instantaneous/current thickness of the sheet material.

The strain increment along the θ and φ direction can be obtained from the incompressibility condition. 279

δε t δε δε φθ =++ 0 Equation A-2

δεφ Let = β Equation A-3 δεθ

Therefore, the strain increment along the θ and φ direction can be obtained from Equation A-2 and Equation A-3 as. − δε − βδε δε = t ,δε = t θ 1+ β φ 1+ β Equation A-4

Substituting equation A-4 and equation A-1 in the Hill’s equivalent strain expression (equation 4-3) and assuming the sheet is oriented such that the major axis of the elliptical die coincides with rolling direction of the sheet corresponding to elliptical test 1, we get equivalent strain increment (δε p ) as

εδ p = ()/ bha 2 2 2 ⎛ 11 − βδε ⎞ ⎛ 11 −δε ⎞ ⎛ 1 −δε 1 − βδε ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜ t t ⎟ h = ⎜ −δεt ⎟ + ⎜ −δεt ⎟ + ⎜ − ⎟ ⎝ rr 090 1+ β ⎠ ⎝ rr 900 1+ β ⎠ ⎝ 90 1+ β rr 0 1+ β ⎠ 2 ⎛ 11 ⎞ Equation A-5 a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠ 2 ⎛ 111 ⎞ b ⎜ ++= ⎟ ⎝ rrrr 900090 ⎠

During the elliptical bulge test, the material flow is restricted using lock bead and the blank holder force therefore, the sheet in the die cavity is stretch formed. Hence the ratio of increment along the θ and φ direction is approximately constant and is equal to the ratio of the major axis to the minor axis and is dependent on anisotropy value. Therefore if the sheet material is assumed to be subjected to proportional loading and the expression for the effective strain ( ε p ) can be obtained by integrating equation A-5

δε φ assuming that the anisotropy values and the ratio ( = β ) do not change with the δεθ strain. 280

r0 = Anisotropy rollingin direction

r90 = Anisotropy in transev erse direction Otherwise the total strain is given as sum of increments as

p = ∑ εδε p Equation A-7

Alternative expression for strain increment along the θ and φ direction can also be obtained by assuming the shape of the deformed sheet as a part of circle/sphere at the center of the major axis and the minor axis as shown in Figure A-1.

δh δh δεθ ,δεφ == Equation A-8 Rθ Rφ

Rθ Then β = Equation A-9 Rφ However at higher dome height, the strain gradients along the curvilinear length of the dome are high. Therefore, the deformed sheet may not be able to be fit into a circle/sphere as we assumed. Therefore strain calculations based on the thickness strain are more accurate (Rees et al. 1995). More information on the calculation of strain increments by this method can be obtained from Rees et al. 1995. Calculation of strain

Rθ increments for the circular bulge test is similar to ellipse test 1 except for β == 1 . Rφ

281

A.2 Stress calculations at the apex of the dome

Consider a small element on the apex of the dome as shown in Figure A-2. Let Rθ and

Rφ be the radius of curvature of the sheet material at the apex of the dome along the major axis and minor axis of the ellipse respectively. Let σ θ and σ φ be the stress in the sheet material along the major axis and minor axis of the sheet material, P be the normal pressure acting on the sheet and t is the current thickness of the sheet material. The rolling direction of the sheet material was assumed to coincide with the major axis of the ellipse as in elliptical test 1. Applying the radial equilibrium on the small element, we get

⎛ δθ ⎞ ⎛ δφ ⎞ δφδθ 2 tR σδφ θφ sin⎜ ⎟ + 2 tR σδθ φθ sin⎜ ⎟ = φ RPR θ Equation A-10 ⎝ 2 ⎠ ⎝ 2 ⎠ 22 ⎛ ⎞ ⎛ ⎞ δφδφδθδθ Assuming that δθ and δφ be small such that sin⎜ ⎟ = sin, ⎜ ⎟ = , we get ⎝ ⎠ 22 ⎝ ⎠ 22

σ σ φθθφ =+ RPRtRtR θφ Equation A-11

σ φ Let = α σ θ We get ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ P 1 P α σ = ⎜ ⎟,σ = ⎜ ⎟ θ t ⎜ 1 α ⎟ φ t ⎜ 1 α ⎟ Equation A-12 ⎜ + ⎟ ⎜ + ⎟ ⎝ RR θθ ⎠ ⎝ RR θθ ⎠ Substituting equation A-11 in the Hill’s equivalent stress expression (equation 4-3) for plane stress conditions and assuming the sheet is oriented such that the major axis of the elliptical die coincides with the rolling direction of the sheet, we get equivalent stress (σ ) as

282

σ = ()/ ac ()2 ()σσ 2 rrrr (−++ σσ )2 c = 90 θ 0 φ 090 φθ rr 090 + )1( 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ ⎜ ⎟ Equation A-13 3 ⎝ rr 900 ⎠

r0 = Anisotropy rollingin direction

r90 = Anisotropy in transev erse direction

Relationship between the stress ratio (α) and the strain ratio (β) for elliptical bulge test 1 Relationship between the stress ratio (α) and the strain ratio (β) for elliptical bulge test 1 could be established using the flow rule as follows.

The flow rule is given by

∂f ij ∂=∂ λε Equation A-14 ∂σ ij Applying equation A-14 for Hill’s yield criteria (equation 4-3) we get

X λε []H()σσ YX +−∂=∂ GσY

Y λε []H()σσ YX +−∂=∂ FσY Equation A-15 Z λε []Gσ X +∂=∂ FσY

XY ∂=∂ λε Nσ XY

Substituting equation 4-5 in A-15 and assuming that major axis of the ellipse coincides with the rolling direction as in elliptical bulge test -1, we get

⎛⎛ 2 ⎞ ⎞ δλδε ⎜⎜ += ⎟ − 22 σσ ⎟ θ ⎜⎜ ⎟ θ φ ⎟ ⎝⎝ r0 ⎠ ⎠ Equation A-16 ⎛⎛ 2 ⎞ ⎞ δλδε ⎜⎜ += ⎟ − 22 σσ ⎟ φ ⎜⎜ ⎟ θφ ⎟ ⎝⎝ r90 ⎠ ⎠

283

⎛ 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ + ⎟ φ − 22 σσ θ ⎜ + ⎟α − 22 δεφ ⎝ r90 ⎠ ⎝ r90 ⎠ β == = Equation A-17 δε ⎛ 2 ⎞ ⎛ 2 ⎞ θ ⎜ ⎟ ⎜ ⎟ ⎜ + ⎟ θ − 22 σσ φ ⎜ + ⎟ − 22 α ⎝ r0 ⎠ ⎝ r0 ⎠

The stress equations obtained for the small element in the apex of the dome in the elliptical bulge test can be used to estimate the flow stress of the material from the elliptical bulge test1. Stress calculations for circular bulge test is same elliptical bulge

Rθ test 1 but with β == 1 . However, for elliptical test 2 and 3, relation ship between Rφ between the stress ratio (α) and the strain ratio (β) are different and are provided in Appendix B and Appendix C, respectively.

Figure A.2 Schematic of the small element at the apex of the dome in the bulged specimen of the elliptical die assumed for analytical calculations

284

APPENDIX B

ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 2

The solution procedure for elliptical bulge test 2 is similar to bulge test 1 except for substitution of stress and strain components into the Hill’s equivalent stress and strain relations. In elliptical bulge test 2, rolling direction is parallel to minor axis. Accordingly, Hill’ s equivalent strain expression is given by

ε p = ()/ bha 2 2 2 ⎛ 11 − βε ⎞ ⎛ 11 − ε ⎞ ⎛ 1 − ε 1 − βε ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜ t t ⎟ h = ⎜ − εt ⎟ + ⎜ − εt ⎟ + ⎜ − ⎟ ⎝ rr 900 1+ β ⎠ ⎝ rr 090 1+ β ⎠ ⎝ r0 1+ β r90 1+ β ⎠ 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠ 2 ⎛ 111 ⎞ b ⎜ ++= ⎟ ⎝ rrrr 900090 ⎠ t εt = ln Equation B-1 t0 t = Current sheet thickness t0 = Initial sheet thickness r0 = Anisotropy in rolling direction r90 = Anisotropy in transevers e direction

Hill’s equivalent stress expression is given by

285

σ = ()/ ac r ()σ 2 r ()σ 2 rr ()−++ σσ 2 c = 90 φ 0 θ 090 φθ rr 090 + )1( 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ Equation B-2 3 ⎝ rr 900 ⎠

r0 = Anisotropy in rolling direction

r90 = Anisotropy in transevers e direction

Relationship between the stress ratio (α) and the strain ratio (β) could be established using the flow rule as follows.

The flow rule is given by

∂f ij ∂=∂ λε …….Equation B-3 ∂σ ij Applying equation A-14 for Hill’s yield criteria (equation 4-3) we get

X λε []H()σσ YX +−∂=∂ GσY

Y λε []H()σσ YX +−∂=∂ FσY Equation B-4 Z λε []Gσ X +∂=∂ FσY

XY ∂=∂ λε Nσ XY

Substituting equation 4-5 in B-4 and assuming that minor axis of the ellipse coincides with the rolling direction as in elliptical bulge test 2 we get

⎛⎛ 2 ⎞ ⎞ δλδε ⎜⎜ += ⎟ − 22 σσ ⎟ θ ⎜⎜ ⎟ θ φ ⎟ ⎝⎝ r90 ⎠ ⎠ Equation B-5 ⎛⎛ 2 ⎞ ⎞ δλδε ⎜⎜ += ⎟ − 22 σσ ⎟ φ ⎜⎜ ⎟ φ θ ⎟ ⎝⎝ r0 ⎠ ⎠ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ + ⎟ φ − 22 σσ θ ⎜ + ⎟α − 22 δεφ ⎝ r0 ⎠ ⎝ r0 ⎠ β == = Equation B-6 δε ⎛ 2 ⎞ ⎛ 2 ⎞ θ ⎜ ⎟ ⎜ ⎟ ⎜ + ⎟ θ − 22 σσ φ ⎜ + ⎟ − 22 α ⎝ r90 ⎠ ⎝ r90 ⎠

286

APPENDIX C

ANALYTICAL SOLUTION FOR THE ELLIPTICAL BULGE TEST 3

C-1 Strain calculations at the apex of the dome

Let δt be the change in the thickness at the apex of the dome, then the strain increment along the thickness of the sheet is given by δt δε = t t Equation C--1 Where t is the instantaneous/current thickness of the sheet material.

The strain increment along the θ and φ direction can be obtained from the incompressibility condition.

δε t δε δε φθ =++ 0 Equation C-2

δεφ Let = β Equation C-3 δεθ

Therefore, the strain increment along the θ and φ direction can be obtained from Equation C-2 and Equation C-3 as. − δε − βδε δε = t ,δε = t θ 1+ β φ 1+ β Equation C-4

In elliptical test 3, the sheet is oriented such that rolling direction is 45o to the minor axis of the elliptical. Therefore, the strain along the orthotropic material axis X and Y can be obtained from strain transformation equations as follows.

287

∂+∂ εε ∂ε ε =∂ φθ −= t x 2 2 ∂+∂ εε ∂ε ε =∂ φθ −= t y 2 2 Equation C-5

∂−∂ εε φθ ∂εt ⎛ 1+− β ⎞ ε xy −=∂ −= ⎜ ⎟ 2 2 ⎝ 1+ β ⎠

Substituting equation C-5 in the Hill’s equivalent strain expression (equation 4-4) we get equivalent strain increment (δε p ) as

εδ p = ()/ bha 2 2 2 ⎛ 11 ∂− ε ⎞ ⎛ 11 ∂− ε ⎞ ⎛ 1 ∂− ε 1 ∂− ε ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜ t t ⎟ h = ⎜ ∂− ε t ⎟ + ⎜ ∂− ε t ⎟ + ⎜ − ⎟ ⎝ rr 900 2 ⎠ ⎝ rr 090 2 ⎠ ⎝ r0 2 r90 2 ⎠ 2 ⎛ ⎛ 1+− β ⎞ ⎞ ⎜− ⎜ ⎟∂ε ⎟ ⎜ ⎜ ⎟ t ⎟ Equation B-6 ⎝ ⎝ 1+ β ⎠ ⎠ + ⎛ 11 ⎞ r 12 ⎜ ⎟ ()45 ⎜ ++ ⎟ ⎝ rr 900 ⎠ 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠ 2 ⎛ 111 ⎞ b ⎜ ++= ⎟ ⎝ rrrr 900090 ⎠

δε φ assuming that the anisotropy values and the ratio ( = β ) do not change with the δεθ strain.

288

ε p = ()/ bha 2 2 2 ⎛ 11 − ε ⎞ ⎛ 11 − ε ⎞ ⎛ 1 − ε 1 − ε ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜ t t ⎟ h = ⎜ − εt ⎟ + ⎜ − εt ⎟ + ⎜ − ⎟ ⎝ rr 900 2 ⎠ ⎝ rr 090 2 ⎠ ⎝ r0 2 r90 2 ⎠ 2 ⎛ ⎛ 1+− β ⎞ ⎞ ⎜ ⎜ ⎟ε ⎟ ⎜− ⎜ ⎟ t ⎟ ⎝ 1+ β ⎠ + ⎝ ⎠ ⎛ 11 ⎞ ⎜ ⎟ ()r45 12 ⎜ ++ ⎟ ⎝ rr 900 ⎠ 2 ⎛ 11 ⎞ a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠ 2 ⎛ 111 ⎞ b ⎜ ++= ⎟ ⎝ rrrr 900090 ⎠ t εt = ln t0 t = Current sheet thickness t = Initial sheet thickness 0 Equation C-7 r0 = Anisotropy in rolling direction r90 = Anisotropy in transevers e direction

C-2 Stress calculations at the apex of the dome

Consider a small element on the apex of the dome as shown in Figure C-1. Let Rθ and

Rφ be the radius of curvature of the sheet material at the apex of the dome along the major axis and minor axis of the ellipse respectively. Let σ θ and σ φ be the stress in the sheet material along the major axis and minor axis of the sheet material, P be the normal pressure acting on the sheet and t is the current thickness of the sheet material. Applying the radial equilibrium on the small element, we get

⎛ δθ ⎞ ⎛ δφ ⎞ δφδθ 2 tR σδφ θφ sin⎜ ⎟ + 2 tR σδθ φθ sin⎜ ⎟ = φ RPR θ Equation C-8 ⎝ 2 ⎠ ⎝ 2 ⎠ 22 ⎛ ⎞ ⎛ ⎞ δφδφδθδθ Assuming that δθ and δφ be small such that sin⎜ ⎟ = sin, ⎜ ⎟ = , we get ⎝ ⎠ 22 ⎝ ⎠ 22

289

σ σ φθθφ =+ RPRtRtR θφ Equation C-9

σ φ Let = α σ θ We get ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ P 1 P α σ = ⎜ ⎟,σ = ⎜ ⎟ θ t ⎜ 1 α ⎟ φ t ⎜ 1 α ⎟ Equation C-10 ⎜ + ⎟ ⎜ + ⎟ ⎝ RR θθ ⎠ ⎝ RR θθ ⎠

+ σσ φθ p 1+ α σ x = = 22 t ⎛ 1 α ⎞ ⎜ + ⎟ ⎜ ⎟ ⎝ R Rφθ ⎠

+ σσ φθ p 1+ α σ y = = Equation C-11 22 t ⎛ 1 α ⎞ ⎜ + ⎟ ⎜ ⎟ ⎝ R Rφθ ⎠

− σσ φθ p 1− α σ xy −= −= 22 t ⎛ 1 α ⎞ ⎜ + ⎟ ⎜ ⎟ ⎝ R Rφθ ⎠

Substituting equation C-11 in the Hill’s equivalent stress expression (equation 4-1) for plane stress conditions we get equivalent stress (σ ) as σ = ()/ ac r ()σ 2 r ()σ 2 rr (σσ )(2 )+++−++ rrr )(12 σ 2 c = 90 X 0 Y 090 YX 45 900 xy rr 090 + )1( 2 ⎛ 11 ⎞ Equation C-12 a ⎜ ++= 1⎟ 3 ⎝ rr 900 ⎠

r0 = Anisotropy in rolling direction

r90 = Anisotropy in transevers e direction o r45 = Anisotrop iny 45 to the rolloing direction

290

A relationship between the stress ratio (α) and the strain ratio (β) could be established using the flow rule as follows.

The flow rule is given by

∂f ij ∂=∂ λε Equation C-13 ∂σ ij Applying equation C-13 for Hill’s yield criteria (equation 4-1) we get

X λε []H()σσ YX +−∂=∂ GσY

Y λε []H()σσ YX +−∂=∂ FσY Equation C-14 Z λε []Gσ X +∂=∂ FσY

XY ∂=∂ λε Nσ XY

Substituting equation C-5 and C-11 in equation C-14 we get

∂ε 1+− β p 1− α − t λN −∂= 2 1+ β 2t ⎛ 1 α ⎞ ⎜ + ⎟ ⎜ ⎟ Equation C-15 ⎝ R Rφθ ⎠

∂ε p 1+ α − t ∂= λ ()+ FG 22 t ⎛ 1 α ⎞ ⎜ + ⎟ ⎜ ⎟ Equation C-16 ⎝ R Rφθ ⎠

Dividing equation C-15 by C-16 we get

1+− β N 12 − α = Equation C-17 1+ β + FG 1+ α

Substituting equation 4-5 in equation C-17 we get

1+− β 1− α ()r += 12 Equation C-18 1+ β 45 1+ α

291

APPENDIX D

BULGE TEST EXPERIMENTAL RESULTS - MATERIAL DDS STEEL

R.D. R.D.

Circular bulge test Elliptical bulge test 1

R.D. R.D.

Elliptical bulge test 2 Elliptical bulge test 3

Figure D.1 Schematic of the deformed DDS samples in circular bulge test, elliptical bulge test 1, 2, and 3 292

100 90 Sample-1 80 Sample-2 70 Sample-3 60 50 40

Pressure (bar) Pressure 30 20 10 0 0 5 10 15 20 25 30 35 Dome height (mm)

Figure D.2 Comparison of the pressure versus dome height curve obtained for three different sample of DDS sheet material in circular bulge.

180 160 Sample- 1 140 Sample- 2 Sample- 3 120 Sample- 4 100 80 60 Pressure (bar) Pressure 40 20 0 0 5 10 15 20 25 Dome height (mm)

Figure D.3 Comparison of the pressure versus dome height curve obtained for three different sample of DDS steel material in elliptical test –1

293

160 Sample - 1 140 Sample - 2 120 Sample - 3

100

60 Pressure (bar) Pressure 40

0 0 5 10 15 20 25 Dome height (mm)

Figure D.4 Comparison of the pressure versus dome height curve obtained for three different sample of DDS steel material in elliptical test –2

160 Sample- 1 140 Sample- 2 120 Sample- 3 100 Sample- 4 80 60 Pressure (bar) Pressure 40 20 0 0 5 10 15 20 25 Dome height (mm)

Figure D.5 Comparison of the pressure versus dome height curve obtained for three different sample of DDS steel material in elliptical test –3

294

180 Elliptical test 1 160 140

120 Elliptical test 3 100 Elliptical test 2 80 Elliptical test- 1 60 Pressure (bar) Pressure Elliptical test- 2 40 Elliptical test- 3 20 0 0 5 10 15 20 25 Dome height (mm)

Figure D.6 Comparison of the pressure versus dome height curve obtained from elliptical test 1, elliptical test 2 and elliptical test 3 for DDS sheet material.

700

600

500

400 Circular Bulge test 300 Ellipse test 1 Ellipse test 2 Stress (MPa) Stress 200 Ellipse test 3 100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

Figure D.7 Comparison of the flow stress curve obtained from circular test, elliptical test 1, elliptical test 2 and elliptical test 3 for DDS sheet material assuming the material as isotropic.

295

2500

2000

1500

1000

Objective function 500

0 01234 Iteration Number

Figure D.8 Change in the objective function during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for DDS sheet material

1.6 1.4 1.2 1 0.8 r0 0.6 r90 0.4 Anisotropy values Anisotropy r45 0.2 0 01234 Iteration Number

Figure D.9 Change in the anisotropy values during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for DDS sheet material

296

600

500

400 Circular test 300 Elliptical test 1 Elliptical test 2 200 Stress (MPa) Stress Elliptical test 3 100

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Strain

Figure D.10 Flow stress of the material obtained from the circular and elliptical bulge tests for the estimated optimum anisotropy values for DDS sheet material

297

APPENDIX E

BULGE TEST EXPERIMENTAL RESULTS - MATERIAL DP600

R.D. R.D.

Circular bulge test Elliptical bulge test 1

R.D. R.D.

Elliptical bulge test 2 Elliptical bulge test 3

Figure E.1 Schematic of the deformed DP600 samples in circular bulge test, elliptical bulge test 1, 2, and 3

298

140 Experiment-1 120 Experiment-2 100 Experiment-3

Pressure (bar) Pressure 40

0 0 5 10 15 20 25 30 35 Dome height (mm)

Figure E.2 Comparison of the pressure versus dome height curve obtained for three different sample of DP600 sheet material in circular bulge.

140 Ellipse test - 1 120 Ellipse test - 2

100 Ellipse test - 3

Pressure (bar) Pressure 40

0 0 2 4 6 8 10 12 14 Dome height (mm)

Figure E.3 Comparison of the pressure versus dome height curve obtained from elliptical test 1, elliptical test 2 and elliptical test 3 for DP600 material.

299

140 Experiment-1 120 Experiment-2

100

60 Pressure (bar) 40

0 024681012 Dome height (mm)

Figure E.4 Comparison of the pressure versus dome height curve obtained for three different sample of DP600 material in elliptical test -1.

140 Experiment-1 120 Experiment-2 Experiment-3 100

60 Pressure (bar) Pressure 40

0 02468101214 Dome height (mm)

Figure E.5 Comparison of the pressure versus dome height curve obtained for three different sample of DP600 material in elliptical test -2.

300

140 Experiment-1 120 Experiment-2 Experiment-3 100

60 Pressure (bar) Pressure 40

0 02468101214 Dome height (mm)

Figure E.6 Comparison of the pressure versus dome height curve obtained for three different sample of DP600 material in elliptical test -3.

1200

1000

800 Circular test 600 Elliptical test 1,2,3

Stress (MPa) Stress 400

200

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Strain

Figure E.7 Comparison of the flow stress curve obtained from circular test, elliptical test 1, elliptical test 2 and elliptical test 3 for DP600 material assuming the material as isotropic.

301

2500

2000

1500

1000

Objective functionObjective 500

0 01234 Iteration number

Figure E.8 Change in the objective function during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for DP600 sheet material

1.6 1.4 1.2 1

0.8 r0 0.6 r90 0.4 r45 Anisotropy values Anisotropy 0.2 0 01234 Iteration number

Figure E.9 Change in the anisotropy values during the optimization to estimate flow stress and anisotropy from the circular and elliptical bulge tests for DP600 sheet material

302

900 800 700

600 Circular test 500 Elliptical test 1 Elliptical test 3 400 Elliptical test 2 300 Flow stress (MPa) stress Flow 200 100 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Strain

Figure E.10 Flow stress of the material obtained from the circular and elliptical bulge tests for the estimated optimum anisotropy values for DP600 sheet material

303

APPENDIX F

BULGE TEST EXPERIMENTAL RESULTS - MATERIAL A5754-O

R.D. R.D.

Circular bulge test Elliptical bulge test 1

R.D. R.D.

Elliptical bulge test 2 Elliptical bulge test 3 Figure F.1 Schematic of the deformed A5754-O samples in circular bulge test, elliptical bulge test 1, 2, and 3

304

90 80 70 60 50 40

Pressure (bar) 30 Experiment-1 20 Experiment-2 Experiment-3 10 0 0 5 10 15 20 25 30 35 Dome he ight (mm)

Figure F.2 Comparison of the pressure versus dome height curve obtained for three different sample of A5754-O sheet material in circular bulge.

120 Sample-1 100 Sample-2 Sample-3 80

40 Pressure (bar) Pressure

0 0 5 10 15 20 Dome height (mm)

Figure F.3 Comparison of the pressure versus dome height curve obtained for three different sample of A5754-O material in elliptical test -1.

305

120 Sample-1 100 Sample-3 Sample-2 80

40 Pressure (bar) Pressure

0 0 2 4 6 8 101214161820 Dome height (mm)

Figure F.4 Comparison of the pressure versus dome height curve obtained for three different sample of A5754-O material in elliptical test -2.

120 Sample-1 100 Sample-3 Sample-4 80 Sample-2

40 Pressure (bar) Pressure

0 0 2 4 6 8 1012141618 Dome height (mm)

Figure F.5 Comparison of the pressure versus dome height curve obtained for three different sample of A5754-O material in elliptical test –3

306

120 Elliptical test 2 Elliptical test 3 100

80 Elliptical test 1 60

Pressure (bar) 40

0 0 5 10 15 20 Dome height (mm)

Figure F.6 Comparison of the pressure versus dome height curve obtained from elliptical test 1, elliptical test 2 and elliptical test 3 for A5754-O sheet material.

350 Circular test 300

250

200 Circular test Ellipse test 1 150 Ellipse test 2 Stress (MPa) Stress Elliptical test 1,2,3 Ellipse test 3 100

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Strain

Figure F.7 Comparison of the flow stress curve obtained from circular test, elliptical test 1, elliptical test 2 and elliptical test 3 for A5754-O material assuming the material as isotropic.

307

APPENDIX G

COMPARISON OF BULGE TEST EXPERIMENTAL RESULTS WITH FE

SIMULATION PREDICTIONS USING THE MATERIAL PROPERTIES

OBTAINED FROM TENSILE TEST AND BULGE TEST.

G.1 Material DDS steel

100 Experiment- 1 90 Experiment- 2 80 Experiment- 3 Experiment- 4 70 Simulation- Bulge test 60 Simulation - Tensile test 50 40 Pressure (bar) Pressure 30 20 10 0 0 5 10 15 20 25 30 35 Dome height (mm)

Figure G.1 Comparison of the pressure versus dome height curve obtained from circular bulge test experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

308

180 Experiment- 1 160 Experiment- 2 Experiment- 3 140 Experiment- 4 120 Simulation- Bulge test Simulation - Tensile test 100

Pressure (bar) 60

0 0 5 10 15 20 25 Dome height (mm)

Figure G.2 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -1 experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

160 Experiment- 1

140 Experiment- 2 Experiment- 3 120 Simulation- Bulge test 100 Simulation-Tensile test

60 Pressure (bar) Pressure 40

0 0 5 10 15 20 25 Dome height (mm)

Figure G.3 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -2 experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

309

160 Experiment- 1 140 Experiment- 2 Experiment- 3 120 Experiment- 4 100 Simulation- Bulge test data

80 Simulation-Tensile test

60 Pressure (bar)

0 0 5 10 15 20 25 Dome height (mm)

Figure G.4 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -3 experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

45 Experiment 40 Simulation- Bulge test Simulation-Tensile test 35 30 25

20 0

Thinning % Thinning 15 10

5 65 90 0 0 1020304050607080 Curvilinear length (mm)

Figure G.5 Comparison of thinning distribution obtained from circular bulge test experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 31.3 mm.

310

60 Experiment-1 50 Experiment-2 Simulation- Bulge test 40 Simulation-Tensile test 0 30

Thinning % Thinning 20

10 65 90

0 0 1020304050607080 Curvilinear length (mm)

Figure G.6 Comparison of thinning distribution along the major axis obtained from elliptical bulge test –1 experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 21.2 mm.

60 Experiment 50 Simulation- Bulge test Simulation-Tensile test 40 0

Thinning % Thinning 20 65 90 10

0 0 102030405060 Curvilinear length (mm)

Figure G.7 Comparison of thinning distribution along the minor axis obtained from elliptical bulge test –2 experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 21.2 mm.

311

G.2 Material DP600 steel

140

120

100

60 Experiment-1

Pressure (bar) Pressure Experiment-2 40 Experiment-3 20 Simulation- bulge test data Simulation- tensile test data 0 0 1020304050 Dome height (mm) Figure G.8 Comparison of the pressure versus dome height curve obtained from circular bulge test experiment for DP 600 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

140 Experiment-1 120 Experiment-2 Simulation- Bulge test data 100 Simulation- Tensile test data 80

60 Pressure (bar) Pressure 40

0 02468101214 Dome height (mm)

Figure G.9 Comparison of the pressure versus dome height curve obtained from elliptical bulge test -1 experiment for DP 600 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 312

140 Experiment-1 120 Experiment-2 Experiment-3 100 Simulation- Bulge test data Simulation- Tensile test data 80

60 Pressure (bar) Pressure 40

0 02468101214 Dome height (mm)

Figure G.10 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -2 experiments for DP 600 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

140 Experiment-1 120 Experiment-2 Experiment-3 100 Simulation- Bulge test data Simulation- Tensile test data 80

60 Pressure (bar) Pressure 40

0 02468101214 Dome height (mm)

Figure G.11 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -3 experiment for DP 600 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

313

40 Experiment Simulation-Tensile test 35 Simulation-Bulge test 30 25 20 0 15 10 Thinning % Thinning 5 90 0 65 -5 0 102030405060708090 -10 Curvilinear length (mm)

Figure G.12 Comparison of thinning distribution obtained from circular bulge test experiment for DP600 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 29 mm. G.3 Material:A5754-0 90 Experiment-1

80 Experiment-2 Experiment-3 70 Simulation - Bulge test 60 Simulation - Tensile test 50 40

Pressure (bar) 30

0 0 5 10 15 20 25 30 Dome height (mm)

Figure G.13 Comparison of the pressure versus dome height curve obtained from circular bulge test experiment for A5754-0 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 314

120 Experiment-1 Experiment-2 100 Experiment-3 Simulation - Bulge test data 80 Simulation - Tensile test

40 Pressure (bar)

0 0 5 10 15 20 Dome height (mm)

Figure G.14 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -1 experiment for A5754-0 sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

Experiment-1 120 Experiment-2 100 Experiment-3 Simulation - Bulge test data 80 Simulation - Tensile test

40 Pressure (bar) Pressure

0 0 5 10 15 20 Dome height (mm)

Figure G.15 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -2 experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

315

120 Experiment-1 Experiment-2 100 Experiment-3 Experiment-4 80 Simulation- Bulge test data Simulation-Tensile test 60

Pressure (bar) Pressure 40

0 0 5 10 15 20 Dome height (mm)

Figure G.16 Comparison of the pressure versus dome height curve obtained from ellipse bulge test -3 experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests 40 Experiment 35 Simulation - Bulge test 30 Simulation-Tensile test 25

20 0 15 Thinning % Thinning 10

5 65 90 0 0 1020304050607080 Curvilinear length (mm) Figure G.17 Comparison of thinning distribution obtained from circular bulge test experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 29.2 mm .

316

18 Experiment 16 Simulation - Bulge test 14 Simulation-Tensile test 12 10 8 0 6 Thinning % Thinning 4

2 65 90 0 -2 0 1020304050607080 Curvilinear length (mm) Figure G.18 Comparison of thinning distribution along the major axis obtained from elliptical bulge test –1 experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 15.6 mm . 16 Experiment 14 Simulation - Bulge test 12 Simulation-Tensile test 10

8 0 6 Thinning % Thinning 4

2 45 90 0 0 1020304050 Curvilinear length (mm) Figure G.19 Comparison of thinning distribution along the minor axis obtained from elliptical bulge test –2 experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests at dome height of 15.6 mm

317

APPENDIX H

COMPARISON OF ROUND CUP DEEP DRAWING EXPERIMENTAL

MEASUREMENTS WITH FE PREDICTIONS

Experiment-1 Experiment-2 160 Simulation - Bulge test data Simulation - Tensile test 140

120

100

60 Punch force (kN) force Punch 40

0 0 20406080100120 Stroke (mm)

Figure H.1 Comparison of the punch force obtained from round cup deep drawing experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

318

150

100

Punch Rolling diameter direction 0 -150 -100 -50 0 50 100 150 DDS Experiment - 1 DDS Experiment - 2 -50 DDS Experiment - 3 DDS Experiment - 4 -100 Simulation - Bulge test Simulation - Tensile test -150

Figure H.2 Comparison of the draw-in obtained from round cup deep drawing experiment for DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests

319

160 180 5 145 0 0 50 100 150 200 -5 75 Thinning % Thinning -10 Simulation - Bulge 0 55 Simulation-Tensile -15 Experiment -20 Curvilinear length (mm)

Figure H.3 Comparison of the thinning distribution along rolling direction in the formed round cup from DDS sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

160

140

120

100

80 Experiment-2 60 Experiment-3

Punch force (kN) Experiment-1 40 Simulation - Bulge test Simulation - Tensile test 20

0 0 20 40 60 80 100 120 Stroke (mm)

Figure H.4 Comparison of the punch force obtained from round cup deep drawing experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests. 320

150

100

Punch Rolling diameter direction 0 -150 -100 -50 0 50 100 150

-50 Experiment-1 Experiment-2 Experiment-3 -100 Experiment-4 Simulation - Bulge test Simulation - Tensile test -150

Figure H.5 Comparison of the draw-in obtained from round cup deep drawing experiment for A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

321

20 15

10 160 180 5 145 0 -5 0 25 50 75 100 125 150 175 200

Thinning % Thinning -10 75 Experiment -15 Simulation - Tensile 0 55 -20 Simulation-Bulge -25 Curvilinear length (mm)

Figure H.6 Comparison of the thinning along rolling direction in the formed round cup from A5754-O sheet material with FE predictions using the material properties obtained from the tensile test and bulge tests.

322

APPENDIX I

OPTIMIZATION RESULTS FOR HISHIDA PART FROM BH210 STEEL

250

200

150

pin (kN) pin 100

Blank holder force applied by each by each applied force holder Blank 0 012345 Iteration no

Figure I.1 Evolution of the design variable during the optimization for the estimation of optimum BHF constant in all the cylinders and in stroke/time for BH210 sheet material

323

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Objective function - Thinning (%) - function Objective 0 012345 Iteration no

Figure I.2 Evolution of the objective function during the optimization for the estimation of optimum BHF constant in all the cylinders and in stroke/time for BH210 sheet material

1.5 Constraint - 1 Constraint - 2 1

0.5

Constraint function Constraint 012345 -0.5

-1 Iteration no

Figure I.3 Evolution of the constraint function during the optimization for the estimation of optimum BHF constant in all the cylinders and in stroke/time for BH210 sheet material

324

Bspline control point -1 Bspline control point - 2 Bspline control point - 3 Bsplinecontrol point - 4 250

200

150

pin (kN) pin 100

50 Blank holder force applied by each by each applied force holder Blank

0 0246810 Iteration no

Figure I.4 Evolution of the design variable during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for BH210 sheet material

5 Objective function - Thinning (%) Thinning - function Objective 0 0246810 Iteration no

Figure I.5 Evolution of the objective function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for BH210 sheet material

325

2 1.8 1.6 1.4 1.2 1 0.8 0.6

Constraint function Constraint 0.4 0.2 0 -0.2 0246810 Iteration no

Figure I.6 Evolution of the constraint function during the optimization for the estimation of optimum BHF constant in all the cylinders and variable in stroke/time for BH210 sheet material

Pin 1 Pin 2 Pin 3 Pin 4 Pin 5 250

200

150

100

Blank holder force (kN) force Blank holder 50

0 051015 Iteration no

Figure I.7 Evolution of the design variables (pin 1- pin 5) during the optimization for the estimation of optimum BHF variable in space/cylinders/pins and constant in stroke/time for BH210 sheet material

326

Pin 6 Pin 7 Pin 8 Pin 9 Pin10 250

200

150

100

Blank holder force (kN) 50

0 0 5 10 15 Iteration no

Figure I.8 Evolution of the design variables (pin 6 - pin 10) during the optimization for the estimation of optimum BHF variable in space/cylinders/pins and constant in stroke/time for BH210 sheet material

0.3

0.25

0.2

0.15

0.1

0.05 Objective function - Thinning (%) Thinning - function Objective 0 02468101214 Iteration no

Figure I.9 Evolution of the objective fucntion during the optimization for the estimation of optimum BHF variable in space/cylinders/pins and constant in stroke/time for BH210 sheet material

327

6 Constraint - 1 5 Constraint - 2 Constraint - 3 4 Constraint - 4 3 Constraint - 5 Constraint - 6 2 Constraint - 7 1 Constraint - 8

Constraint function Constraint 0

-1

-2 02468101214 Iteration no

Figure I.10 Evolution of the constraint function during the optimization for the estimation of optimum BHF variable in space/cylinders/pins and constant in stroke/time for BH210 sheet material

328

APPENDIX J

DRAWBEAD SECTIONS EXTRACTED FROM LIFT GATE DIE SCANNED

DATA

Bead in the die

Z Bead in the blankholder - outer X

Figure J.1 Schematic of the drawbead 2 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

X Bead in the blankholder

Figure J.2 Schematic of the drawbead 3 cross section extracted from scanned data of the lift gate tooling die and the blankholder

329

Bead in the die

Z Bead in the blankholder - outer X

Figure J.3 Schematic of the drawbead 6 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Bead in the blankholder - outer

Figure J.4 Schematic of the drawbead 7 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Y Bead in the blank holder - outer

Figure J.5 Schematic of the drawbead 8 cross section extracted from scanned data of the lift gate tooling die and the blankholder 330

Bead in the die

Y Bead in the blankholder - outer

Figure J.6 Schematic of the drawbead 9 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Bead in the blankholder - inner Z

Y Figure J.7 Schematic of the drawbead 10 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Z Bead in the blankholder - inner X

Figure J.8 Schematic of the drawbead 11 cross section extracted from scanned data of the lift gate tooling die and the blankholder 331

Bead in the die

Z Bead in the blankholder - inner X

Figure J.9 Schematic of the drawbead 12 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Bead in the blankholder - inner

Figure J.10 Schematic of the drawbead 13 cross section extracted from scanned data of the lift gate tooling die and the blankholder

Bead in the die

Y Bead in the blankholder - inner

Figure J.11 Schematic of the drawbead 14 cross section extracted from scanned data of the lift gate tooling die and the blankholder

332

Bead in the die

Y Bead in the blankholder - inner

Figure J.12 Schematic of the drawbead 15 cross section extracted from scanned data of the lift gate tooling die and the blank holder

333

APPENDIX K

DRAWBEAD FORCES FOR BH210 SHEET MATERIAL

Table K.1 Draw bead forces for the draw beads in the lift gate die obtained for BH210 steel sheet material from draw bead simulation for two different clearances between the blankholder and the die

Clearence=sheet thickness Clearence=sheet thickness + 1 (0.78 mm) mm (1.78 mm) Bead Information Normal force Pulling force Normal force Pulling force kN/10 mm kN/10 mm kN/10 mm kN/10 mm Bead 1 0.80 0.65 0.60 0.50 Bead 2 0.70 0.60 0.60 0.45 Bead 3 0.50 0.55 0.30 0.35 Bead 4 0.30 0.20 0.28 0.20 Bead 5 0.55 0.41 0.49 0.39 Bead 6 0.80 0.63 0.70 0.58 Bead 7 0.60 0.50 0.50 0.40 Bead 8 0.65 0.43 0.55 0.38 Bead 9 0.53 0.46 0.41 0.38 Bead 10 0.40 0.50 0.28 0.38 Bead 11 0.60 0.60 0.45 0.40 Bead 12 0.50 0.50 0.30 0.38 Bead 13 0.40 0.35 0.30 0.25 Bead 14 0.50 0.40 0.38 0.33 Bead 15 0.35 0.35 0.25 0.25

334