Osu1143214139.Pdf (10.83

HIGH VELOCITY FORMABILITY AND

FACTORS AFFECTING IT

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree of Doctor of Philosophy in the Graduate

School of The Ohio State University

Mala Seth Dehra, B.E, M.S.

*******

The Ohio State University

2006

Dissertation Committee: Approved by

Dr. Glenn S. Daehn, Adviser

Dr. Peter Anderson Adviser

Dr. Kathy Flores Graduate Program in Materials Science and Engineering

Mala Seth Dehra

2006

ABSTRACT

High velocity forming methods successfully address problems faced in conventional forming techniques. They can be effectively used for forming metals with low formability like aluminum alloys and high strength steel. They can be instrumental is manufacturing of lighter vehicles with higher fuel efficiency. Electromagnetic forming

(EMF) is an HVF method that is gaining wide acceptance due to its advantages and scope for commercialization.

A number of experimental studies were carried out with EMF with the main goal of exploring fundamentals about material formability at high velocities, which can be used to establish practical design guidelines and to make models of high velocity formability.

Thus the main factors that influence high velocity formability – inertia / size effects; changes in constitutive behavior; impact; and dynamic failure modes, were studied mainly with experiments. The role of changes in constitutive behavior in improving formability was studied from existing studies and new theoretical studies involving High velocity Forming Limit Diagram (FLD) and through solving an inverse problem of ring expansion.

Tube free-expansion experiments were carried out to demonstrate enhanced metal formability even in the absence of die impact. To further establish the significance of inertia, electromagnetic ring free-expansion experiments with rings of different aspect ratios were carried out. A higher aspect ratio sample had better formability in terms of uniform and total elongation and also had fewer necks than a low aspect ratio (more slender) ring at the same velocity. The results clearly demonstrated the influence of sample aspect ratio (dimensions) and hence inertia on high velocity formability.

Die impact experiments were carried out with tubes and rings to show the beneficial influence of die arrest of a moving sample. It was revealed that die impact in an appropriate range of velocities can significantly suppress failure and reduce the number of tears and fractures in the samples. Further a new mode of failure in die impacted samples, spall-like dynamic rupture was observed, which had characteristics similar to classic spall failure.

Thus through all these studies, the important factors influencing high velocity formability was studied and it was shown that it can be more complex than quasi-static formability.

Boundary conditions for each forming operation can play a more significant role and hence simple tools like FLDs might not be practical tools for studying high velocity formability.

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Dedicated to the memory of my mother,

Mrs. Komal Seth

ACKNOWLEDGMENTS

I express my sincere gratitude and appreciation to my advisor, Dr. Glenn S. Daehn, for

his continuous guidance, inspiration and support throughout my research. His pursuit for

‘looking at the big picture’, creativity and enthusiasm have been a big inspiration for me.

His intellectual guidance, encouragement and understanding made it possible for me to complete this dissertation while I was away from my family. I would like to thank Dr.

John Bradley at General Motors USA, my project sponsors, for financial support and creative discussions about my research.

I am deeply indebted to my parents and the rest of my family for their constant support. I would have never come so far but for the love and inspiration of my mother Mrs. Komal

Seth, my father Mr. Anil Seth, brother Ashish and my grandparents, who encouraged me throughout. My husband, Amit has been a constant source of positive energy for me. I thank him for his understanding, love and support during my prolonged absences.

I would also like to thank Dr. Vincent J. Vohnout and my other High Rate Forming group

members, Manish Kamal, Kinga Unocic, Jianhui Shang, Edurne Iriondo, Scott Golowin

and Yuan Zhang, for their help.

VITA

May 30, 1977…………………………..Born - Shimla, India

1999…………………………………….B.E., Metallurgical Engineering Punjab Engineering College, Chandigarh, India

2003…………………………………….M.S. Materials Science and Engineering, The Ohio State University, Columbus OH, USA

2000 – Present………………………….Graduate Research Associate, Dep’t of Materials Science and Engineering, The Ohio State University, Columbus OH, USA

PUBLICATIONS

1. Mala Seth, V.J. Vohnout and G.S. Daehn, “Formability of steel sheet in high velocity impact”, J. of Materials Processing Technology, Vol.168 (2005), pp. 390-400.

2. Mala Seth and Glenn Daehn, “Effect of Aspect Ratio on High Velocity Formability of Aluminum Alloy”, Symposium: Trends in Materials and Manufacturing Tech. for Transportation Industries, TMS, San Francisco (2005).

3. Glenn S. Daehn., Manish Kamal , Mala Seth, Jianhui Shang, “Strategies for Sheet th Metal Forming Using Mechanical Impulse”, 6 Global Innovations Symposium: Trends in Materials and Manufacturing Technologies for Transportation Industries: Sheet Metal Forming, TMS Annual Meeting, San Francisco CA, (2005).

4. G.S. Daehn, E. Iriondo, M. Seth, M. Kamal, J. Shang, Electromagnetic and High Velocity Forming: Opportunities for Reduced Cost and Extended Capability in Sheet Metal Forming, Society of Manufacturing Engineers Summit Conference, August, Wisconsin, (2005). vi

TABLE OF CONTENTS

Abstract ……………………………………………………………………….……….....ii

Dedication…………………………………………………………………………...... iv

Acknowledgements……………………………………………………………………….v

Vita……………………………………………………………………………...... ……...vi

List of Figures………………………………………………………………………..….xiv

List of Tables…………………………………………………………………...... xxi

Chapters:

1. INTRODUCTION ...... 1

2. HIGH VELOCITY FORMING ...... 6

2.1 INTRODUCTION...... 6 2.2 HIGH VELOCITY FORMING METHODS ...... 7 2.2.1 Explosive forming ...... 7 2.2.2 Electrohydraulic forming (EHF) ...... 11 2.2.3 Electromagnetic forming...... 12 2.2.4 Gun forming ...... 17 2.3 Advantages and challenges of using aluminum alloys...... 18 2.3.1 Advantages of using Al alloys vehicles ...... 18 2.3.2 Challenges associated with using Al alloys with conventional forming...... 19

2.4 Advantages of High velocity forming ...... 20 2.4.1 Formability enhancement...... 21 2.4.2 Wrinkling is suppressed ...... 27

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2.4.3 Springback is reduced ...... 28 2.4.4 Economic advantages...... 28

2.5 SUMMARY ...... 29 BIBLIOGRAPHY…………………………………..……………………………..30

3. LITERATURE REVIEW ...... 36

3.1 INTRODUCTION...... 36 3.2 MATERIAL AND OTHER FACTORS AFFECTING FORMABILITY AT CONVENTIONAL FORMING VELOCITIES ...... 38 3.2.1 Strain hardening ...... 38 3.2.2 Strain rate hardening ...... 39 3.2.3 Anisotropy...... 40 3.2.4 Inhomogeneities ...... 40 3.2.5 Damage accumulation ...... 41 3.2.6 Superimposed hydrostatic pressure...... 43

3.3 FORMING LIMIT DIAGRAMS ...... 44 3.4 THEORETICAL MODELS FOR CALCULATING FLD’S...... 48 3.4.1 Theoretical models for determining FLD’s based on necking theory...... 48 3.4.2 Theoretical models for determining FLD’s based on sheet non- homogeneity...... 49 3.4.3 Theoretical models for determining FLD’s based on incorporation of damage evolution...... 52

3.5 SUMMARY FOR CONVENTIONAL FORMING...... 56 3.6 HIGH VELOCITY FORMABILITY...... 57 3.7 EFFECT OF INERTIA ON HIGH VELOCITY FORMABILITY ...... 58 3.7.1 Models based on initial inhomogeneity and study of growth of a neck...... 59 3.7.2 Models based on Instability / Perturbation Analyses ...... 64

3.8 EFFECT OF DIE IMPACT ON HIGH VELOCITY FORMABILITY...... 65 3.9 EFFECT OF CHANGES IN CONSTITUTIVE BEHAVIOR ON HIGH VELOCITY FORMABILITY ...... 67

3.10 SUMMARY FOR HIGH VELOCITY FORMING ...... 74 viii

BIBLIOGRAPHY…………………………………………………………………76

4. TUBE EXPANSION EXPERIMENTS...... 84

4.1 BACKGROUND AND MOTIVATION...... 84 4.2 EXPERIMENTAL PROCEDURE...... 89 4.2.1 Capacitor Bank...... 90 4.2.2 Actuator...... 90 4.2.3 Workpiece ...... 92 4.2.4 Rogowski Probes...... 97

4.3 METHODOLOGY ...... 98 4.4 RESULTS AND DISCUSSION – FOUR-TURN COIL ...... 99 4.4.1 Four-turn coil with sample of the same length (3.17 cm) at 6.72 kJ...... 99 4.4.2 Four-turn coil with sample of length (8.51 cm) greater than the coil at 6.72 kJ ...... 102

4.5 RESULTS AND DISCUSSION – TEN-TURN COIL ...... 104 4.5.1 Ten-turn coil with sample of same length (8.51cm) at 13.92 kJ:...... 104 4.5.2 Ten-turn coil with short (3.17cm) sample at 8kJ...... 108

4.6 RESULTS AND DISCUSSION – TWO-TURN COIL...... 111 4.7 SIMULATION OF ELECTROMAGNETIC FIELDS WITH MAXWELL 2D...... 112 4.7.1 Background about MAXWELL 2D ...... 113 4.7.2 Simulations for the coil – short sample configuration ...... 114 4.7.3 Simulations for the coil – long sample configuration ...... 117 4.7.4 Simulations for the coil – same sample length configuration ...... 119

4.8 NUMERICAL SIMULATION OF TUBE EXPANSION EXPERIMENTS ...... 120 4.8.1 Code description...... 121 4.8.2 Numerical code results for EM expansion of a 3.17cm sample with 4-turn coil at 7.04 kJ ...... 122

4.9 COMBINED RESULTS FROM CURRENT TRACES AND CIRCUIT PARAMETERS FOR ALL SAMPLE – COIL CONFIGURATIONS...... 124

4.10 DISCUSSION: COMBINED TRUE STRAIN DATA FOR DIFFERENT SAMPLE LENGTH – COIL CONFIGURATIONS ...... 126

4.11 CONCLUSIONS ...... 129 BIBLIOGRAPHY………………………………………………………………..131

5. SIMULATION OF FORMING LIMITS FOR ELECTROMAGNETIC TUBE EXPANSION...... 133

5.1 BACKGROUND AND MOTIVATION...... 133 5.2 PROBLEM FORMULATION ...... 135 5.2.1 Localization of deformation analysis ...... 135 5.2.2 Constitutive response of the metal sheet ...... 137 5.2.3 Selection of material constants...... 139 5.2.4 Strain, Strain rate and Current density profiles ...... 141 5.2.5 Numerical algorithm ...... 144

5.3 COMPARISON OF EXPERIMENTAL AND SIMULATION RESULTS ...145 5.4 DISCUSSION...... 151 5.5 CONCLUSIONS ...... 156 BIBLIOGRAPHY………………………………………………………………..158

6. TUBE DIE IMPACT EXPERIMENTS...... 160

6.1 BACKGROUND AND MOTIVATION...... 160 6.2 EXPERIMENTAL PROCEDURE...... 166 6.2.1 Capacitor bank...... 167 6.2.2 Actuator...... 167 6.2.3 Workpiece ...... 168 6.2.4 Rogowski probes ...... 168 6.2.5 Die Arrangement ...... 169 6.2.6 Contact plates ...... 169

6.3 METHODOLOGY ...... 170 6.4 RESULTS AND DISCUSSION...... 172

6.4.1 Sample profiles at increasing energy levels ...... 172 6.4.2 Current traces ...... 176

6.5 TUBE FLANGING EXPERIMENTS...... 177 6.6 CONCLUSIONS ...... 181 BIBLIOGRAPHY………………………………………………………………..183

7. RING EXPANSION EXPERIMENTS...... 184

7.1 BACKGROUND...... 184 7.1.1 Advantages of ring geometry and trends about ductility ...... 185 7.1.2 Multiple fragmentation...... 187 7.1.3 Effect of aspect ratio on formability...... 195

7.2 EXPERIMENTAL PROCEDURE...... 199 7.2.1 Capacitor bank...... 200 7.2.2 Actuator...... 200 7.2.3 Workpiece ...... 203 7.2.4 Rogowski and Pearson Probes ...... 206 7.2.5 High Speed Camera...... 206

7.3 METHODOLOGY ...... 208 7.4 RESULTS AND DISCUSSION- POSITION AND VELOCITY PROFILES ...... 209 7.4.1 Position and Velocity profiles for 4x4 rings ...... 210 7.4.2 Position and Velocity profiles for 2x2 rings ...... 216 7.4.3 Position and Velocity profiles for 1x1 rings ...... 218 7.4.4 Peak velocities for all three ring types ...... 220

7.5 RESULTS AND DISCUSSION – TRUE STRAINS ...... 221 7.6 RESULTS AND DISCUSSION – NECKS AND FRACTURES...... 228 7.7 RESULTS AND DISCUSSION- SAMPLE PICTURES...... 230 7.8 RESULTS AND DISCUSSION – CURRENT TRACES...... 233 7.9 RESULTS AND DISCUSSION – COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS ...... 235

7.10 CONCLUSIONS ...... 243

BIBLIOGRAPHY………………………………………………………………..245

8. RING DIE IMPACT EXPERIMENTS ...... 247 8.1 BACKGROUND AND MOTIVATION...... 247 8.2 EXPERIMENTAL PROCEDURE...... 248 8.2.1 Capacitor bank...... 249 8.2.2 Actuator...... 250 8.2.3 Workpiece ...... 251 8.2.4 Rogowski probes ...... 251 8.2.5 Die arrangement ...... 251

8.3 METHODOLOGY ...... 253 8.4 RESULTS AND DISCUSSION...... 253 8.4.1 Die impact of 2x2 rings...... 254 8.4.2 Die impact of 4x4 rings...... 256

8.5 CONCLUSIONS ...... 259

9. DYNAMIC FAILURE MODES…………………………………………………….260

9.1 NECKING ...... 260 9.2 SHEARING...... 262 9.3 SPALL-LIKE DYNAMIC RUPTURE ...... 263 9.3.1 What is spall? ...... 263 9.3.2 Fundamentals about spalling...... 266 9.3.3 Microstructural aspects of spall...... 269

9.4 EXPERIMENTAL STUDIES FOR SPALL-LIKE FAILURE...... 274 9.4.1 Sample1- AA2024 disc formed with EHF ...... 274 9.4.2 Sample 2- High strength steel formed with EMF...... 279 9.4.3 Sample 3 – Copper sheet impacted with EMF...... 281 9.4.4 Sample 4 – Electromagnetic welding of AA6061 rod with tube...... 283 9.4.5 Comparison of cross-sections of undeformed, free-formed and die impacted AA5754-O ring samples...... 286

9.5 DISCUSSION AND CONCLUSIONS...... 287

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BIBLIOGRAPHY………………………………………………………………..290

10. CONCLUSIONS AND FUTURE WORK…………………………………………292

10.1 ISSUE: Role of constitutive behavior in enhancing formability at high velocities ...... 292 10.2 ISSUE: Influence of Inertia in High velocity formability and size effects in fragmentation and necking ...... 294 10.3 ISSUE: Influence of high velocity impact on formability...... 296 10.4 ISSUE: Dynamic failure modes ...... 298 10.5 ISSUE: Appropriateness of a Forming Limit Diagram (FLD) for depicting high velocity formability...... 300

BIBLIOGRAPHY……………………………………………………………………...302

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LIST OF FIGURES

Figure 2.1 Methods of high velocity forming based on type of energy used [3]……………... 8 2.2 Schematic examples of typical explosive forming operations. (a) Use of detonation cord to form a cylinder (b) Method forming a flat panel [1, 7]. ………9 2.3 Actual results from typical explosive forming scenarios: (a) 10-foot diameter domes formed explosively from AA2014-0 [6], (b) a 310 stainless steel spark guide and its perform, (c) 321 stainless steel aircraft panel formed in a single operation (d) 6 in. diameter corrugated cylinder formed from 0.01 inch thick A- 286 stainless steel [7]. …………………………………………………………...10 2.4 Schematic of a general electromagnetic forming system ……………………….13 2.5 Typical axisymmetric parts made by EMF [41]. ………………………………...16 2.6 (a) Schematic of the set-up used in hydroforming and electrohydraulic forming (b) Comparison of forming AA2024 T4 sheet into a conical die using a hydroforming process (left) and using electrohydraulic forming (right) [10]. ………………….21 2.7 EMF experiments with high strength steel sheets. (a) Experimental setup schematic and picture, (b) Pictures of samples of two steels deformed by the small and large axisymmetric and medium wedge punches. These are samples in which failure had just initiated [37]. ……………………………………………………………23 2.8 Plot indicating the aggregate high velocity and quasi-static ductilities of the five steels. …………………………………………………………………………….24 2.9 Gun forming (a) Experimental setup schematic with die profile, (b) Pictures of sample formed with a polymeric bullet. 0.38mm sheet formed without tear (left), while 0.15mm failed sheet. (c) Strain distribution for one steel (quasi-static ductility 3.7%) with different bullet geometries. ………………………………..26

3.1 Stages in void nucleation, growth and coalescence in ductile metals [9]. ……….42 3.2 Fracture strain vs. superimposed hydrostatic pressure for a few Al alloys [17]. ...43 3.3 A typical Forming Limit Diagrams along with different strain states [3]…….....45 3.4 The forming window for plane stress forming of sheet [5]…………………..….48 3.5 Schematic of sample with groove ‘b’ and uniform region ‘a’ [1]……….……....50 3.6 (a) Right hand side of a tensile test specimen with associated (b) Velocity, (c) force and (d) stress profiles [73]…………………………………………………62 3.7 Flow stress of annealed 0.9999 copper measured at a strain of 15% as a function of strain rate [87]………………………………………………………………...68

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3.8 Comparison of experimental results with finite volume simulation ●, ‘thick’ specimen; ▲, ‘thin’ specimen; o, finite volume with fitted power law. Results indicate that ‘increased’ strain rate sensitivity at high strain rates is an artifact [94]………………………………………………………………………...…….70 3.9 Variation of microhardness of 6061 T4 sheet formed with high velocity impact as a function of strain at low and high rates [62]………………………………..….72 3.10 Stress strain curves generated with simulation in LS-DYNA with electromagnetism module [96].….……………………….…………………………………….…...72

4.1 Engineering Major vs. Minor strain for AA5754 samples electromagnetically formed along with pictures of samples to indicate the areas of maximum strain. (a) Free formed samples formed at 5.8kV, (b) Samples formed with conical die impact at 8kV [6]. ………………………………………………………………..86 4.2 FLD for AA5754 samples electromagnetically free formed at three charge voltages. The sample at 5.5kV did not fail; the sample at 6kV had just split while the sample formed at 7KV had cracked and failed [7]. ………………………….87 4.3 Experimental setup for electromagnetic tube expansion. (a) Schematic, (b) Picture with capacitor bank, actuator, tube workpiece and Rogowski probes. ………….90 4.4 Schematic of a four-turn actuator used for Electromagnetic expansion of tubes. (a) Front view (b) Top view. ………………………………………………………...93 4.5 Pictures of a bare 10, 4 and 2-turn coils.…………………………………………94 4.6 Different types of deformations possible with the grid circles. (a) Uniaxial Tension (b) Plane Strain (c) Biaxial Tension …………………..……………….95 4.7 Stress vs. Strain plot for 6063- T6 tube samples cut in the longitudinal and transverse directions. …………………………………………………………….96 4.8 Schematic showing the arrangement of Rogowski probes used for measuring primary and induced currents. …………………………………………………...97 4.9 Picture of 3.17cm tall sample deformed with a 4-turn coil at 6.72 kJ ………….100 4.10 True strain FLD for 3.17cm sample deformed with a four turn coil at 6.72 kJ ...101 4.11 Current vs. Time plot for 3.17cm sample deformed with 4-turn coil at 6.72 kJ. Peak current = 128 kA, Rise time = 18 µs. …………………………………….101 4.12 Tall (8.51cm) sample deformed with a 4-turn coil at 6.72 kJ………………….. 102 4.13 True strain FLD for a tall sample (8.51cm) deformed with a four-turn coil at 6.72 kJ ……………………………………………………………………………….103 4.14 Current vs. time plot for tall (8.51cm) sample deformed with 4-turn coil at 6.72 kJ. Peak current = 138 kA and Rise time = 16µs. ……………………………..104 4.15 Picture of tall sample (length 8.51cm) (a) front view, (b) inside view, deformed with ten turn coil at 13.92 kJ. The formation of intersecting necking bands is clearly evident. …………………………………………………………………105 4.16 True strain FLD for 8.51cm sample electromagnetically expanded by a 10 turn coil at 13.92 kJ …………………………………………………………………107 4.17 Current vs. Time plot for 8.51cm tall sample electromagnetically launched with ten turn coil at 13.92 kJ. Peak current = 124 kA, Rise time = 24 µs. ………….108

4.18 Picture of short sample (3.17cm) electromagnetically expanded with a ten-turn coil at 8kJ ………………………………………………………………………109 4.19 True strain FLD for electromagnetic expansion of a short (3.17cm) sample with a ten-turn coil at 8kJ. ……………………………………………………………..110 4.20 Current vs. Time plot for short sample (3.17cm) electromagnetically expanded with a ten-turn coil at 8kJ. Peak current = 94 kA, Rise time = 24 µs.………….110 4.21 Picture of 1.74 cm sample electromagnetically expanded with a two-turn coil at 6.4 kJ. The skewed shape of the sample can be clearly seen. ………………….111 4.22 Current vs. Time plot for 1.74 cm sample deformed with two-turn coil at 6.4kJ. Peak current = 170 kA, Rise time =16 µs. ……………………………………..112 4.23 Results from Maxwell 2D for the case of a 4-turn coil with a short tube (2.22cm). (a) Flux lines, (b) Magnetic filed and (c) Current Density ……………………..116 4.24 Picture of a 2.22cm sample electromagnetically expanded with a four-turn coil at 4.8kJ. …………………………………………………………………………...116 4.25 Results from Maxwell 2D simulations for the case of four-turn coil with a long tube (8.51cm). (a) Flux lines, (b) Magnetic field and (c) Current density ……..118 4.26 Results from Maxwell 2D simulations for the case of four-turn coil with same length (3.17 cm) tube. (a) Flux lines, (b) Magnetic field and (c) Current density ………………………………………………………………………………….120 4.27 Results from Mathematica code for the case of 4-turn coil with sample of same length at 7.04 kJ. (a) Velocity, (b) Strain rate, (c) Radius, (d) Stress, (e) Temp. Rise and (f) Current vs. time. …………………………………………………..123 4.28 Comparison between experimental and calculated current traces for the case of a 4-turn coil electromagnetically expanding a 3.17cm sample at 7.04 kJ.……….124 4.29 Variation of Peak primary current with root of energy for all samples tested with two, four and ten-turn coils. ……………………………………………………126

5.1 The weak band model showing band orientation in reference configuration ….136 5.2 AA6063-T6 uniaxial quasi-static true stress vs. true strain plot: experimental data and corresponding best fit ……………………………………………………...140 5.3 Comparison of the experimentally determined current density profile with the simulated current density profile. Simulated dimensionless strain profile also shown. These plots correspond to a 31.7mm tube deformed with a 4-turn coil at 6.72 kJ of energy (case (a)). ……………………………………………………144 5.4 Comparison of simulated and experimental forming limits for an AA6063-T6 tube of length 3.17 cm electromagnetically expanded with a 4-turn coil at 6.72 kJ of energy (case (a)). ……………………………………………………………….147 5.5 Comparison of simulated and experimental forming limits for an AA6063-T6 31.7 mm tube deformed with a 10-turn coil at 8 kJ of energy (case (b)). …………...148 5.6 Comparison of simulated and experimental forming limits for an AA6063-T6 85.1 mm tube deformed with a 4-turn coil at 7.52 kJ energy (case (c))……………..148 5.7 Comparison of simulated and experimental forming limits for an AA6063-T6 85.1 mm tube deformed with a 10-turn coil at 13.92 kJ energy (case (d)) ………….149

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5.8 The localization angle Φ as a function of strain ratio, for the simulation of the 31.7 mm tube deformed with a 4-turn coil at 6.72 kJ (experiment (a)). …………….150 6.1 FLD for AA6061 and added low and high rate forming data [4] ………………162 6.2 Predicted hydrostatic stress and void volume fraction history for failed elements in the free form simulations. The hydrostatic stress is normalized by yield stress. Both values keep increasing till failure [1]. ……………………………………164 6.3 Hydrostatic stress and void volume fraction history for the top surface of a sheet undergoing die impact. Reversal in void volume fraction and hydrostatic stress after impact is evident [1]. ……………………………………………………...165 6.4 Schematic of the tube die impact experimental setup ………………………….167 6.5 Die arrangement with steel die, spacers and actuator used in tube die impact experiments. Tube sample has not been shown in the picture. ………………...170 6.6 Velocity vs. Radius profile schematic for an expanding tube specimen, with and without impact. …………………………………………………………………171 6.7 Picture of sample of length 3.17cm, impacted with a cylindrical die at 8.4kJ. Necking and tearing due to the sample not completely filling the die in some areas is evident. ………………………………………………………………...174 6.8 Picture of sample of length 3.17cm, impacted with a cylindrical die at 11.2kJ. No necking or tearing is evident. Sample completely filled the die cavity. ……….175 6.9 Pictures and forming conditions for electromagnetically formed samples with and without impact. …………………………………………………………………176 6.10 Current vs. Time trace for sample impacted at 11.2 kJ. Peak primary current = 176 kA, Rise time = 18µs. ………………………………………………………….177 6.11 Picture of experimental setup for tube flanging ………………………………..178 6.12 (a) Flange die side view and (b) top view. ……………………………………..178 6.13 Pictures of flanged samples at increasing energy levels. ……………………….179 6.14 Limit strains in tubes flanged at different energy levels. ……………………….179

7.1 (a) Ductility as a function of launch velocity. Open symbols indicate the average uniform elongation and the solid ones represent the measured total elongation. (b) Photographs of the original ring geometry and fragmentation after a total strain of 45% [2, 3]. ……………………………………………………………………...186 7.2 (a) Schematic of the tensile test geometry and failure morphology. Right side of sample is driven and failure produced at driven end at high velocity beyond the second critical velocity. (b) Plot of experimentally observed strain to failure as a function of velocity with predicted behavior. [5]. ……………………………..188 7.3 Neck development in terms of level curves of q, a normalized inelastic strain rate, over the θ-time plane [11]. First (left) a non-local zone of slow thinning appears (large parabola), which gives way to (right) a cascade to instabilities with different levels of strain. Some of them result in fracture. ……………………..192 7.4 Influence of sample aspect ratio of tensile sample on ultimate strain [19]……..196 7.5 Circumferential extension without failure in free form electromagnetic expansion of AA6061-T4 and OFHC Cu samples vs. Ring height [20]. ………………….197

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7.6 Schematic of experimental ring expansion setup with camera and oscilloscope. 201 7.7 Electromagnetic ring expansion experimental setup picture with capacitor bank, actuator and ring specimen. …………………………………………………….201 7.8 Schematic of five-turn brass wire actuator used for EM ring expansion experiments. (a) Front view, (b) Top view ……………………………………..202 7.9 Picture of insulation covered five-turn ASTM B16 brass wire actuator ……….202 7.10 Picture of 1x1, 2x2 and 4x4 AA5754-O ring specimens ………………………204 7.11 Engineering Stress vs. Strain plot for longitudinal and transverse sections of the AA5754-O tube used for making the rings …………………………………….205 7.12 Picture of experimental setup with the actuator-ring assembly and the Pearson and Rogowski probes used for measuring primary and induced currents. …………207 7.13 Picture of experimental setup with high speed camera and light assembly.……207 7.14 Experimental results for a 4x4 ring electromagnetically expanded at 2.96 kJ. (a) Pictures of expanded ring, (b) High velocity image with camera parameters: Width = 19µs, Delay = 3µs and number of cycles = 5, (c) Position vs. time plot, (d) Velocity profile …………………………………………………………….212 7.15: High velocity images shown to scale for 4x4 rings electromagnetically expanded at (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ , (e) 4 kJ and (f) 5.6 kJ. ……...213 7.16 Position vs. Time plots for 4x4 rings measured from high velocity images for ring expanded at energy of (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ , (e) 4 kJ and (f) 5.6 kJ……………………………………………………………………215 7.17 Velocity vs. Time profile for 4x4 rings measured from high velocity images for ring expanded at energy of (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ , (e) 4 kJ and (f) 5.6 kJ……………………………………………………………… ...216 7.18 Experimental results for a 2x2 ring electromagnetically expanded at 1.04 kJ. (a) Picture of fragmented ring, (b) High velocity image with camera parameters: Width = 17µs, Delay = 4µs and number of cycles = 5, (c) Position vs. time plot, (d) Velocity profile……………………………………………………………. 218 7.19 Experimental results for a 1x1 ring electromagnetically expanded at 0.56 kJ. (a) Picture of fragmented ring, (b) High velocity image with camera parameters: Width = 14µs, Delay = 4µs and number of cycles = 4, (c) Position vs. time plot, (d) Velocity profile. …………………………………………………………….220 7.20 Experimentally measured peak velocity vs. energy of launch plot for 4x4, 2x2 and 1x1 rings. ……………………………………………………………………….221 7.21 True uniform and total strains for a number of 4x4 rings electromagnetically launched at increasing energy levels. …………………………………………..222 7.22 True uniform and total strains for a number of 2x2 rings electromagnetically launched at increasing energy levels. …………………………………………..223 7.23 True uniform and total strains for a number of 1x1 rings electromagnetically launched at increasing energy levels. …………………………………………..223 7.24 Combined true strain vs. peak velocity plot for all aspect ratio rings.………….226 7.25 Number of necks and fractures for rings of all three aspect ratios, as a function of the peak velocity attained by them during electromagnetic expansion. ………..229

xviii

7.27 Current vs. Time trace for a 1x1 ring electromagnetically launched at 0.8 kJ. Peak primary current = 32 kA, Peak induced current = 18kA. ………………………234 7.28 Current vs. Time trace for a 2x2 ring electromagnetically launched at 1.04 kJ. Peak primary current =39.12 kA, Peak Induced current = 56 kA. ……………..234 7.29 Current vs. Time trace for a 4x4 ring electromagnetically launched at 2.16 kJ. Peak primary current = 55.2 kA, Peak Induced current = 106 kA. …………….235 7.30 (a) Number of necks qmax , and (b) critical stress or extension sc, as a function of extension velocity Vo / VP for different aspect ratios α [12]. …………………..239 7.31 Comparison of experimentally measured and numerically [12] calculated number of necks as a function of non-dimensional velocity. …………………………...240 7.32 Fracture strain vs. non-dimensional velocity for all three aspect ratio rings for comparison with numerical results. …………………………………………….241

8.1 Schematic of ring die-impact experimental setup………………………………249 8.2 Picture of ring die-impact experimental setup ………………………………….250 8.3 Front view of cylindrical die placed on the coil with the help of spacers and screws. ………………………………………………………………………….252 8.4 Velocity vs. Radius profile schematic for an expanding ring specimen, with and without impact. …………………………………………………………………254 8.5 Picture of 2x2mm rings. Innermost ring is undeformed ring, followed by free formed rings launched at 0.96 kJ (one failure, circumferential strain = 22.9%) and 1.76 kJ (eleven failures, circumferential strain = 34.8%). The outermost ring is the die impacted ring (no failures, circumferential strain = 40.3%).…………...255 8.6 Picture of 4x4mm rings. Innermost ring is undeformed ring. The middle ring is a freely expanded ring launched at 3.36 kJ (one failure, circumferential strain = 39.2%). Outermost ring is die impacted ring at 5.6kJ (no failure, circumferential strain = 39.5%). ………………………………………………………………...257

9.1 Three broad classifications of failure modes in high velocity forming. (a) Necking, (b) Shearing and (c) Spall-like dynamic rupture with little macro plasticity …..261 9.2 Few pictures of necked samples, formed at high velocities ……………………261 9.3 Pictures of few samples sheared at high velocities. …………………………….262 1.1 A dynamic deformation system with different areas of a target impacted by a projectile, experiencing dynamic fracture [3]………………………………….265 9.5 Dynamic fracture for A, plate impact; B, explosive loading and C, expanding ring [5]. ……………………………………………………………………………...267 9.6 Reflection of a shock wave from a free surface and development of a spall fracture [1]. ……………………………………………………………………………...267 9.7 (a) Calculated and observed velocities of free surface of target indicating spall and no spall conditions, (b) Void volume distribution in the central region of a 6.5mm thick target [6]. …………………………………………………………………269

xix

9.8 (a) Spalling by brittle fracture in low-carbon steel (b) Spalling by ductile void formation in nickel [3]. …………………………………………………………270 9.9 Micrographs of spall in AA6061-T6. (a)Void growth indicative of ductile spall, (b) Cracks indicative of brittle spall [8, 9]. ………………………………………...270 9.10 (a) Cracked second phase particles and voids in spalled AA2024. (b) SEM micrograph showing ductile fracture of matrix [10]. …………………………..271 9.11 SEM micrograph of spall in Cu containing second-phase particles [11].………272 9.12 (a) Complete spalling of AISI 1008 steel and nickel plate. (b) Incipient spall damage in copper [3]. …………………………………………………………..273 9.13 (a) Electrohydraulic forming setup and truncated conical die geometry [18], (b) Picture of sample formed at 9.4 kJ, (c) Picture of disc (with slanting fracture surface), which popped out from the central region of the sample. ……………276 9.14 Optical micrographs of sections of disc sample at various magnifications. ……277 9.15 SEM micrographs of the fracture surface of the disc sample ………………….278 9.16 SEM micrographs from another disc sample show a crack. ……………………279 9.17 (a) Experimental setup, (b) Wedge shaped punch used for impact, (c) Picture of high strength steel sample formed with impact against the punch. …………….280 9.18 Optical micrographs at various magnifications for Sample 2. …………………281 9.19 SEM micrographs of the fracture surface of Sample 2. ………………………..282 9.20 (a) Experimental setup schematic, (b) Picture of copper sheet impacted with a flat plate at 10.4 kJ. …………………………………………………………………283 9.21 Optical micrographs of cross-sections of copper sheet at various magnifications. ………………………………………………………………………………….284 9.22 Picture of mounted sample showing spall in AA6061 rod welded with AA6061 tube [20]. ……………………………………………………………………….285 9.23 Optical cross-sections of the spalled AA6061 rod. …………………………….285 9.24 Optical micrographs of cross-sections of AA5754-O rings. (a) Undeformed, (b) Free formed at 3.6 kJ and necked, and (c) Die impacted at 5.6 kJ …………….287

LIST OF TABLES

Table

2.1 Electrical resistivity values of common metals ...... 15

4.1 Chemical Composition of ASTM B16 Brass wire used for making actuators [11]…………………………………………………………………………...….91

4.2 Electrical and Mechanical properties of ASTM B16 Brass wire used for making the actuators...... 91

4.3 Different configurations of coils and sample lengths used in experiments……....93

4.4 Average values of mechanical properties of AA6063-T6 as determined from tensile tests………………………………………………………………………………96

5.1 Material parameters of AA6063-T6 tube obtained from experiments...... 139

5.2 Material parameters used from AA6061-T6 [yadav]...... 139

5.3 Density, specific heat, plastic work conversion factor and resistivity of Al...... 140

5.4 Material parameters for FLD of AA6061-T6 ...... 141

5.5 Parameters from tube expansion experiments for four test cases...... 143

7.1 Dimensions of the three different cross-section rings...... 203

7.2 Mechanical properties of the AA5754-O tube, used for making ring specimens.205

xxi

CHAPTER 1

INTRODUCTION

Formability is a fundamental material behavior, by virtue of which, a metal can undergo plastic deformation up to failure, under a suitable loading in a process. The ability of a sheet material to be bent or formed into intricate parts along with its strength, availability and cost are some factors responsible for its widespread use in a large variety of industrial and consumer products. Formability under conventional forming regime, is a function of a number of factors like material properties (e.g. strain hardening coefficient, strain rate sensitivity, anisotropy ratio) and process parameters (e.g. strain rate, temperature). For conventional sheet metal forming, Forming Limit Diagrams (FLD’s) are a realistic and efficient diagnostic tool for evaluating formability and estimating the technological possibilities of producing a component without design problems.

There is significant interest in the automobile industry to make body components and panels with aluminum or high strength thin steel sheets, with the main advantages of weight savings and the accompanied fuel efficiency in mind. However the propensity of

these metals to neck and tear at relatively low strain levels makes it difficult to use them for making geometrically complex parts, with conventional forming processes. As a result there is renewed interest in High Velocity Forming (HVF) techniques, which have been proven to address many issues of conventional metal forming processes through dramatic improvements in formability of metals. These methods were discovered in the late 1800’s and saw some application in forming thick plates in the 1930’s. Between about 1950 and the early 1970’s the U.S. government funded numerous studies on the development and application of high velocity forming. There are a number of HVF methods based on the source of energy. These are all similar in the respect that the workpiece is given kinetic energy early in the process and then forming is largely inertial as kinetic energy is dissipated as plastic deformation.

Electromagnetic forming (EMF) is a non-contact HVF method that technique where large forces can be imparted to a conductive metallic work piece by pure electromagnetic interaction. It can be used to accelerate the sheet to velocities on the order of 200 m/s over a distance of a few millimeters. This technique like some other HVF methods leads to improved formability, improved strain distribution, reduction in wrinkling, control of springback and interference fit between the sheet and the die in addition to other economic benefits like lighter tooling and equipment, lower die tryout time, one-sided dies etc. It can be effectively used to form intricate shapes out of materials with low quasi-static ductility like aluminum alloys and high strength steels, thereby overcoming the limitations imposed by conventional forming techniques.

The main purpose of this work is to provide experimental insight which can be used for

developing models of high velocity formability. High velocity formability of sheet metals

and the factors that influence it were studied. We found that high velocity formability is

different and in some ways more complex than quasi-static formability. The factors that

need to be considered in high velocity formability are inertia, impact, sample dimensions,

changes in constitutive behavior, boundary conditions, dynamic fracture with

fragmentation and possibly new failure modes. As a result a possible Forming Limit

Diagram under high velocity forming conditions will be different from quasi-static

forming. However more data and understanding are needed to be predictive. We have

made only a start. It is clear from these studies that there are ways to use high velocity

forming to exceed conventional forming limits.

A large number of studies have been performed that analyze the fundamentals of quasi-

static formability. This study contributes to doing the same for high velocity formability.

HVF methods have the potential to become the future of the automobile industry.

However more fundamental research needs to be done so that ultimately practical

guidelines can be established and so that these technologies can be utilized in an effective

way to form metals that are difficult to form conventionally. Greater practical insight into

the fundamentals of high velocity formability is needed. This has been attempted in this

study by performing experiments which can establish the factors that affect high velocity formability and their contribution to it.

Chapter 2 discusses high velocity forming in detail. It covers the basics about HVF, the

different HVF techniques based on the source of energy. The advantages of high velocity

forming and how it is different from conventional forming techniques is also discussed.

In Chapter 3, a review of the parameters which greatly influence formability at

conventional forming velocities is presented. The concept of Forming Limit Diagrams,

their use, factors that effect it and the broad theories that are in use for analytically predicting them are studied with the aim of understanding the main features of ductility

like plastic instability and necking at conventional sheet metal forming velocities and to

lay a foundation for understanding formability at high velocities. The main factors that

influence high velocity formability like inertia and impact, changes in constitutive

behavior, are presented along with studies in the literature which examine them.

In Chapter 4 tube expansion experiments have been presented. The main goal here was to

study basic concepts of high velocity formability and generate data including different

strain states, in the absence of impact, which was used to develop an analytical model for

generating forming limit diagrams under high velocity forming conditions. This study has

been presented in Chapter 5. These experiments were further extended to include die

impact in order to illustrate the benefits of arresting an expanding sample by die impact on the number of failures in it. Tube flanging experiments were also performed to further

illustrate the influence of impact (by inertial ironing) on formability. These studies are

given in Chapter 6.

Ring expansion experiments were conducted to understand the influence of sample slenderness (aspect ratio) and inertia on formability. These experiments and their results are presented in Chapter 7. They emphasize the significance of inertia in stabilizing deformation and postponing failures, in the high velocity regime. Comparison with mathematical results, further illustrate this point. Ring free expansion experiments were also extended to include die impact to quantify the beneficial influence of die impact.

This study is discussed in Chapter 8.

Chapter 9 is an introduction to the different modes of failure commonly observed in the high velocity regime. Few samples from previous experiments in the High velocity forming group at OSU, were analyzed metallographically to understand the failure observed in a few specimens. It is believed that a new mode of failure – ‘spall-like dynamic rupture’ is seen in some samples after die impact that is different from the conventional modes of necking and shearing. Micrographs of a few such samples along with background about spall fracture are presented.

Chapter 10 lists the conclusions and future work for the different studies undertaken. The motivation for those studies, the method used in them and the new insights gained from them are summarized.

Throughout this document, chapters which describe each experiment have been written in a self-sufficient manner to include studies in the literature, motivation, experimental technique, results, conclusions and bibliography. 5

CHAPTER 2

HIGH VELOCITY METAL FORMING

2.1 INTRODUCTION

High velocity forming (HVF) is the shaping of materials by rapidly conveying energy to them during short time durations [1]. They derive their name from the high workpiece velocities that are characteristic of them. The workpiece is quickly accelerated away from the energy source as it acquires kinetic energy, which is dissipated as plastic work during metal deformation. High velocities, typically ranging from 50-300 m/s, are acquired. The factor that distinguishes the HVF techniques from the conventional metal working processes is the magnitude and time of application of pressure.

In the case of HVF, a very high pressure is applied in very short time duration such that inertial forces and kinetic energy in the workpiece are significant [2]. Owing to the rapid release of energy involved in them, they are also referred to as High Energy Rate

Forming (HERF) processes. Under dynamic forming conditions, physics is very different

and inertia becomes an important factor. In conventional forming conditions, inertia is

neglected as the velocity of forming is typically less than 5 m/s while typical high

velocity forming operations are carried out at workpiece velocities of about 100 m/s [2].

2.2 HIGH VELOCITY FORMING METHODS

There are a number of methods for HVF mainly based on the source of energy used for

obtaining high velocities. The common methods are explosive forming, electrohydraulic forming (EHF), electromagnetic forming (EMF) and gun forming. These are all similar in respect that the workpiece is given kinetic energy early in the process and then forming is largely inertial as kinetic energy is dissipated as plastic deformation. The physics that determines the as-launched velocity profile of the workpiece is quite different in each of the forming methods. Figure 2.1 shows a schematic of a few common methods based on the source of energy. These processes have been in use since the early 1950’s and in the

old literature there is evidence of different techniques like pneumatically driven

hammers, exploding gasses, burning of propellants among others [1]. A brief description

of the popular HVF methods will be provided next.

2.2.1 Explosive forming

Explosive forming [4-8] is a high velocity forming technique that utilizes the chemical

energy of explosives to generate shock waves through a medium and use them for

workpiece deformation.

Figure 2.1: Methods of high velocity forming based on type of energy used [3].

Depending on the position of the explosive charge relative to the workpiece, this technique is divided into two categories – standoff and contact. In a Standoff operation, energy is released at some distance from the workpiece and is propagated through an intervening medium (typically water), in the form of a pressure pulse. On the other hand, in a contact operation the explosive charge is detonated while it is in intimate contact with the workpiece. Thus there are differences in energy requirements and the mechanical behavior of the workpiece. In general large standoff distances produce greater amounts of stretch forming while lower standoff distances increase the amount of draw.

The average strain rate in an explosive forming operation is about 10-100/s. Common explosives include TNT, RDN, Pentolite and smokeless powder. Their deformation velocities range from <1 m/s to 8300 m/s which can create pressures up to 23 GPa [2]. 8

(a) (b)

Figure 2.2: Schematic examples of typical explosive forming operations. (a) Use of detonation cord to form a cylinder (b) Method forming a flat panel [1, 7].

Figure 2.2 shows some common configurations in which explosives are used. Explosive

forming was one of the most widely used high rate forming technique for large and bulky

components, typically for military operations. It was mostly used for low-volume

production of complex parts of tough metals. The use of explosives facilitated the

fabrication of these parts without having to build massive and complicated machine tools.

It is one of the only affordable methods of fabricating large sections from thick plates like sections of ships, large nuclear reactor components and heat exchangers [9]. Figure 2.3 shows pictures of few specimens created by explosive forming. The main advantages of this technique over more conventional techniques are that one-sided dies are used which reduces the time needed for creating prototypes and also the cost of the system. Forming can be achieved with or without a die [8].

(a) (b)

Figure 2.3: Actual results from typical explosive forming scenarios: (a) 10-foot diameter domes formed explosively from AA2014-0 [6], (b) a 310 stainless steel spark guide and its perform, (c) 321 stainless steel aircraft panel formed in a single operation (d) 6 in. diameter corrugated cylinder formed from 0.01 inch thick A-286 stainless steel [7].

2.2.2 Electrohydraulic forming (EHF)

Electrohydraulic forming is a high velocity forming technique in which an electrical arc discharge in a liquid is used to convert electrical energy to mechanical energy for metal deformation. A large amount of energy stored in a capacitor bank (typically 10-100 kJ) is discharged across a spark gap or wire bridging an electrode gap, submerged in a liquid

(usually water) bath, over a very short time (typically < 100 µs). This vaporizes the surrounding fluid and creates a high intensity shock wave in it that provides transient pressures, which force the workpiece in contact with the fluid, into an evacuated die.

Studies [1, 10] have depicted that the important variables to be controlled in this process are the electrode configuration, spark plug, bridge wire and discharge energy. The different electrode configurations needed to form different types of components as well as a detailed description of the other process parameters is listed in [4]. Various applications like bulging, forming, flanging, drawing and piercing can be accomplished by Electrohydraulic forming technique if the design requirements are not achievable with conventional equipment. Unlike electromagnetic forming (discussed later) however, electrohydraulic forming is not suitable for radial compression or swaging operations, as the energy source is usually a point or a line. On the other hand electrohydraulic methods are not limited to forming metals that are good electrical conductors. Electrohydraulic and explosive forming are conceptually similar as in both the processes, high intensity shock waves are transmitted to the workpiece. Electrohydraulic forming has various advantages over explosive forming like its substantially lower noise level and less hazard.

The potential for this process was recognized in the mid 1940’s. Interest was shown in

EHF by several U.S automakers but due to the inherent process control issues like the

consumable nature of the discharge electrodes, need for media containment and its

inherent irreproducibility, it was not widely used for mass production.

2.2.3 Electromagnetic forming

Electromagnetic forming (EMF) is one of the most attractive high velocity forming

techniques that gained significant acceptance in the commercial metal forming industry.

In this non-contact technique, large forces can be imparted to a workpiece by pure

electromagnetic interaction. It uses the energy of strong magnetic field to plastically

deform metal at high speed. Figure 2.4 shows a schematic of a general electromagnetic system. It in principle uses a capacitor bank, a forming coil and an electrically conductive workpiece. A significant amount of energy (usually between 5 and 200 kJ,) is stored in a large capacitor by charging to a high voltage (usually between 3,000 and 30,000 volts).

When the capacitors, connected in series with the forming coil, are charged and switched, the large current transient causes a high intensity magnetic field around the coil. The currents take the form of a damped sine wave and can be understood as a ringing

Inductance-Resistance-Capacitance (LRC) circuit.

L2 , R2 , I2 Lc , Rc Probe 2 d Q1 Workpiece (L1I1 + MI 2 ) + R1I1 + = 0 dt C1 Coil d (L2 I 2 + MI 1 ) + R2 I 2 = 0 I1 Rb dt 1 dM Probe 1 P = I I L m 1 2 p A dh

Where L1=Lp+Lc , R1=Rb+Rc

Figure 2.4: Schematic of a general electromagnetic forming system

The key governing equations are

d Q1 ()L1I1 + MI2 + R1I1 + = 0 ……. (2.1) dt C1

d ()L I + MI + R I = 0 ……. (2.2) dt 2 2 1 2 2

Where L1, R1 and C1 are the inductance, resistance and capacitance of the bank and actuator. L2, R2 and are the inductance and resistance of the workpiece. M is the mutual

inductance between the actuator and workpiece. Detailed calculations of the coupled

system are complex [10-13]. The peak current is typically between about 104 to 106

amperes and the time to peak current is on the orders of tens of microseconds. This

transient magnetic field induces eddy currents in the nearby conductive workpiece, which

run in a direction opposite to the primary current in the actuator. These eddy currents in 13

turn produce an associated secondary magnetic field in the workpiece. This causes the two conductors – coil and workpiece, to repel each other with the Lorentz repulsion force and rapidly move the workpiece away from the restrained coil [1, 10-13]. When this force is greater than the workpiece material’s yield strength, permanent plastic deformation results. The magnetic pressure is proportional to the volume of the field and square of the field strength. Different types of coils like solenoids, single turn or multi- turn pancake, hand-wound or machined actuators can be fabricated for different applications.

A primary requirement of the process is that the metal to be formed should be an electrical conductor. The efficiency of the conversion from electrical to mechanical energy is a function of the electrical conductivity of the workpiece. Materials with an electrical resistivity of 15 µΩ.cm or less are ideal materials for the process. Table 2.1 shows the electrical resistivity values of common metals. Poor conductors can be formed with this technology only with the use of a highly conducting metal as driver to push the workpiece. Materials of lower conductivity demand higher ringing frequency for effective forming. For these reasons metals with high conductivity such as aluminum and copper are very well suited to electromagnetic forming and also developing a relatively high system ringing frequency is often quite important [2].

There was wide interest in the research on Electromagnetic forming techniques in the

1960’s. The first commercial magnetic forming machine was marketed in 1962 [1].

Later, the research efforts were largely abandoned. However, companies like Maxwell

Magnaform continued marketing the equipment, mostly for assembly operations. The

EMF group at the Ohio State University revived interest in the technology in the early

‘90s and has published various papers and thesis [14-40], demonstrating the use of the

EMF process in metal forming.

Electrical resistivity Metal (µΩ cm)

Aluminum alloys 3-6

OFHC copper 1.7

Steels 15-30

Pure iron 9.8 Titanium 160 Magnesium 9.2

Brass 6-10

Table 2.1: Electrical resistivity values of common metals

EMF is mainly used to expand, compress or form tubular shapes. It has been commercially used for the past thirty years mostly for the joining and assembly of concentric parts and compression crimp sealing. Such assemblies can be designed to optimize axial or torsional strength such that the joint strength can exceed the strength of the parent tube. Figure 2.5 shows pictures of few parts made by EMF.

Figure 2.5: Typical axisymmetric parts made by EMF [41].

EMF has various advantages like enhanced formability, better surface finish, reduced springback and wrinkling, exceptional dimensional tolerance, inexpensive tooling, high reproducibility and potential for automation. The magnetic field used for forming requires no lubrication, leaves no tool contact marks and requires no clean up after forming. With manual feed equipment, production rates of 600-1200 assemblies per hour are typical while with fully automated equipment; a rate of 12,000 operations per hour has been demonstrated in production.

Details of the present applications [42] of EMF, physical concepts and calculations [43-

45] related to electromagnetic metal forming can be found in past publications. Baines et al [45] in 1965 published one of the earliest studies on EMF of thin walled copper and aluminum tubes with solenoidal coils and flat circular diaphragms using a flat spiral

pancake coil. Several experimental works for studying tube bulging [46, 47] and

developing a process control technique for EMF [48] can be found. In EMF, the

distribution of forming pressure on the workpiece is dependant on the coil configuration

[49, 50]. The correspondence between electromagnetic field intensity and magnetic pressure can be exploited in the design of electromagnetic forming coils. The final workpiece morphology after electromagnetic forming is related to the coil shape [30]. For any situation, there are an optimum number of coil windings [34]. Numerical studies [32,

34, 51, 52] model the electromagnetic forming processes.

2.2.4 Gun forming

There is considerable evidence in the literature [53-61] about the use of free flying projectiles to penetrate, perforate and even weld metal plates. However there have been very few attempts to use this as a practical metal forming process. G.G Corbett et al [53] effectively summarized the recent research in the wide range of projectile-target configurations in the field of impact dynamics. The work of a large number of people in the area of dynamic loading of plates and shells has been reported. Typically they involve tests in which plates are struck by hard steel spheres at velocities ranging from 150-2700 m/s [54] ; spot welding different materials by high-speed water jets [56-59] ; use of polymeric projectiles for impact spot welding of thick and very high strength plates with an industrial stud driver gun at impact velocity around 750 m/s [56]. The effect of projectile nose geometry [57, 58, 60, 61] on the weld interface has also been studied by the use of different nose geometries. High velocity projectiles launched with a rifle gun 17

have also been used for spot impact welding of aluminum – steel sheets wherein it has been demonstrated that the strength of the joint is even higher than the sheets [31, 35].

Seth [36] used the same rifle for forming high strength steel sheets against a die with a hemispherical cavity.

2.3 Advantages and challenges of using aluminum alloys

Conventional forming processes are often limited by ‘forming window’ due to problems like wrinkling, springback and low formability of materials. Thin sheets are particularly difficult to form via the conventional route as even small compressive stresses in the plane of the sheet produces wrinkling. Thus difficult to form materials like high strength steels and some aluminum alloys, create problems when formed conventionally.

2.3.1 Advantages of using Al alloys vehicles

The use to aluminum alloys over some heavy steels, in the automobile sector can lead

to various advantages:

• Aluminum parts can be twice as thick as steel but still 40% lighter and 60%

stiffer. Their lower mass leads to improved fuel economy, acceleration, and

braking performance. Up to 8% fuel savings can be realized for every 10%

reduction in weight from substituting aluminum for heavier metals [62]. For

example, the Audi A2 has been designed to use 3 liters of fuel per 100

kilometers. Its body weight is 43% less than a conventional steel body [63].

• Aluminum parts have excellent collision energy management characteristics

and can be designed to absorb the same energy as steel at only 55% of the

weight thereby leading to safety in automobiles in the instance of a crash [62].

• Aluminum’s unique combination of lightweight, high-strength and corrosion-

resistance characteristics make it the ideal alloy for developing marine

applications like high-speed aluminum ferries, Bicycle frames, baseball bats,

golf clubs etc. [64].

• Aluminum alloys also have superior recycling ability which becomes

increasingly important in terms of the total life cost of vehicles [27, 65].

2.3.2 Challenges associated with using Al alloys with conventional forming

Formability of aluminum is poor in conventional stamping methods. There is a cost

penalty associated with using aluminum alloys compared to steels. There is a 100% to

200% cost premium for weight reduction of 20% to 40% in vehicles [66] due to the

following:

• Al alloys have low formability (approximately 2/3rd) in comparison to most steels

[66]. They have a tendency to neck and tear at relatively low strain levels, making

it difficult to use them to make geometrically complex parts conventionally. It

also leads to a high scrap rate expectancy which is a cost premium.

• The press forming of aluminum alloys has problems in comparison to steel

principally due material parameters like low strain rate hardening, normal

anisotropy, strain rate sensitivity; and a high galling tendency [67-69]. The

significance of these parameters will be explained in Section 3.2.

• In addition to that, in conventional processes the die tryout with mating male and

female dies is always slow and expensive. Heavy and expensive tooling and large

number of press operations add to the expenses. The typical die design and tryout

time with Al alloys is more than 50% higher than that of steel.

• Al alloys have high springback due to a low elastic modulus (approximately 1/3rd

of steels). This in turn adds to the die tryout time and cost [66].

Due to all these factors, virtually all Al vehicle construction so far has been relatively low volume! Audi A2 and A8 are the only Al intensive vehicles in mass production. Low volume vehicles include the Lotus Elise, Acura NSX and Plymouth Prowler.

2.4 Advantages of High velocity forming

There is great interest in the automobile sector for using Al alloys in vehicles but it is

challenging to use them with conventional forming techniques as outlined in the previous section. HVF methods accelerate the metal sheet at velocities which are 100 to 1000

times greater than the deformation rates of conventional quasi-static forming such as the

sheet metal stamping (~0.1 m/s to ~100 m/s) [27]. They have been proven to address

many of these issues, which is responsible for the recent resurgence of interest in developing these technologies.

2.4.1 Formability enhancement

Over the past several years it has been shown that that the formability of metals improves

dramatically at high velocities. This extended ductility at high velocity is referred to as

‘hyperplasticity’. A few examples of this improvement in formability with HVF methods

will be discussed.

Balanethiram et al [14-16] performed EHF experiments with AA6061 T4, copper and interstitial free iron sheets, forcing the metals into a conical die with an apex angle of 90° at velocities near 150 m/s. Experiments were also performed with quasi-static hydraulic pressure and it was observed that while the metals completely filled the cone and showed increased ductility in the EHF technique, the same was not possible with the quasi-static, conventional technique. This is shown in Figure 2.6. At high velocities, plane strain stretching to the amount of 100% was observed without failure as compared to 20-40% strains at low velocities.

To vacuum sample Die

water

Bridge wire (a) (b)

Figure 2.6 (a) Schematic of the set-up used in hydroforming and electrohydraulic forming (b) Comparison of forming AA2024 T4 sheet into a conical die using a hydroforming process (left) and using electrohydraulic forming (right) [10]. 21

Altynova et al [24] also showed that strains to failure in ring expansion of 6061-T4,

6061-T6 and annealed OFHC copper can be increased to about two-fold relative to quasi- static ductility. Seth et al. [36, 37], examined the high velocity formability of high strength steel sheets formed on impact with massive curved punches of different configurations. The sheets were launched with electromagnetic force at velocities ranging from 50-220 m/s, using a flat spiral electromagnetic coil and an aluminum driver sheet.

The schematic of the experimental setup is shown in Figure 2.7a.

The fifteen steel materials were in the form of sheets of thickness ranging from 0.15-0.38 mm. They had quasi-static ductilities ranging from 1.3 - 25.6% while their tensile strengths varied from 330 - 675 MPa. The sheets were launched at increasing energies onto the axisymmetric and wedge shaped punches, till a sample in which failure had just initiated was obtained. Figure 2.7b shows pictures of a few of these samples.

Punch Work piece with driver sheet below

Coil

To capacitor bank (a)

(b)

Figure 2.7: EMF experiments with high strength steel sheets. (a) Experimental setup schematic and picture, (b) Pictures of samples of two steels deformed by the small and large axisymmetric and medium wedge punches. These are samples in which failure had just initiated [37].

It was observed that the failure strains of all the steels were dramatically increased beyond those obtained in tensile tests. Formability seemed to depend largely on local boundary conditions as dictated by the punch/tool geometry used. 23

Steel 1 : High velocity Steel 2: High velocity 40 Steel 3 : High velocity

ngg %) Steel 4 : High velocity

n Steel 5 : High velocity i 30 a

r Steel 1 : Quasi-static t Steel 2 : Quasi-static Steel 3 : Quasi-static 20 Steel 4 : Quasi-static

Major s Steel 5 : Quasi-static 10

0 -20 -10 0 10 20 30 Minor strain (engg %)

Figure 2.8: Plot indicating the aggregate high velocity and quasi-static ductilities of the five steels.

The strain distribution obtained depended on the shape of the punch used for impact. The forming limits of steels with both very low and high quasi-static ductility were similar in

HVF thus indicating that in high velocity impact, the quasi-static ductility of the material is not of primary importance to the material’s formability. Figure 2.8 shows the forming limits obtained from all the steels. In the plot, solid symbols have been used to indicate high velocity data while open symbols have been used to indicate corresponding quasi- static ductilities. Although there are large differences in quasi-static ductilities, such is not the case with the high velocity ductilities of these steels. All the points for the high velocity data lay approximately in the 20–55% strain range. The high velocity formability of these materials is 2-30 times higher than the corresponding quasi-static values.

In another study, Seth [36] used different nose-shaped projectiles to press sheet metal

(same high strength steels as in the previous example) into a 9.5 mm hemispherical die cavity. A 9mm caliber commercial air rifle was used to launch the projectiles at velocities ranging from 135 - 205 m/s as measured by a chronograph.

Die Chronograph

Safety Cage 9 mm air rifle

2.7mm 9.5 mm (a)

(b)

40 v1-soft lead-541 ngg %) 30

(E v1-hollow end-352 n i ra t 20 v1-round end-147 r s o j v1-polymer-776 a

M 10

0 -40 -20 0 20 40 Minor strain (Engg %)

(c)

Figure 2.9: Gun forming (a) Experimental setup schematic with die profile, (b) Pictures of sample formed with a polymeric bullet. 0.38mm sheet formed without tear (left), while 0.15mm failed sheet. (c) Strain distribution for one steel (quasi- static ductility 3.7%) with different bullet geometries.

Bullets were launched onto the workpiece at decreasing velocities till the lowest velocity achievable by the gun was reached and a sample with minimal or no tear was obtained.

Figure 2.9a shows a schematic of the experimental setup along with the profile of the die cavity. Figure 2.9b shows pictures of samples formed with a polymer bullet, with a pointed nose. It was observed that while the relatively thick (0.38mm) steels could be formed into the die cavity without any failure, the thin (0.15mm) steels were impossible to form without tearing. The greater inertial stabilization of the thicker material was responsible for this behavior. Once again from these experiments, high strains to failure were observed from all steels tested, despite differences in their quasi-static ductilities. 26

Figure 2.9c shows that limit strains of up to 40% were observed from steels with quasi- static ductility of 3.7% which is almost a ten times increase in formability. Boundary conditions, like the bullet nose geometry were also important.

From the above examples, it is clear that formability can be exceptionally enhanced under high velocity conditions, especially under the influence of a high velocity impact. Thus these technologies are ideal for forming materials with low inherent ductility, like aluminum alloys and high strength sheets and can very well overcome the limitations imposed by conventional forming techniques.

2.4.2 Wrinkling is suppressed

Wrinkling is caused by the presence compressive stresses in the sheet and excess material in the die during a forming operation. It necessitates a change in direction of the sheet metal from the original launch path. At high velocities, wrinkling is suppressed as this change in direction of the material, is inhibited by its momentum. The deviation from the smooth-shape velocity profile requires acceleration, which is resisted by inertial forces.

With the help of electromagnetic ring compression and sheet – conical die impact experiments [25], it has been shown that the number of wrinkles monotonically reduce as the launch energy is increased. Thus high velocity forming techniques make metals like aluminum more formable by widening the forming window.

2.4.3 Springback is reduced

The origin of springback stress lies in the differential elastic strains through the thickness of the sheet while forming. Springback is reduced in high velocity forming due to through-thickness compressive stresses that act in the sheet, at impact with the die that cause the residual elastic strains in the sheet to be minimized. If adequate energy is provided to the workpiece, it impacts the die in all areas while still possessing sufficient kinetic energy and experiences reduced springback [25]. Upon impact the large pressures cause the struck surface to displace and the rebound wave tends to produce an interference fit between the sheet component and the die [2]. This leads to higher dimensional tolerance and hence reduced die tryout time.

2.4.4 Economic advantages

High velocity forming methods also have other economic advantages that make them preferable to conventional forming processes. Cost of die-tryouts is reduced, as only one- sided dies are needed. The need and cost for binders and lubricants is eliminated. These processes involve lighter tools and fewer press operations. Thus much lighter and smaller equipment is required with dynamic phenomena even to produce high surface pressures needed for applications like coining. High velocity forming processes have potential for automation and also for the combination of forming and assembly operations.

2.5 SUMMARY

High velocity forming methods are gaining popularity due to the various advantages associated with them. They overcome the limitations of conventional forming and make it possible to form metals with low formability into complex shapes. This in turn has high economic and environmental advantages linked due to potential weight savings in vehicles. The main methods of HVF and the advantages associated with them were discussed in this chapter. The main reasons responsible for the huge improvement in formability are inertial stabilization of necks, inertial ironing on impact and changes in constitutive behavior. These factors along with the factors that affect formability in conventional forming conditions will be studied in details in the next chapter.

BIBLIOGRAPHY

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CHAPTER 3

LITERATURE REVIEW

3.1 INTRODUCTION

Formability is the intrinsic ability of materials to be formed into various shapes in a forming operation. It is possibly the most important material property in metal forming and can be quantitatively defined by the effective strain at failure [1, 2]. It is a function of a number of factors like material properties (e.g. strain hardening coefficient, strain rate sensitivity, anisotropy ratio) and process parameters (e.g. strain rate, temperature). For conventional sheet metal forming, Forming Limit Diagrams (FLD’s) are a realistic and efficient diagnostic tool for evaluating formability and estimating the technological possibilities of producing a component without design problems.

The main purpose of this chapter is to review studies in the literature about the formability of sheet metals and the factors that influence it. It starts with a review of how material parameters like strain rate sensitivity, strain rate hardening, and anisotropy; and

superimposed hydrostatic pressure effect formability. Then the concept of Forming Limit

Diagrams, their use and the broad theories that are in use for analytically predicting them are studied with the aim of understanding the main features of ductility like plastic instability and necking at conventional sheet metal forming velocities. Thereon the factors which effect formability at high forming velocities, like inertia and impact, are studied. Once again the analytical models for understanding the importance of inertia at high velocities have an underlying emphasis on how a change in necking pattern

(multiple necks) manifests itself as formability enhancement at high velocities.

Although good advances have been made towards understanding the reasons behind enhancement of formability at high velocities, they are just steps in the learning curve for developing a comprehensive understanding of the same. The theoretical models for determining FLD’s have been studied here to develop a working knowledge of the limits to sheet metal formability and to lay a foundation for understanding formability at high velocities. The existing models for studying the influence of inertia at high velocities help to understand the dynamics of neck formations at high velocities. They also indicate the areas where further experimental evidence is needed. Future electromagnetic forming experiments can be designed to provide support for existing and future studies. The ultimate aim is to get practical insights into the factors that affect formability so that high velocity forming processes can be utilized in an effective way to form metals that are difficult to form conventionally.

3.2 MATERIAL AND OTHER FACTORS AFFECTING

FORMABILITY AT CONVENTIONAL FORMING VELOCITIES

It has been a well known fact that material and some external parameters are important factors that effect formability at conventional forming velocities. The effect of these parameters at high velocities has not been that widely studied and in some cases has been shown to have different effect on formability. Most of the studies in the literature cater to the importance of these factors only at conventional forming velocities and will be presented next.

3.2.1 Strain hardening

Strain hardening, or work hardening, is the intrinsic ability of materials to strengthen or harden with increasing strain level [3]. It is one of the most important properties influencing the formability of sheet metals. When a material is cold worked, its strength and hardness increase by virtue of strain hardening. However, a consequence of that is also lessening of the remaining ductility. [1]. During plastic deformation, a region undergoing thinning can resist further deformation by virtue of strain hardening and can spread deformation to its neighboring regions, thus promoting uniform thinning. The parameter reflecting this behavior is strain hardening index, n = d(lnσ )/ d(lnε ). The uniform elongation (strain on the material until a neck forms) is mostly controlled by n.

Thus a high n delays the onset of necking and improves formability. According to

Considere criterion, the end of uniform elongation occurs when the true work-hardening

rate, equals the true strain [4]. The greater the strain-hardening of a sheet, the better it performs in processes where there is considerable stretching as the straining will be more uniformly distributed and the sheet will resist tearing [5]. For most metals, n has values between 0.1 and 0.5 [6].

3.2.2 Strain rate hardening

Strain rate hardening is the intrinsic resistance of materials to strain localization. The parameter reflecting this behavior is strain rate sensitivity index, m = d(lnσ )/ d(lnε&) ε

[3]. Positive values of m lower the rate of growth of strain rate gradients between the region of instability (like a diffuse neck) and the uniform region and thus increases post- uniform elongation by postponing the localization of deformation into a sharp neck [1].

Conditions that promote high m values also promote high failure strains in the necked region. Rate sensitivity of most sheet materials is small (between 0 and 0.3) at room temperature. For steel it is slightly positive and for aluminum, zero or slightly negative

[5]. Rate sensitivity is temperature dependant and at temperatures greater than half the absolute melting point, the rate sensitivity climbs rapidly [6]. Superplastic alloys have a very high value of m, in the range of 0.2 to 0.4 and have post-uniform elongation up to thousands of percent [3]. It has been shown that there is strong synergy between values of m and n in determining the total elongation and this interaction becomes more important for large values of m and n. Total elongation increases almost linearly with n, in the region of small m, and the rate of increase is larger at higher m values [7]. 39

3.2.3 Anisotropy

Anisotropy of sheet metal is the variation in its plastic behavior with orientation. It can be

expressed by the normal anisotropy coefficient, R = ε w /ε t . Thus it is the ratio of plastic strain in the width direction to that in the thickness direction in the specimen [1]. The difference in properties for a sheet aligned with the rolling, transverse or 45o directions is referred to as planar anisotropy. The value of R averaged over measurements taken at different angles to the rolling direction is expressed by R . If R is greater than unity, it indicates that the material has a high thinning resistance due to greater strength in the through-thickness direction. It also implies high strength in biaxial tension, while a low

R-value indicates easy thinning and hence a low biaxial strength [1]. In deep drawing parts, a high value of R allows deeper parts to be drawn [6]. In shallow, smoothly contoured parts like auto body outer panels, its higher value reduces wrinkling [5].

3.2.4 Inhomogeneities

Inhomogeneities in the form of variations in composition, strength or thickness have a strong effect on formability. Uniform elongation is strongly dependent on the inhomogeneity factor f, which is expressed as the ratio of the areas or thickness of the inhomogeneous and uniform regions. The biggest imperfection (a region of lowest thickness or strength) becomes a potential site for the onset of instability and localized necking thereby leading to reduction in formability. The strain in an imperfection region accelerates ahead of the uniform region after the tension maximum is reached thereby resulting in pre-mature failure [5]. 40

3.2.5 Damage accumulation

Damage accumulation in ductile metals limits formability in most bulk forming operations. It involves the generation of porosity via nucleation and growth of voids. The porosity of a solid is characterized by the volume fraction of voids, Cv =Vv /V where Vv and V represent the volume of void and total volume of the voided solid respectively. For conventional metallic alloys, at the initial stages of deformation, the void volume

-5 -4 fraction, Cv is very small (~10 for Al alloys and ~10 for steels [8]) however it can increase drastically with deformation. In general, factors that increase the strength of the matrix, like cold work, solid solution strengthening, precipitation etc., reduce ductility because of the higher stresses encountered in forming a material with high flow strength.

The stages involved [9] in void coalescence are shown in Figure 3.1 and are as follows:

a) Nucleation of voids: A void forms around an inclusion or second phase particle when sufficient stress/strain is applied to either cause it to debond at the matrix interface, or fracture. Fracture of hard non-metallic inclusions or interfacial decohesion of particles from the matrix cause voids to form. Large non-metallic inclusions are often damaged during fabrication and thus may crack or debond even prior to plastic deformation, making void nucleation relatively easy [9]. The presence of hydrostatic stress, σm =

(σ1+σ2+ σ3)/3, where σ1, σ2 and σ3 are the principal stresses, is also instrumental in the nucleation of voids [10]. The volume fraction, nature, distribution and shape of the inclusion effect formability of metals [6, 11]. It has been shown that the true strain to fracture decreases rapidly with increasing volume fraction of second phase particles.

b) Growth and coalescence of voids: Once voids form, further plastic strain and hydrostatic stress cause them to grow. If the initial volume fraction of voids is low, each void can be assumed to grow independently. Upon further growth, voids interact and eventually neighboring voids coalesce either by direct impingement or by localization of strain [11]. Accelerated linking of voids can take place along a narrow band joining the large voids or an accelerated necking due to the growth of a second population of microvoids between the first populations of large voids. Several models have been proposed for to establish the criterion for void coalescence [10]. The rate of damage progress may vary greatly in different materials and depends on the stress state in the process being used [12].

Figure 3.1: Stages in void nucleation, growth and coalescence in ductile metals [9]. 42

3.2.6 Superimposed hydrostatic pressure

Superimposed hydrostatic pressure can affect the mechanical behavior of metals as has been revealed by experimental observations [13] of tensile testing under compressed fluid

(water or oil) i.e. in the presence of hydrostatic pressure. For most structural materials like steel and aluminum while superimposed hydrostatic pressure has a very minor influence on their yield and ultimate tensile strengths, there is a large effect of pressure on ductility, obtained from reduction in area measurements. Experiments also demonstrated an increase in the strain hardening coefficient for a high strength aluminum alloy under the influence of hydrostatic pressure [14].The work of Bridgman [15] on a variety of steels, through metallographic investigations of cross-sections of fractured specimens revealed that the pressure induced ductility changes were due to pressure- induced suppression of damage. The work of French [16] showed that increased levels of pressure inhibit the total number of voids present at equivalent levels of strain.

Al Al-1Si-0.7Mg-

AA6061

AA2124

AA7075-

Figure 3.2: Fracture strain vs. superimposed hydrostatic pressure for a few Al alloys [17]. 43

A study [17] done on Al 6061 and brass revealed that superimposed pressure suppresses damage (reduced area fraction of voids) associated with inclusions and that the size of voids reduces with even moderate amounts of pressure increase. Figure 3.2 shows the dependence of fracture strain of a few Al alloys on the superimposed hydrostatic pressure. The different alloys behave differently under the influence of hydrostatic pressure due to different heat treatments and failure mechanisms. Pressure retards void nucleation, growth and coalescence by counteracting the detrimental hydrostatic tensile stresses that evolve during deformation and promote internal damage [13]. In general, for any forming operation, the maintenance of a compressive mean stress increases formability. A few industrially important forming processes which utilize the beneficial aspects of negative mean stress are extrusion, wire drawing, rolling and forging.

3.3 FORMING LIMIT DIAGRAMS

The forming limit of sheet metal is defined to be the state at which a localized thinning of the sheet initiates during forming, ultimately leading to a split in the sheet [2]. In a sheet metal forming process, it is essential to gauge the forming severity with respect to necking. The most commonly used method for the same is the Forming Limit Diagram

(FLD), developed by Keeler [18] and Goodwin [19] and is a representation of limiting in- plane principal strains, withstood by the sample without failure. FLDs are constructed by plotting the minor and major strain combinations at different strain states and drawing a curve through the maximum values of major-minor strain combinations observed without failure. 44

Figure 3.3: A typical Forming Limit Diagrams along with different strain states [3].

Figure 3.3 displays a typical FLD [3]. As shown in the figure, the strain combinations below the forming limit curve are considered to be safe while the one above it are considered to be associated with failure. To analyze an actual sheet forming operation, during die-tryout stage potential trouble points can be measured and compared to the

FLD and it can be estimated if fracture or necking will appear during forming. If the strains are near the failure curve, problems are likely to occur in production due to tool wear and day-to-day variations in other aspects like lubrication, sheet properties or tool alignment. Thus modifications in the working conditions (e.g. lubrication), part design

(e.g. fillet radii or angles) or material can be made [2].

FLD’s can be determined experimentally by stretching circle gridded specimen strips of different widths over a hemispherical punch to create different strain states ranging from uniaxial tension for the left side of the FLD to balanced biaxial tension for the right side

[1]. The sheets are stretched until the first perceptible neck is observed. Surface strain is measured by comparing the grid marked on the sheet surface before and after the forming operation. The values of circles wholly or partially in the neck are considered ‘failed’ while the strains in the circles one or more diameters away from it are considered ‘safe’.

The final shapes of the grid circle for uniaxial, plane strain and biaxial strain conditions are also shown in Figure 3.3. The level of the FLD is denoted by FLDo, which corresponds to the plane strain-forming limit (the lowest point on the curve) and represents the most severe forming condition.

A few factors that affect the FLD are the sheet metal’s mechanical properties, its thickness and the strain path or the amount of pre-strain. FLDo increases with n and m.

The value of FLDo is approximately given by n for a material with less strain rate sensitivity. However, the forming limit curve for a material with high rate sensitivity intercepts the major strain axis at a value higher than n. This is because a high value of m slows down the rate of growth of a neck, especially in biaxial tension in which necking is a gradual process beyond maximum tension. Imperfections in the material like inclusions, segregates, thickness variations etc., lower the limit strain and hence the forming limit curve. A thickness effect is also observed experimentally with the level of the FLD rising as the thickness increases.

An experimental observation is that the shape of the FLD is insensitive to the level of the anisotropy ratio R [1]. However some numerical analyses for prediction of FLD show that the shape of the yield locus is influenced by R which in turn influences the biaxial forming limit. An increasing R value increases FLDο and enhances formability. History dependant material properties have a significant influence on forming limits [20]. It has also been suggested that high R and n values as a measure of good deformation characteristics is not sufficient [21]. Tests that consider the state of stress occurring in individual zones of the drawing part are better for the evaluating sheet metal formability.

Although the FLD method has been proven to be a useful tool in the analysis of forming severity, it might not always be accurate in complicated forming operations because of different modes of deformation experienced by the part at different stages of a single forming operation. Even a change in lubrication can affect the path of deformation. It has also been reported that a criterion based on stress, instead of strain is better for evaluating forming limits [21]. Despite complications, the FLD method largely forms a guideline for practical sheet metal forming applications. In summary, the sheet metal forming process can be limited by various factors like local necking, tearing, fracture before necking, wrinkling etc. Practically, a sheet can only be deformed by tensile forces. Typically, a forming window [5] in which plane stress sheet forming is possible is identified and shown in Figure 3.4.

Figure 3.4: The forming window for plane stress forming of sheet [5].

3.4 THEORETICAL MODELS FOR CALCULATING FLD’S

A number of analytical models have been proposed to predict FLD’s. The broad classifications, models and studies based on them, with the inherent aim of understanding formability, will be discussed further.

3.4.1 Theoretical models for determining FLD’s based on necking theory

The earliest analysis of plastic instability in uniaxial tension was conducted by Considère

[23]. According to him, for a rate independent material, diffuse necking (length of neck approximately equal to width of sample) initiates at maximum load, where the load increment caused by strain hardening is equal to the load decrement caused by geometric softening. The critical strain for diffuse necking was given by є* = n. 48

Swift [24] used the Considère criterion to determine limit strains in biaxial tension. In some early works for calculation of FLD Swift’s criterion of diffuse necking was used in conjunction with Hill’s [25] criterion of localized necking (length of neck approximately equal to thickness of sheet), which assumed that for uniaxial tension, the necking direction is coincident with the direction of zero elongation. According to Hill’s theory, which neglects strain-rate sensitivity, the local necking strain, є* = 2n at uniaxial tension and є* = n at plane strain conditions. His theory adequately predicted the observed behavior for failures in the negative minor strain region. However, this theory could not explain the strain localization phenomena in the positive minor strain region of the FLD.

Also, the predicted FLD with these theories was lower than the experimental one.

3.4.2 Theoretical models for determining FLD’s based on sheet non- homogeneity

The hypothesis that localized necking initiates from a pre-existing non-homogeneity was first proposed by Marciniak and Kuczynski (M-K model) [26, 27], to predict the occurrence of localized necking under biaxial tension. The defect, which could be structural or geometric, is characterized for mathematical analysis as though it is thinner than the rest of the sheet, and is assumed to be in the form of a linear groove oriented perpendicular to the axis of the largest principal stress. The two zones in the material: uniform region ‘a’ having a thickness ta and the groove region ‘b’ having the thickness tb, are shown in Figure 3.5 [1].

a b a

tb t a

Figure 3.5: Schematic of sample with groove ‘b’ and uniform region ‘a’ [1].

The coefficient of geometrical non-homogeneity is expressed as f = tb /ta. The strains parallel to the groove in both the regions are assumed to be same. Deformation outside the groove is assumed to occur such that the ratio of stresses and strains remain constant.

M-K showed that deformation within the groove occurs at a much faster rate than the rest of the sheet and the concentration of strain in the groove eventually approaches plane strain. Failure is taken to occur when the ratio of strain in the groove to the strain in the uniform region becomes too high (infinity in theory, about 10 in practice) [2].

Thus the inhomogeneity leads to an unstable growth of strain in the weaker regions subsequently leading to localized necking and failure. The principal strains in the uniform region when the strain in the groove is localized represent the limit strains and define a point in the forming limit curve. By varying the strain ratio different points on the FLD can be obtained. The level of the predicted FLD is fairly sensitive to the assumed value of the initial imperfection f, which is an adjustable fitting parameter in the theory.

Early calculations of the FLD based on M-K theory in conjunction with Hills

(anisotropic) quadratic yield function [29-31] resulted in discrepancies with experimental data and usually the theoretical FLD had too high a slope and over predicted the equal- biaxial (є1= є2) limit strain. Sowerby and Duncan [31] using Hill’s quadratic yield function showed a strong dependence of the right side of the FLD on value of R, a fact which has not been observed experimentally. Hosford [32, 33] made calculations for the right side of the FLD with a proposed non-quadratic anisotropic yield function and showed that with his yield function a much better fit with experimental FLD is obtained and the right hand side of the FLD becomes independent of R. Thus the yield criterion used has a significant influence on the forming limits in biaxial tension [34].

Other works [35-38] have also shown that M-K theory can be used successfully to calculate FLD’s if it is suitably coupled with an appropriate yield function, which accurately describes the plastic behavior of the material. Butuc et al. [37] developed a flexible code for FLD prediction based on M-K model, which could integrate different yield functions and hardening laws in the form of sub-routines. They observed that better results are obtained when the M-K analyses is used with some yield functions [38] other than the typical ones like Von Mises or Hills and thus the success of the model depends on the choice of the applied constitutive laws. Some studies [33, 34] have utilized the M-

K model to show the beneficial effects of material properties on forming limit strains. It has been shown that a higher value of n delayed the onset of instability to a higher strain value while a higher value of m resulted in a more gradual localization to the plane strain condition thereby increasing the limit strain. 51

In the M-K analysis, a linear imperfection of infinite length is assumed. Some works [39,

40] have challenged this assumption. Burford and Wagoner [39], with the help of finite element analysis showed that both the size and aspect ratio (length/width) of the imperfection affect the localized necking process. The rate of strain localization decreased for a reduced aspect ratio defect (thus higher limit strains) and this influence was found to be of comparable magnitude to the inhomogeneity factor f. It has also been stated [40] that the simplifications made in the M-K model of a defect of infinite length contribute to overstating the impact of a real material defect and thus the M-K curve represented a more severe forming condition than the finite-length notch.

3.4.3 Theoretical models for determining FLD’s based on incorporation of damage evolution

In sheet metal forming processes, formability can be limited by the occurrence of internal damage (void) evolution, which can eventually lead to localized necking failure. The description of plastic deformation of voided solids requires theories of plasticity in which yielding is dependant on hydrostatic stress, as opposed to classical plasticity theories which assume that yielding is independent of hydrostatic stress. In the past three decades a number of theories have been proposed with significant simplifications, in which the material containing voids of different sizes is assimilated to a porous material with a uniform distribution of voids of cylindrical or spherical shapes. The efforts in this direction were largely pioneered by Gurson [41].

Based on an approximate analysis, of a spherical unit cell of matrix material with a spherical void in the center, such that the matrix obeys the Von Mises yield criterion,

Gurson incorporated the aspects of a progressively cavitating solid into a constitutive framework which includes the effect of hydrostatic stress (which is instrumental in the formation and growth of voids), to determine the stresses required to initiate or sustain plastic flow within the plastically dilating porous solid. The original model overestimated ductility, as it did not consider the coalescence of voids. Tvergaard and Needleman [42,

43] modified the Gurson yield function to give the Gurson-Tvergaard yield function in which calibration coefficients based on comparisons with three-dimensional finite element solutions were included. The void volume fraction as included in the original model was modified such that when the volume fraction of voids reaches a critical value, the effect of void coalescence can be included and the onset of plastic instability is accelerated. Other modifications to the Gurson yield function, by integrating it with other anisotropic yield criterion, have also been proposed [44-47] to include the effects of normal anisotropy in sheet metals. Using the modified and unmodified forms of Gurson’s constitutive relation, a number of studies have been conducted to carry out analytical and finite element simulations of failure in porous ductile materials [48-54] and also for the determination of FLD’s [44, 48, 53-58].

Most of the studies for the determination of FLD’s have been proposed as improvements to the M-K model by integrating the aspect of void growth (through Gurson’s yield function) in the analysis. Metallographic analysis [52] as well as analytical studies [46,

47, 53, 54] of void growth in sheets deformed under different loading conditions have 53

revealed a larger growth rate and volume fraction of voids in sheets deformed under equibiaxial conditions as compared to other conditions. This is because of a higher tensile condition of stress in biaxial loading, which promotes void growth. Thus the consideration of void growth in a model for determining FLD’s, leads to predictions of earlier necking in stretch forming thereby reducing the slope of the predicted FLD in the positive minor strain region. This leads to analytical predictions being closer to the experimental FLD’s [53-58]. Another motivation for these studies is the fact that in the original M-K model, an unrealistically high value of inhomogeneity (if it is assumed to be associated with the surface roughness) has to be assumed to obtain reasonable agreement with experimental data.

One approach for predicting FLD’s is to consider a region of inhomogeneity, which has a volume fraction of particles (that eventually nucleate voids), higher than the average and thus has a lower flow stress than the rest of the sheet. Melander [53] followed this approach for predicting FLD’s for copper alloys. In his model the sheet was considered to have a uniform thickness and contained just one fitting parameter, namely the ratio of fraction of particles in the neck to the average fraction of particles. The predicted FLD was in good correspondence with experimental data. However a drawback of this model was that quite a high value of the fitting parameter had to be assumed and it was obtained empirically in relation to the average particle fraction in the copper based alloys studied, and was not valid for other alloys with different particle distributions.

Thus in order to obtain accurate analytical predictions with a realistic inhomogeneity parameter, the geometric instability should be assumed to result from the combined effects of a thickness imperfection and damage accumulation. This also reduces the imperfection sensitivity of the M-K model. This approach has been followed in a number of studies [54-58] for predicting FLDs. Ragab et al. [55] used this approach and demonstrated that assuming a higher initial void volume fraction caused a reduction in limit strains. Also, the calibration coefficients in the Gurson model had a great influence on limit strains especially under equibiaxial loading conditions. In the original M-K model an unrealistic inhomogeneity (f =0.990 to 0.995 instead of f = 0.999) had to be assigned to bring the calculated limit strains within the order of magnitude of their formulation. Good agreement with experimental FLD’s for copper alloys was obtained with realistic values of assumed thickness imperfection (f = 0.999) and initial void volume fraction (~10-4).

In another study [56] for FLD prediction with the same basic model but a different yield criterion for voided materials, Ragab et al. obtained agreeable results with experimental

FLD but that model was very sensitive to the value of an empirical fitting parameter whose value when changed from 2 to 1.8 resulted in a drop by 50% in the values of the limit strain under equibiaxial conditions. Brunet and Morestin [57, 58] incorporated the

Gurson's yield function for FLD prediction and compared the theoretical FLD with and without the consideration of damage, with experimental ones. While the curve without damage globally overestimated the limit strains, the curve with damage, was in better agreement with the experimental FLD. 55

3.5 SUMMARY FOR CONVENTIONAL FORMING

Formability in conventional forming is greatly effected by material parameters like strain hardening, strain rate sensitivity and anisotropy. High values of n, m and R are desirable for improved formability in sheet metal forming. Inhomogeneities and damage from existing inclusions and second-phase particles can limit formability. Superimposed hydrostatic pressure can improve formability by reduction in damage. In general a forming window limited by necking, tearing and wrinkling, exists for plane stress sheet metal forming.

FLD is an effective and easy to use tool for predicting formability limitations under quasi-static forming conditions. From the models for the determination of FLD’s, it can be gathered that integration of damage parameters (through Gurson's model) in the M-K model (with a very small thickness inhomogeneity), yields better results in terms of agreement, quantitatively as well as qualitatively with experimental data. The consideration of damage evolution produces flattening of the FLD for positive strains and also the need for assuming an exaggerated value of inhomogeneity (which cannot even be linked to material parameters) is avoided.

A major shortcoming of all these models seems to be their sensitivity to different empirical parameters, which tend to change the results dramatically. In most of these models these empirical parameters are determined by best fitting with experimental results and are not linked to measurable material parameters. Thus in the absence of valid 56

experimental results, they are of not much practical significance for replacing the need for doing experiments for obtaining FLD’s. However these models are quite useful in understanding the onset and development of instabilities leading to failure; which in turn is very important in understanding formability.

3.6 HIGH VELOCITY FORMABILITY

High velocity forming is characterized by imparting a high velocity to a workpiece and converting this energy to plastic deformation by constraint of the part or impact with a die. The factor that distinguishes the HVF techniques from the conventional metal working processes is the magnitude and time of application of pressure. In the case of

HVF, a very high pressure is applied in very short time duration such that inertial forces and kinetic energy in the workpiece are significant [59].

The methods used to impart velocity to a sheet are quite distinct from those traditionally used in that launch does not usually involve contact with a hard punch or tool and typically only a single tool is used. In electromagnetic forming, body forces are produced by the interactions of magnetic fields without any contact and can be effectively used to create bulges that take on roughly spherical section geometries although for great dimensional precision, die impact is needed. These forming devices can also be integrated with conventional tooling thereby greatly expanding the capabilities of traditional stamping. Also, these unusual modes of imparting energy to a body along with the short loading duration offer the ability to use very lightweight forming systems [60]. 57

As described in Section 3.4, in the past years, estimation of quasi-static (strain rate < 1s-1) ductility has been extensively studied in terms of plastic instability and flow localization, i.e., necking. In conventional sheet metal forming processes the amount of stretching or set of available strain states that can be attained, are described by the FLD and was discussed in details above. In general, in high velocity forming one is not constrained by the usual limits prescribed by the FLD. It does not include inertial effects and therefore cannot describe formability in high velocity forming. Instead, if launch and boundary conditions are properly chosen, ductility far beyond typical quasi-static ductility can be achieved [60].

Extensive analytical and experimental investigations [61-69] have revealed that the formability of many materials increases at high deformation velocities and these observations were discussed in Chapter 2. Some factors – inertia, impact and changes in constitutive behavior, which become more important to formability at high forming velocities and are responsible for a change in the observed necking pattern and for enhancing formability. A few details about the reasons for enhanced ductility along with existing analytical and experimental studies, will be presented next.

3.7 EFFECT OF INERTIA ON HIGH VELOCITY FORMABILITY

Inertia is the inherent property of a body that makes it oppose any force that would cause a change in its motion. At high velocity, as bodies like to maintain their launch velocity profile, inertia becomes an important factor that stabilizes deformation. In order to 58

understand how inertia aids formability enhancement and change in necking pattern at high velocities, a number of numerical analyses [70-84], mostly of high velocity expansion of rings have been carried out. These analyses can be roughly placed into two categories based on the way instability is assumed and studied.

3.7.1 Models based on initial inhomogeneity and study of growth of a neck

In these studies [70-77], an initial inhomogeneity in the form of a constriction or a taper in the wall thickness or the width of the sample is introduced and the emphasis is on studying the growth of a single neck at an imperfection under dynamic conditions. These models are fairly predictive of ductility in the broad sense and help to explain the difference in the conditions of a growing neck under quasi-static and dynamic conditions and to understand the influence of inertial stabilization on a growing neck. The broad outcome of these studies is that inertia slows down the rate at which a neck grows at an imperfection. A few individual studies will now be discussed.

Fyfe and Rajendran [70] studied the effect of inertia on the failure of rings expanding at high strain rates of about 104 s-1. They assumed the thickness inhomogeneity region

(constriction), B, to have had a slightly higher volume fraction of voids as compared to the uniform region, A. Their analysis revealed that the growth of local strain in the non- uniform region was inhibited at increasing strain rates due to inertia such that the quasi- static failure criterion of ‘dєB /dєA → ∞’ was no longer applicable. They also carried out experiments of ring expansion under dynamic conditions and showed that the predicted 59

uniform strain at failure was in good correspondence to the observed values. However, their theoretical results were very sensitive to the value of the assumed thickness imperfection and the difference in void volume fraction in the two regions and they stipulated that the latter factor could only be an ‘educated guess’.

Xu and Daehn [71] used a one-dimensional dynamic numerical simulation of sheet tensile and expanding ring tests to study the influence of velocity on ductility. A taper in the specimens, representative of a geometric defect, was also assumed. For their system

(discretized into elements), the one-dimensional momentum equation on each node was given by

Mi üi + Fi = 0 ……(3.1)

where Mi üi is the inertial force, üi and Mi being the nodal acceleration and mass respectively and Fi is the resultant nodal internal force. At low velocities, as material acceleration is small, the inertial forces are insignificant. For both the tensile and ring specimens, the total elongation at failure was observed as a function of extension and expansion velocities respectively. The results showed that the ductility for both the specimen geometries was invariant to velocity at low test velocities. However, in the case of tensile samples, beyond a critical velocity, the total elongation at failure increases, but drops beyond a second critical velocity, such that failure occurs at the mobile end of the specimen. This velocity is a result of wave propagation effects and is often referred to as the ‘Von Karman velocity’. On the other hand in the case of the axisymmetric ring expansion, beyond a critical velocity, elongation increases (without a drop) 60

monotonically as a function of velocity. Thus it was revealed that varied sample launch conditions, i.e., sample geometry, could produce very different results. Formability enhancement with velocity occurs only in a particular range of velocities. Hence the establishment of a proper velocity field is essential.

Using a similar model, Xu et al. [72] deduced that the total elongation of the samples is increased by inertia at high velocities primarily due to an increase in the post-uniform elongation because significant acceleration is developed only after necking commences.

The primary influence of inertia in improving formability at high velocities is by stabilizing deformation against neck growth. Balanethiram and Daehn [73] used a one- dimensional model of a tensile test specimen, to explain how inertia suppresses neck growth. They analyzed the velocity distribution in a tensile test sample before and after necking. During uniform deformation, the velocity of the tensile sample varies linearly with its position. However, at localization most of the deformation takes place in a vanishing narrow zone and necking produces a change in the velocity profile to an approximate step function. The change in velocity over time, i.e., acceleration, is resisted by inertia. The inertial forces were calculated as:

L 2 ∆v VAρ x 0 Fin = ρA dx = (x − ) …..(3.2) ∫ L 0 ∆t ∆t 2L

Where ρ, ∆v, A, L are the density, local change in velocity, specimen cross-section area and half-gauge length respectively and V is the velocity of the sample end.

Figure 3.6 shows the right hand side of the tensile specimen with the associated velocity, inertial force and stress profiles. The inertial force is tensile and is not present in the quasi-static deformation. The maximum inertial forces occur at the mobile end of the tensile specimen, thereby leading to the maximum increase of stress in the uniform part of the specimen gauge length. There is a corresponding extra strain increment that develops in the region outside the neck, which reduces the strain gradient between the uniform and localized regions thereby retarding neck development. Thus inertial forces diffuse deformation throughout the specimen, leading to stabilization against neck growth, by increasing the stress at the gripped end which represents the uniform part of the sample. The specimen can receive additional elongation while failure is delayed.

Also, it is the velocity rather than the strain rate that is important in determining if there will be increased formability due to inertial effects.

L F inertial force

(a) Inertial force Position (x) L (c) localized V

y ∆v dynamic

stable static Velocit

V/2 Eng. Stress Position (x) L Position (x) L (b) (d)

Figure 3.6: (a) Right hand side of a tensile test specimen with associated (b) Velocity, (c) force and (d) stress profiles [73]. 62

Although these models were somewhat limited and ignored changes in constitutive behavior, adiabatic heating, void growth, and were one-dimensional, they successfully examined that at high velocities inertia is a first order variable that controls localization and can as much as double the strains to failure.

Tvergaard et al. [74] performed a numerical simulation for dynamic ring expansion to understand the influence of inertia. Using FEM, the formation of necks in ring segments with an imperfection in the form of a taper in their wall thickness was studied at different loading rates. They showed that while at low strain rates (<100 s-1), necking appeared at relatively low strains and in the thinnest cross-section, while at high strain rates (>1000 s-

1), necking was delayed and more than one neck appeared simultaneously, with additional necks appearing in regions without initial imperfections. The lower the loading speed, the earlier the necking began because of a lower inertial effect. Under high rate of loading, the onset of necking was also delayed (i.e. occurred at higher strain levels) by considering a sample with a higher ratio of thickness to radius. Although this model too neglected m, temperature rise and considered only the effect of inertia, the importance of inertial effects and sample thickness to high velocity formability was revealed.

Using a model, very similar to the above [74], Sorenson and Freund [75] studied the development of necks with time in a part of a ring specimen undergoing high velocity radial expansion. Their results revealed that even in the dynamically loaded specimen, in the early stages of deformation, necks evolve only from the sites around the ring

circumference at which the wall thickness is minimum. However, as deformation progresses, the mean strain rates in these sites are only slightly higher than the background rates due to inertial resistance, thus indicating that the background velocity is too large to allow neck formation at these sites. They showed that imperfection sensitivity of the material is suppressed at high strain rates.

3.7.2 Models based on Instability / Perturbation Analyses

The following studies [78-84] are mathematical analyses in which the growth of periodic instabilities with different spacings is considered. Their main objective is to capture multiple neck formation and fragmentation behavior at high velocities and to observe the number of necks that form at a particular velocity. The results of these analyses are typically compared with ring fragmentation experiments [68, 69]. These studies show that at high velocities, multiple necks with an intermediate spacing form and the growth of large-spaced instabilities (infinite spacing - one or very few necks), as observed in low velocity deformation, is suppressed due to inertia. Although the detailed results of these studies will be discussed in Chapter 7, a few individual studies will be mentioned here.

Shenoy and Freund [79] studied the dynamic necking during rapid plane strain extension of a block of strain hardening material, at a constant end velocity. They showed that at high velocities, an intermediate spacing of necks is observed and with increasing velocity, the observed spacing between necks decreases in magnitude. The analysis indicated that for small aspect ratio blocks (height or thickness/width), the number of 64

necks per unit length is an increasing function of deformation velocity and decreases with the aspect ratio of the block. With increasing velocities, an increase in ductility is shown by an increase in the dynamic stress or strain a material can support. Thus dynamic formability is influenced by the aspect ratio of the block and the deformation velocity and inertial forces are the dominating factor in deciding the necking pattern. Their importance is also reflected in the effect of aspect ratio on formability as a large aspect ratio block also has higher inertial resistance. Gurduru and Freund [80] adapted the above analysis

[79] to a homogenously deforming cylindrical rod in order to facilitate comparison with ring expansion experimental results [69]. Both these analyses neglected the curvature of the rings and other effects like m and attributed all the results to inertia.

Other studies by Fressengeas and Molinari [81], Mercier and Molinari [82 - 83] also established the stabilizing influence of inertial parameters and rate sensitivity on dynamic necking instability. There analytical results compared well with the experimental results of Altynova et al. [66, 68].

3.8 EFFECT OF DIE IMPACT ON HIGH VELOCITY

FORMABILITY

A number of experimental [62-65] and numerical studies [63, 85, 86 ] have demonstrated tremendous increase in high velocity formability when the material is formed with a high velocity impact. Balanethiram’s experiments involving high velocity sheet impact into a

conical die [62, 63] showed closer to a 5-fold increase in limit strain. Similarly experiments of electromagnetic forming of high strength steel sheets [64, 65] involving high-speed impact with relatively massive punches of different configurations, have revealed almost a 10-fold increase in formability. While inertia could be held accountable for as much as doubling the limit strain, in these experiments the material is formed significantly during high velocity die-strike. The rapid deceleration of a workpiece as it makes contact with a tool in high velocity forming produces through-thickness compressive stresses, which promote stretching of the material instead of localization.

A fairly unique aspect of high velocity forming is that when two solid bodies impact with significant velocity very high pressures are created [60]. At lower velocities the impact may be fully elastic, using linear elasticity, the impact pressure, Pi, that is developed when two semi-infinite elastic bodies labeled 1 and 2 collide at an impact velocity Vi is given as:

ρ1ρ2C1C2 Pi = Vi ρ1C1 + ρ2C2 ………..(3.3)

Here for each material ρ represents density and C is the longitudinal wave speed.

Longitudinal wave speeds are on the order of 7,000 m/s for most structural metals. For aluminum-steel and steel-steel couple’s impact pressures of 2 GPa and 5.6 GPa are generated for a 200 m/s impact. Higher pressures are available by modifying the strike material or by increasing the impact velocity. Even at modest impact speeds, it is easy to develop pressures large enough to produce plastic deformation. Thus, in addition to

inertial stabilization against neck growth, ‘inertial ironing’ due to die-strike can contribute to enhanced formability at high velocity.

Imbert et al. [85, 86] studied the effect of tool/sheet interaction on damage evolution in the electromagnetic forming of aluminum alloy sheet. They observed a very modest increase in formability over the conventional FLD in the case of forming into an open cavity (free forming) versus forming into a conical die involving die strike. They developed a model to study the change in void volume fraction with time, as the sheet metal impacted the conical die. Their analysis revealed a complex stress state including compressive hydrostatic stresses develops on the surface of the sheet during and after die impact, which result in reduction of void volume fraction. On the other hand in the case of a free-formed sample, the void volume fraction increases steadily as deformation progresses and no such reduction of damage is observed. Metallographic analysis also revealed higher porosity levels in the free-formed sample as compared to the die- impacted sample. Details about the study are discussed in Section 6.1.

3.9 EFFECT OF CHANGES IN CONSTITUTIVE BEHAVIOR ON

HIGH VELOCITY FORMABILITY

It is well know that the fundamental constitutive behavior (stress, strain, strain-rate relations) for most metals change qualitatively at strain rates above about 1000s-1 to

10,000 s-1 [87]. At these high strain rates, the apparent strain rate sensitivity of the material increases markedly [87, 88] which is shown in Figure 3.7.

As noted by Gourdin and Lassila [89], Follansbee and Kocks [87], Regazonni et al. [88], this increase in rate sensitivity can also be considered to be an artifact of comparing flow stress at constant strain, which is not a valid state parameter. If the comparison is made at constant structure with the flow stress at 0 K as the structure parameter, no such increase is observed. It is also suggested that these high rates increase the rate of work hardening and at a given strain this gives the appearance of increased rate sensitivity.

Figure 3.7: Flow stress of annealed 0.9999 copper measured at a strain of 15% as a function of strain rate [87].

Gorham et al. [90] conducted high strain rate compression tests and noted that value of strain rate at which the observed strain rate sensitivity increases, is highly dependent upon specimen size and so it may not be fully representative of inherent material properties. An estimate of inertial stress shows that inertia may be an important mechanism behind this observation. Dioh et al. [91, 92], in compression tests for strain rates up to 104 s-1 obtained using a split Hopkinson pressure bar apparatus, showed that the choice of specimen thickness in these tests significantly affects the measured flow stresses at high strain rates, with the thicker specimens showing an apparent enhancement of flow stress at high rates. Michel et al., [93] also observed a size effect on the constitutive behavior of brass sheet metal during tensile and hydraulic bulge test.

Changing the sample thickness in these tests affected the formability limits and flow stress behavior which decreased on reducing the specimen size and thickness.

Oosterkamp et al. [84], analyzed the strain rate sensitivity of two commercial aluminum alloys AA6082 and AA7108 in T6 and T79 tempers, over a wide range of strain rates from 0.1 to 3000 s-1. Uniaxial compression tests and Split Hopkinson Pressure Bar

[SHPB] tests were used for conducting experiments. Stress wave propagation effects which strongly influence test results above strain rates of 100 s-1 were also taken into account. Specimens of two different thicknesses – thick (4.5 mm) and thin (2.6mm) were tested. Numerical simulations were also performed and the results are shown in Figure

3.8 which shows the variation of flow stress with strain rate for thin and thick specimens test. It is clear that increased strain rate sensitivity at high strain rates depends on the

geometry of the samples. It should be noted though that flow stress is an inherent material property of homogenous material and cannot be geometry dependant. This increase in rate sensitivity is more pronounced for thick specimens and could be a consequence of stress wave propagation effects in tests. Similar results were also borne out by Dioh [91].

Figure 3.8: Comparison of experimental results with finite volume simulation: ●, ‘thick’ specimen; ▲, ‘thin’ specimen; o, finite volume with fitted power law. Results indicate that ‘increased’ strain rate sensitivity at high strain rates is an artifact [94].

Temperature rise during deformation could also lead to changes in constitutive behavior.

Non-isothermal temperature effects such as adiabatic shear can also intervene and produce low instability strains. Since flow stress, decreases with temperature, severe localization can take place due to dominating effects of thermal softening over the hardening influence of strain rate sensitivity or strain rate hardening. Since at high strain

rates, there is insufficient time for the heat to flow away, adiabatic localization is a possibility [95]. However, the strain rates and velocities commonly encountered in high velocity forming processes are much lower than those accompanying adiabatic localization.

Earlier experimental studies with electrohydraulic die impact forming [62, 63] and electromagnetic ring expansion [66, 68] compared the hardness measurements in high velocity formed samples, at increasing velocities, with those in a quasi-statically formed sample for different materials. This was done to study how velocity affects material constitutive behavior. Figure 3.9 shows the variation of the terminal micro hardness measurement as a function of strain for a 6061T4 sheet formed with impact at low and high velocity [62]. There is not a big difference in the increase in hardness with strain for the sample formed with high velocity impact or quasi-statically. If changes in constitutive behavior were dominating factors in HVF, the intensified work hardening would have resulted in very large differences in hardness values. Based on these results, it seems unlikely that a big jump in strain hardening or strain rate sensitivity parameters is primarily responsible for enhancement of formability under high velocity conditions.

However there is also evidence contrary to the above studies which suggest that enhanced strain rate sensitivity is just an artifact. A number of works have simulated stress strain curves at different strain rates which show a higher flow stress at high strain rates.

Among them, recent results by Pierre L’Eplattenier [96] are very convincing and are shown in Figure 3.10. 71

Figure 3.9: Variation of microhardness of 6061 T4 sheet formed with high velocity impact as a function of strain at low and high rates [62].

Figure 3.10: Stress strain curves generated with simulation in LS-DYNA with electromagnetism module [68].

These stress strain curves, were obtained by solving the inverse problem of simulating electromagnetic ring expansion experiments (Chapter 7), using the recently added electromagnetism module in a dynamic FEM software, LS-DYNA. The model considered both constitutive behavior at high strain rates (Johnson-Cook model) and also included inertial effects. This makes these results more convincing as compared to other studies which typically study one of the two parameters. These results indicate that there is higher rate sensitivity at high strain rates making the material harder. The conventional forming influence of higher m value would lead to higher resistance to localization.

Thus there is still some ambiguity regarding the influence of constitutive behavior parameters at high velocities. There is evidence supporting both schools of thought about whether or not heightened rate sensitivity is an artifact. The general belief is that heightened rate sensitivity may be real however the results need to be carefully analyzed.

These factors could be considered contributory factors that aid inertia in increasing formability at high velocities. It should be kept in mind that many high velocity-forming operations do not always include high strain rate effects. For example tube expansion at

200 m/s of a tube of diameter of 4 cm will sustain a strain rate of 104 s-1 while a 1 m diameter tube will only have a strain rate of 400 s-1 [60]. Thus, while these changes in constitutive behavior can be important, they are not universal to all forming operations, even if they are carried out at high velocity.

3.10 SUMMARY FOR HIGH VELOCITY FORMING

It was noted earlier that formability can be significantly enhanced at high velocities. The factors that may be responsible for this observation were explored. Different studies in the literature were discussed. Analytical models based on different criterion were presented to understand the influence of inertia on high velocity formability. They revealed that inertia is a dominating factor influencing high velocity formability. It acts to diffuse deformation and prevents localization. It is also responsible for the different necking pattern of multiple necking observed at high velocities. Inertial forces result in necking at intermediate spacing and prevent localized necking and failure from one intense neck as observed in quasi-static conditions.

High velocity impact is also a strong factor and has a stabilizing influence through inertial ironing and also through reduction in damage parameters due to compressive stresses that develop due to tool-sheet interaction. Strain-rate induced changes in constitutive behavior, can contribute to the enhanced formability. There is ambiguity regarding this issue and although high flow stress and increased rate sensitivity has been shown by some studies, other studies also suggest that the observed increase in strain rate sensitivity at high strain rates is an artifact. Thus these factors can be considered contributory factors which aid inertia in enhancing high velocity formability and not the primary factor.

There is a change in the failure mode at high velocities with the formation of multiple necks which ultimately lead to fragmentation but this occurs at a much higher strain than the strain at which a single neck tends to localize in quasi-static conditions. High velocity formability is very dependant on boundary conditions like the specimen geometry, the punch/die geometry, presence of impact, pressure distribution of impulse, deformation velocity etc.. Most of these factors are not as important in quasi-static conditions. Thus relatively simple tools like FLD’s might not be a useful parameter in determining formability at high velocities. The traditional FLD does not include inertial effects and thus cannot represent high velocity formability.

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[96] Unpublished research by Pierrer L’Eplattenier at LSTC, Livermore CA, using LS- DYNA

CHAPTER 4

TUBE EXPANSION EXPERIMENTS

4.1 BACKGROUND AND MOTIVATION

Existing high velocity formability experimental data involves high velocity impact and is limited to only the right side of the FLD. Extensive analytical and experimental investigations have revealed that the formability of many materials increases at after high velocity impact. Balanethiram et al. [1, 2] compared the results of forming aluminum into a conical die using quasi-static fluid pressure versus at high velocity (near 150 m/s) with

EHF of Al 6061, copper and iron sheets, and observed an increase of plane strain failure limit by 3-5 times for these materials. Seth et al. [3, 4] also observed a dramatic improvement in the formability of cold rolled sheet steel as developed in impact with a curved punch, at velocities of 50 – 220 m/s generated with EMF (Section 2.4.1).

Imbert et al. [5, 6] conducted Free form and conical die electromagnetic forming experiments on 1 mm AA5754 sheet. They observed smaller improvements in

formability in the free form case, than in the case with die impact. In the free form experiments, while safe strains beyond the conventional FLD were observed in a very narrow region, in conical die impact experiments, much higher improvement in formability was observed, and that too over a significant region of the part. Figure 4.1 shows a comparison between the FLDs for the free form and impact experiments. Figure

4.1a shows the FLD for free form samples with most of the deformed sample giving strain readings lower than the quasi-static FLD, and the higher strains being reported from a very narrow region close to the tip of the free formed sample. Figure 4.1b shows the FLD for the sample formed with impact against a conical die, in which a bigger portion of the conical sample had strains higher than the quasi-static FLD.

(a)

(b)

Figure 4.1: Engineering Major vs. Minor strain for AA5754 samples electromagnetically formed along with pictures of samples to indicate the areas of maximum strain. (a) Free formed samples formed at 5.8kV, (b) Samples formed with conical die impact at 8kV [6].

In a study by Oliveira et. al. [7], a series of EMF free-form and die impact experiments on 1mm AA5754 and AA5182 were carried out. Successive runs at increasing energy levels were done until the energy level associated with the point of failure was reached.

Figure 4.2 shows the limit strains from these free form experiments at three voltage levels, along with the quasi-static FLD. It is evident from the figure that from the free

formed samples that had not failed (5.5kV) or had just split (6kV), no improvements in formability were seen. Only in the sample that had failed (7kV- the highest energy tested), formability had improved to levels higher than the quasi-static curve. The higher strains from the 7.0 kV sample were attributed to the strain path change due to tearing along the long edge of the workpiece near the die-entry radius. Since all of the ‘safe’ strains form the FLD lay below the forming limit curve, it was deduced that very little

“hyperplasticity” or increase in formability at high velocities was seen from the free- form experiments. However it should be noted that the sample launched at 7kV also had the highest velocity of launch and hence is expected to have higher formability due to higher inertial stabilization.

Figure 4.2: FLD for AA5754 samples electromagnetically free formed at three charge voltages. The sample at 5.5kV did not fail; the sample at 6kV had just split while the sample formed at 7KV had cracked and failed [7]. 87

Also, the higher strains from these samples were all taken from ‘safe’ areas away from the line of failure. Thus there is an observed increase in formability. It should also be noted that the above experimental results were obtained at peak velocities of 250-300 m/s and strain rates of about 3500 /s (which is a reasonable but not a very high level).

Thus the existing studies which indicate a dramatic improvement in formability, typically involve high velocity impact which in itself is a big contributory factor in improving formability. A comprehensive experimental study which provides the formability of materials under high velocity conditions in the absence of impact, and can be attributed mostly to the influence of velocity and inertia, was missing.

Thus in the present study, free forming experiments were designed to study high velocity formability without impact, which included negative minor strains (drawing) in addition to the biaxial (stretching) considered previously [1, 2] in order to achieve, for the first time, a complete experimental data set for the representation of a forming limit diagram under high velocity conditions. It was desired to investigate if improvements in formability obtained from high velocity impact could also be obtained under high velocity conditions in the absence of impact unlike some previously reported studies [7].

These experiments were also designed to generate data like primary and induced current time plots, high velocity strain distribution etc. to support modeling analysis of EMF to include negative and biaxial strain states for the first time and help create a full computed representation of an FLD at high velocities. This Modeling analysis [8] was carried out

in collaboration with University of Michigan. Data gathered from the experimental current traces, like current density and rise times for primary and induced currents, were essential input parameters in the simulations. This study will be discussed in Chapter 5.

4.2 EXPERIMENTAL PROCEDURE

The experimental setup consisted of a capacitor bank connected to a solenoidal coil with a cylindrical workpiece around it. The workpiece and the coil are in close proximity.

When current is discharged through the capacitor bank, the current flowing through the coil induces a secondary current in the workpiece. The repulsion between the two fields generates electromagnetic forces to cause the expansion of the workpiece. Two Rogowski coils were also used to measure the primary and induced currents. Figure 4-2 shows a schematic and picture of the experimental setup. A few details about the experimental setup components will be explained next.

Charging Coil Circuit Capacitor Tube Bank Specimen

Ragowski Probe (R2) Ragowski Probe (R1) (a)

Capacitor bank

Tube Actuator

Rogowski Probes

(b)

Figure 4.3: Experimental setup for electromagnetic tube expansion. (a) Schematic, (b) Picture with capacitor bank, actuator, tube workpiece and Rogowski probes.

4.2.1 Capacitor Bank

A commercial Maxwell Magneform capacitor bank with a maximum stored energy of 16 kJ was used for the experiments. The energy of the bank was stored in 8 capacitors, each with a capacitance of 53.25 µF. The system had a maximum working voltage of 8.66 kV.

Both the number of capacitors and charging voltage could be adjusted to control the discharge energy and voltage.

4.2.2 Actuator

The actuators used for electromagnetic tube expansion were solenoids fabricated by commercial spring winding [9] from a 6.35 mm diameter ASTM B16 Brass wire. The 90

coils had an outer diameter of 54 mm and a pitch (center to center distance between consecutive wire turns) of 9.4 mm. The wire was covered with heat shrink-wrap tubing to provide insulation and then potted in Urethane. The high strength of brass and the epoxy provide better integrity to the coil. Table 4.1 gives the nominal composition of ASTM

B16 Brass which is a free-cutting brass. Table 4.2 lists the mechanical properties of the brass wire used for making the actuator [11].

Copper Zinc Lead Iron

61.5 Nominal 35.4 Nominal 2.5 Minimum 0.35 Maximum

Table 4.1: Chemical Composition of ASTM B16 Brass wire used for making actuators [11].

Electrical Conductivity Tensile Strength Hardness (%IACS) (MPa) (Rockwell B) 26 400 78

Table 4.2 Electrical and Mechanical properties of ASTM B16 Brass wire used for making the actuators.

The actuators were made in three different configurations so that when used along with samples of different lengths, different strain states can be created. The configurations

consisted of two, four and ten turns respectively such that the coils were of different lengths while the pitch, wire and outer diameter in each configuration were identical. The length of the wound portion in the 2, 4 and 10 turn coils was 1.7, 3.4, 8.9 cm respectively.

The length of the leg of the coil which extended beyond the wound portion was 12.5 cm in all three configurations. Figure 4-4 shows a schematic of the front and top view of a four-turn coil along with all the measurements. Figure 4-5 shows a picture of a bare four- turn coil which was later potted in epoxy before using it.

4.2.3 Workpiece

The workpiece material used in these experiments was AA6063 T6. The samples were made from a tube with an inner diameter of 57 mm and a wall thickness of 1.75 mm.

Samples were machined into different lengths using a lathe. Mainly three different configurations of sample lengths were used such that there lengths were approximately equal to the lengths of the three different coil lengths. The three different coil lengths of

1.74 cm (0.687”), 3.17cm (1.25”) and 8.51cm (3.35”) were used. To create different strain states, in addition to testing samples with coils of the same lengths, experiments were also run with some configurations of coils with samples of lengths more or less than the coil. The different configurations used are shown in Table 4.3.

R=R= 1.58 1.25 cm cm 3 . 1 4 . 2 6 c 7 m

c m 54 mm OD

R= 6.75 mm L e g

L e

n R= 1.58 cm g t h

1 2 . 5

c m

(a) (b)

Figure 4.4: Schematic of a four-turn actuator used for Electromagnetic expansion of tubes. (a) Front view (b) Top view.

Sample Length 2-turn 4-turn 10-turn Coil 1.74 cm (0.687”) X 3.17 cm (1.25”) X X 8.51 cm (3.35”) X X

Table 4.3: Different configurations of coils and sample lengths used in experiments

Figure 4.5: Pictures of a bare 10, 4 and 2-turn coils.

With the objective of measuring strains at various locations in the tube specimens after deformation, their surface was electrolytically etched with 2.5mm diameter, non-touching circles. After high velocity deformation, the circles would distort into ellipses. The measurement of the major and minor axes of the ellipse and their change from the original diameter of the circle would indicate the major and minor strains on the specimen. The Electrolytic etching was done using ‘Lectroetch’ Equipment [10]. The possible ways in which the circles can deform is shown in Figure 4.6. If the tube is stretched in uniaxial tension, the grid circles elongate in the direction of the major axis and contracts in the other. In the case of deformation under plane strain conditions, the 94

circles elongate in the major axis direction and remain unchanged in the direction of the minor axis. Under biaxial tension, the circles elongate in both directions.

(a) (b) (c) Figure 4.6

Figure 4.6: Different types of deformations possible with the grid circles. (a) Uniaxial Tension (b) Plane Strain (c) Biaxial Tension

Tensile tests were conducted on the AA6063-T6 tube material by cutting sections and then flattening out the tube carefully in a vice. The tensile samples were then water jet cut according to ASTM standard. They were 2.54 cm long and 0.63 cm wide. Samples were tested in the longitudinal and transverse directions to account for anisotropy. An average of four samples was tested in both the directions. They were tested in an MTS machine at a strain rate of 3.3x10-3.The stress strain plots for two sample runs in the longitudinal and transverse directions are shown in Figure 4.7. As is evident from the figure, there is difference in mechanical properties in the material in the longitudinal and transverse directions. The material appears to be stronger and less ductile in the transverse direction.

This can be attributed to anisotropy and the method of procuring the data which involved flattening of the tubes. Table 4.4 shows the average values of the mechanical properties of AA6063-T6 samples tested in longitudinal and transverse directions.

300

250

200 ) 6063-L1 Pa

M 6063-L4 ( 150 s

s 6063-T3 re 6063-T4 St 100

0 02468101214 Strain (engg %)

Figure 4.7: Stress vs. Strain plot for 6063- T6 tube samples cut in the longitudinal and transverse directions.

Ultimate Tensile Strength Yield Strength % Elongation (MPa) (MPa)

236.7 203 10.41

Table 4.4: Average values of mechanical properties of AA6063-T6 as determined from tensile tests.

4.2.4 Rogowski Probes

Two Rogowski probes from Rocoil [11] were used to measure the primary and induced currents. The schematic for the probe setup for measuring primary and induced currents is shown in Figure 4.8. Here Rogowski Probe R1 measures the primary current. The

Rogowski Probe R2 measures the induced current in the tube in addition to the product of the number of coil turns and the primary current. These probes were 60 cm long and had a peak range of 150, 000 Amperes/Volt. For these experiments, the setting of 100,000

Amperes/Volt was used. The Rogowski probes shown in Figure 4-8 were used in conjunction with a two-channel digital storage scope (Fluke Scopemeter 99 Series II).

The scopemeter can follow input signals from 15 Hz to 50 MHz and a rise time of less than 7 ns. From the scopemeter the current traces were transferred to a computer.

Tube Workpiece

Actuator

Rogowski Probe (R1) Rogowski Probe (R2)

Figure 4.8: Schematic showing the arrangement of Rogowski probes used for measuring primary and induced currents.

4.3 METHODOLOGY

The basic approach used in these experiments was to expand the tubular workpieces using electromagnetic forces generated between the solenoid actuators and the workpiece. The samples were launched at incremental launch velocities (by varying the energy discharged from the capacitor bank). In each case a threshold launch velocity was found that caused tears in the sample that did not reach the boundary i.e. the samples were torn but still in one piece.

Once this launch velocity was obtained, the sample was launched again at a slightly higher velocity to ascertain that if the sample is launched at a velocity higher than that particular threshold velocity, it will be torn into pieces and will not be in one piece or will have a larger tear. In addition to that launches were carried out at lower energy level to ascertain that there will be no failure in the sample below the threshold energy and also to collect experimental data at various energy levels. Thus after a series of launches, one particular sample obtained at the threshold launch velocity was used for strain evaluation.

Strains were measured and reported from areas of high strains immediately adjacent to the failure zone. In some cases, strains were also measured for samples launched at energies lower and higher than the threshold launch velocity.

Samples were launched at incremental energy levels for all the sample-coil configurations listed in Table 4.3. For each of these cases the data gathered like the current traces, strain data and pictures of samples will be presented next.

4.4 RESULTS AND DISCUSSION – FOUR-TURN COIL

With the four-turn coil, the sample lengths tested were the 3.17cm and the 8.51cm which were lengths respectively equal to and greater than the length of the coil. For each of the cases, test results will be presented for energy levels lower than and equal to that needed to initiate tears.

4.4.1 Four-turn coil with sample of the same length (3.17 cm) at 6.72 kJ

Electromagnetic expansion of a sample with length (3.17cm) same as the 4-turn coil at

6.72kJ, resulted in a deformation with minimal tearing at the edges and no big tears.

Figure 4.9 shows the picture of the deformed sample. Failure is caused by the development of a single necking band diagonally across the height of the sample. The small tear at the sample edge is near the neck band. This tear could also have been initiated by inhomogeneities like roughness at the edges. As is evident from the figure, the deformed length of the sample is almost straight. This is due to the fact that the length of the sample and the coil were almost equal. This resulted in a uniform pressure throughout the sample length.

Strains were measured in the deformed sample with the analysis of the circle grids etched on the surface of the sample. Measurements were reported from regions of maximum strain which in this case was a single necking band running diagonally across the height of the sample. The Strains were categorized as ‘safe’ when there were one or two circle

diameters away from the necking band and they were categorized as ‘unsafe’ when they were along the necking band. The measurements were made using a computer software.

Figure 4.9: Picture of 3.17cm tall sample deformed with a 4-turn coil at 6.72 kJ

Figure 4.10 shows the true strain readings from this sample, categorized as safe and unsafe. As is evident from the figure, high (true) strains of the order of 20-45% were obtained from this material which has an intrinsic ductility of 8-12%. As expected the strains were mostly uniaxial because of the straining conditions and the fact that the regions of maximum strain were clustered near the top and bottom edges.

Figure 4.11 shows the experimental current-time trace. From the trace valuable data like the peak current and rise time (time needed for the current to rise from zero to the peak value) can be obtained. These are pertinent parameters needed for numerical simulations.

The rise time is the. In this case the peak current was 128 kA and the rise time was 18µs.

100

50 )

% 40 in ( a r t 30 Safe Strains S Unsafe Strains jor 20 a Tensile test M

ue 10 Tr 0 -40 -30 -20 -10 0 10 20 True Minor Strain (%)

Figure 4.10: True strain FLD for 3.17cm sample deformed with a four turn coil at 6.72 kJ

400 Primary Current 300 Induced Current

) 200 A (k t

n 100 rre u C 0 0 50 100 150 200 250 300 350 400 -100

-200 Time (µs)

Figure 4.11: Current vs. Time plot for 3.17cm sample deformed with 4-turn coil at 6.72 kJ. Peak current = 128 kA, Rise time = 18 µs.

101

4.4.2 Four-turn coil with sample of length (8.51 cm) greater than the coil at

6.72 kJ

When a 4-turn coil was used to electromagnetically launch a tall sample (8.51cm), at 6.72 kJ, it resulted in the sample opening up in the center which was the region of maximum biaxial strain. This tall sample – short coil configuration was the only one which resulted in slightly positive minor strains. Figure 4.12 shows a picture of the tall deformed sample which was launched with the four turn coil at 6.72 kJ. The sample is bulged out in the central region due to concentration of flux in the central region, which ultimately results in the initiation of a tear.

Figure 4.12: Tall (8.51cm) sample deformed with a 4-turn coil at 6.72 kJ 102

The strain distribution in this sample was mostly with positive minor stains. The failure mode was the biaxial bulge at tube mid height. A large crack initiated at the bulge center where the strains were the highest. The strains in this sample were considered to be

‘unsafe’ when they were next to the fracture in the bulge area. They were considered to be ‘safe’ when they were taken form regions away from the fracture, slightly above or below the area of maximum bulge. Figure 4.13 depicts the true strain distribution from this sample on a FLD. The experimental current time trace is shown in Figure 4.14. The recorded values of Peak current and Rise time were 138kA and 16µs respectively.

30 Safe Strains Unsafe Strains 20 Tensile test

10 Major True Strain (%)

0 -40 -30 -20 -10 0 10 20 Minor True Strain (%)

Figure 4.13: True strain FLD for a tall sample (8.51cm) deformed with a four-turn coil at 6.72 kJ

103

300 Primary 200 Induced 100 )

A 0

t (k 0 50 100 150 200 250 300

n -100 rre

u -200 C

-300

-400

-500 Time (µs)

Figure 4.14: Current vs. time plot for tall (8.51cm) sample deformed with 4-turn coil at 6.72 kJ. Peak current = 138 kA and Rise time = 16µs.

4.5 RESULTS AND DISCUSSION – TEN-TURN COIL

4.5.1 Ten-turn coil with sample of same length (8.51cm) at 13.92 kJ:

The electromagnetic expansion of a sample with length 8.51cm (same as a 10 turn coil), with a 10-turn coil resulted in a deformed sample with no big tears, but the initiation of small tears at the edges. Figure 4.15 shows the picture of the sample launched at 13.92 kJ.

As is evident from the figure, the sample profile is straight because the length of the sample and the coil was same. Thus a uniform magnetic pressure was maintained throughout the length of the sample.

104

8.51 cm

(a)

(b)

Figure 4.15: Picture of tall sample (length 8.51cm) (a) front view, (b) inside view, deformed with ten turn coil at 13.92 kJ. The formation of intersecting necking bands is clearly evident.

105

An interesting mode of failure was observed in these tall deformed samples. As can be seen in Figure 4.15, there was formation of necking bands diagonally across the length of the sample. These bands roughly had two orientations at angles of (48 to 52) o and – (52 to 55) o to the circumferential direction. Bands of each particular orientation were seen recurring throughout the sample, parallel to each other. The bands of these two orientations intersected each other recurrently and the locations of intersection of these bands, in some cases were initiation sites for small perforations as shown in the figure.

This pattern of necking bands was seen across the entire circumference of the sample.

These recurrent bands appear to be similar to sinusoidal wave imperfections with a particular mode and wavelength, as is discussed in the literature [12, 13]. Two wavelength modes were seen on this sample. One mode of instabilities had a distance of

~2cm between consecutive bands and it was the more prominent one with a higher degree of localization than the other mode. This second mode had a distance of ~0.5 cm between bands and was less localized than the first one. Thus the sample clearly had one mode of localization, more prominent than the other. It should also be noted that the angle of these bands was very close to the direction of neck propagation in tensile samples (~54 o).

In addition to these necking bands, small tears also initiated at the top and bottom edges.

The edge preparation of the sample could have an influence on these failure initiation sites. There was slight flaring of the sample near the edges due to the length of the sample and the coil not being exactly equal, which also lead to initiation of these tears.

Figure 4.16 shows the true strain FLD for the sample. As expected from the shape of the sample and the mode of deformation, the strains are uniaxial. The highest strains in this 106

case were found along the necking bands. They were considered to be ‘unsafe’ when they were from circles along the necking band, and‘safe’ when they were from circles parallel to the necking band but offset by one or two circles. Once again, a noticeable improvement in formability with strains of the order of 30-50% has been observed.

30 Safe Strains Unsafe Strains 20 Tensile test

10 Major True Strains (%) 0 -50 -40 -30 -20 -10 0 10 20 Minor True Strains (%)

Figure 4.16: True strain FLD for 8.51cm sample electromagnetically expanded by a 10 turn coil at 13.92 kJ

Figure 4.17 shows the current vs. time plot for this sample. For this case, the measured

Peak primary current was 124 kA while the Rise time was 24µs. As expected, in comparison with the four-turn coil, the ten-turn coil has higher rise time and a lower peak current due to the higher inductance of the coil.

107

200 100 0 -100 0 100 200 300 400 500 600

KA) -200 ( t

n -300 e r

r -400

Cu -500 -600 Primary -700 Induced -800 Time (µs)

Figure 4.17: Current vs. Time plot for 8.51cm tall sample electromagnetically launched with ten turn coil at 13.92 kJ. Peak current = 124 kA, Rise time = 24 µs.

4.5.2 Ten-turn coil with short (3.17cm) sample at 8kJ

The electromagnetic launch of a short (3.17cm) sample with a ten-turn coil at 8kJ resulted in a sample with no big tear. Figure 4.18 shows the picture of the sample. The sample profile after launch in this configuration is flared out at the edges. Expansion of a sample at energy slightly higher than this resulted in a big tear in the sample. In this sample, the maximum strains were observed at the flared edges. There was independent nucleation of small tears at the top and bottom edges of the sample. This could be due to inhomogeneities along the sample edges and due to the flared edge geometry of the sample. 108

Figure 4.18: Picture of short sample (3.17cm) electromagnetically expanded with a ten-turn coil at 8kJ

Strains recorded from the outside edge of the stretch flange configuration were considered to be ‘unsafe’ while the strains measured from regions offset by one or two circles were considered to be ‘safe’. As is evident from the figure, the mode of deformation is uniaxial stretching and improvements in formability are seen. As the energy of launch was increased to higher levels than this sample, it resulted in sample with higher strains and the formation of one big tear progressing from one edge to the other. Figure 4.20 shows the experimental current vs. time trace for this sample.

109

40 Safe strains 30 Unsafe strains Tensile test 20

10 Major True strain (%) 0 -50 -40 -30 -20 -10 0 10 20 Minor True strain (%)

Figure 4.19: True strain FLD for electromagnetic expansion of a short (3.17cm) sample with a ten-turn coil at 8kJ.

250 Primary Induced 150

50 A)

0 100 200 300 400 500 600 nt (k -50

Curre -150

-250

-350 Time (µs)

Figure 4.20: Current vs. Time plot for short sample (3.17cm) electromagnetically expanded with a ten-turn coil at 8kJ. Peak current = 94 kA, Rise time = 24 µs.

110

4.6 RESULTS AND DISCUSSION – TWO-TURN COIL

The two-turn coil was used to electromagnetically launch samples of length 1.74 cm which is almost the same length as that of the coil. However after a few runs it was observed that all samples had a skewed geometry due to the configuration of the coil. The pitch of the coil (although same as the four and ten turn coils), was too high for a coil of just two turns, for it to have a uniform field. Thus due to the non-uniform field of the coil, it was not possible to get a straight deformed sample profile. Due to this, additional tests and configurations for this coil were not carried out. The test results for one particular sample of length 1.74 cm, which was electromagnetically launched with the two-turn coil at 6.4 kJ, will be discussed next. Figure 4.21 shows the picture of this sample. The skewed sample profile of this electromagnetically expanded sample is clearly evident. No failures were obtained in the sample. This sample was not electrolytically etched with a circle grid hence a strain distribution could not be obtained. From the measurement of the sample circumference an engineering strain of 10.2% in the circumference and -8% in the height were calculated.

Figure 4.21: Picture of 1.74 cm sample electromagnetically expanded with a two- turn coil at 6.4 kJ. The skewed shape of the sample can be clearly seen. 111

The experimental Current vs. Time trace for this sample is shown in Figure 4.22. The measured Peak current was 170 kA and the Rise time was 16 µs. As this coil has the lowest number of turns/length and hence the lowest inductance, it had the highest peak current and the lowest rise time when compared with the four and ten turn coils.

200 Primary 150 Induced

100

A) 50 k t (

n 0 0 100 200 300 400 rre -50 Cu -100

-150

-200 Time (µs)

Figure 4.22: Current vs. Time plot for 1.74 cm sample deformed with two-turn coil at 6.4kJ. Peak current = 170 kA, Rise time =16 µs.

4.7 SIMULATION OF ELECTROMAGNETIC FIELDS WITH

MAXWELL 2D

The deformed sample profile in each of the coil-sample length configurations tested here is a result of the profile of the electromagnetic field between the coil and the sample.

112

Different lengths of the samples relative to a particular sample length, result in a different distribution of magnetic pressure, which in turn is responsible for the final geometry of the deformed sample. In order to understand this better, simulations were conducted with a software call MAXWELL 2D [15]. This tool was primarily used here for qualitative results and not for the actual values of any parameter.

4.7.1 Background about MAXWELL 2D

Maxwell 2D Field Simulator is a 2D structure electromagnetic field simulator. It is a comprehensive, easy-to-use software tool for design problems requiring an accurate, two- dimensional representation of the electric or magnetic field behavior. It is a powerful simulation software for electromagnetic, electrothermal, and electromechanical analysis.

It can be used to visualize magnetic fields and predict magnetic forces and is very useful in designing magnetic circuits. It can simulate frequency and time domain electromagnetic fields, steady-state thermal fields, strongly-coupled electromagnetic circuits and equivalent circuit model [15, 16].

It is a finite element based field solver and uses automatic, adaptive meshing. It iteratively calculates the desired electrostatic or magnetostatic field solution and special quantities of interest, including force, inductance, capacitance, and power loss. Up to eight different solvers may be selected based on the problem to be simulated. The different solvers include Electrostatic, Magnetostatic, Electrostatic, Eddy Current, DC

Conduction, AC Conduction, and Eddy Axial. In the present study only the eddy current 113

solver has been used. This solver can simulate the effects of time varying currents.

However the calculations are steady state and variation of frequency with time cannot be simulated. Additionally, plots of flux lines, B and H fields, current distribution, and energy densities over the entire phase cycle are available [16]. The eddy current solver assumes that all currents are sinusoidal and oscillate at the same frequency.

Two important boundary conditions were used in this study. In order to effectively isolate the model from any other sources of current or magnetic fields, a Balloon boundary was used which considers the region outside the drawing space as being nearly “infinitely large”. The other boundary condition used in the simulations was the eddy current source.

The total current in the conductor was constrained to the value specified by the user which in this study was 100kA. The eddy current field simulator solves for time harmonic electromagnetic fields governed by Maxwell’s equations. The quantity that the eddy current field simulator actually solves for is the magnetic vector potential.

4.7.2 Simulations for the coil – short sample configuration

Using Maxwell 2D, the electromagnetic field was determined for the case of a four-turn coil electromagnetically coupled with a sample of length 2.22cm (0.875”), that is a sample shorter than a four-turn coil. A section of a four turn coil having the same dimensions, as described in Section 4.2, with a wire diameter of 6.35mm and the center to center distance between adjacent circles of 9.4 mm was used. The section of the tube

114

was taken at a distance of 1.5 mm from the coil and the thickness of the section was taken to be 1.7 mm. Thus all the settings were taken to be the same as the experiments. Using the Eddy current solver with total current of 100kA and a frequency of 10 kHz, the problem was solved to obtain the electromagnetic field flux lines, the magnetic field and the current density. Figure 4.23 shows the results for the Maxwell 2D simulation for the configuration of a four turn coil with a short sample. Figure 4.23a shows the electromagnetic field between the coil and the short sample. The flux lines are shown to be almost straight along the length of the tube. However near the top and bottom end of the short tube, the lines bend over leading to the flaring out of the tube diameter near the ends. Figure 4.23b shows the magnetic field for this configuration. Figure 4.23c shows the current density profile with hot spots near sample edges.

(a)

115

( b ) ( c )

Figure 4.23: Results from Maxwell 2D for the case of a 4-turn coil with a short tube (2.22cm). (a) Flux lines, (b) Magnetic filed and (c) Current Density

Figure 4.24 shows an actual picture of a sample of length 2.22cm, expanded with a four – turn coil at 4.8 kJ. In this sample run the measured Peak current was 118 kA and the Rise time was 18µs. This shape can be explained from the MAXWELL simulations.

Figure 4.24: Picture of a 2.22cm sample electromagnetically expanded with a four- turn coil at 4.8kJ.

116

4.7.3 Simulations for the coil – long sample configuration

Simulations for the four-turn coil with long sample case were done in Maxwell 2D with the same problem set up as described in Section 4.7.2. This was according to the experimental results discussed in Section 4.4.2. The length of the sample was taken to be

8.51cm and a total current of 100 kA and a frequency of 10 kHz was used as an input parameter. Figure 4.25 shows the results from this simulation.

(a)

117

(b) (c)

Figure 4.25: Results from Maxwell 2D simulations for the case of four-turn coil with a long tube (8.51cm). (a) Flux lines, (b) Magnetic field and (c) Current density

Figure 4.25a shows the electromagnetic field with concentration of magnetic flux in the central region of the sample, in the area where it is in close coupling with the coil.

However, the area away from the central region is not in coupling with the coil. Thus this region does not expand as much as the central region with the high magnetic pressure.

This is responsible for the central region bulging out. The same conclusion can be arrived at by looking at Figures 4.25b and c which respectively show the magnetic field and the current density reducing as one moves away from the central region of the sample where these values are the highest and lead to a bulge in its profile after deformation.

118

4.7.4 Simulations for the coil – same sample length configuration

Simulations for the four-turn coil with same length sample case were done in Maxwell

2D with the same problem set up as described in Section 4.7.2, except that the sample length was taken to be 3.17cm. This was according to the experimental results discussed in Section 4.4.1. Figure 4.26 shows the results from this simulation. Figure 4.25a shows a uniform electromagnetic field throughout the length of the sample including the edges, as was not the case in the previous two configurations discussed in Sections 4.7.2 and 4.7.3.

Figures 4.25b and c show the magnetic field and the current density respectively. It is clear that because of the uniform field throughout the sample length, the sample maintains a largely uniform and straight profile after electromagnetic launch. There is no flaring out at the edges or bulging in at the center.

(a)

119

(b) (c)

Figure 4.26: Results from Maxwell 2D simulations for the case of four-turn coil with same length (3.17 cm) tube. (a) Flux lines, (b) Magnetic field and (c) Current density

4.8 NUMERICAL SIMULATION OF TUBE EXPANSION

EXPERIMENTS

A numerical computer code was written to simulate electromagnetic tube expansion. The approach taken was an adaptation of that used by Gourdin [17] and Wenfu Pon [18] with revision for tube geometries largely following Jablonski and Winkler [19]. This program was adapted to simulate the case of coil with sample of equal length configuration.

120

4.8.1 Code description

The code, written in Mathematica software, considers the electromagnetic interaction of a conductive tube and a solenoid. Deformation is considered to be uniform along the length of the tube and current is assumed to run uniformly within the tube. From an electromagnetic point of view, the system is modeled as two LRC circuits coupled through their mutual inductance which decreases as the workpiece and coil separate due to deformation. It is assumed that the material is rate independent and its plastic constitutive law is expressed as, σ = kεN where N is the strain rate sensitivity. The inductance of the solenoid of finite length, where the solenoid length is greater than the diameter the inductance is estimated as:

2 2 10µ0πrs n Ls = ………(4.1) 10l + 9rs where rs, n and l are the effective radius, number of turns and length of the solenoid respectively. The mutual inductance is estimated as:

⎛ 2 2 ⎞ µ0πrsrt (t)n rt (t) − rs M = ⎜1 − 2 ⎟ …………(4.2) l ⎝ rt (t) ⎠

Where rt(t) is the tube’s effective radius. The input parameters in the code include material parameters like material density, resistivity and the constitutive behavior for the material. Geometrical factors like the radius and number of turns of the solenoid, and radius and length of the tube are also some input parameters.

121

4.8.2 Numerical code results for EM expansion of a 3.17cm sample with 4-

turn coil at 7.04 kJ

This code is an effective and easy to use tool for predicting properties and trends of

important values that cannot be measured during tube expansion experiments. By giving

input parameters like the geometric parameters and the energy, trends of values like

primary and induced currents, the capacitor charge, the radius of the sample in motion,

tube velocity and strain rate etc., as a function of time can be predicted and displayed.

Velocity (m/s) Strain rate (/s) 200

6000

150 5000

4000 100 3000

50 2000 1000 Time se c 0.00005 0.0001 0.00015 0.0002 0.00005 0.0001 0.00015 0.0002 Time (s) Time (s) H L (a) (b)

RaRadiusdius cm (cm) Stress (MPa) 175 H L 3.8 H L 150

125 3. 6 100

3.4 75

50 3.2

25 Time (s) Time se c 0.00005 0.0001 0.00015 0.0002 0.00005 0.0001 0.00015 0.0002 Time (s) Time (s) H L (c) (d)

122

TuTebemDTp.C Rise oC CurrentCurrent k(kA)A Current (kA) 25 H L H L 200 20

15 100

10 0.00005 0.0001 0.00015 0.0002 5

-100Time Time (s) 0.00005 0.0001 0.00015 0.0002 Time (s)

(e) (f)

Figure 4.27: Results from Mathematica code for the case of 4-turn coil with sample of same length at 7.04 kJ. (a) Velocity, (b) Strain rate, (c) Radius, (d) Stress, (e) Temp. Rise and (f) Current vs. time.

As an example, Figure 4.27 shows the output plots for the case of a 4-turn coil

electromagnetically expanding a 3.17cm tube at 7.04 kJ. This is an efficient tool for

visualizing the motion of the sample with time. Figures 4.27a and b give the sample

velocity and strain rate as a function to time. As the velocity of the sample has not been

measured in these experiments, this is a helpful tool for calculations. Figure 4.27c shows

the motion of the sample with time. The measured outer radius of the deformed sample in

this case was 4cm which is in agreement with the results from the simulations. Figure

4.27f shows the predicted primary and induced currents. The comparison between the

predicted and experimental current traces is given in Figure 4.28. There is very good and

accurate correspondence between the measured and calculated values, which stands for

the accuracy of the code in predicting other values which have not been measured.

123

400 Calc_primary 300 Calc_Induced )

A 200 Exp_primary Exp_Induced t (k

n 100 e r r

u 0 C -100 0 50 100 150 200 250 300 350 400 -200 Time (µs)

Figure 4.28: Comparison between experimental and calculated current traces for the case of a 4-turn coil electromagnetically expanding a 3.17cm sample at 7.04 kJ.

4.9 Combined Results from current traces and circuit parameters for all sample – coil configurations.

For a simple N-turn solenoid that has an internal area Ao and an amount Ai of this is excluded by enclosing a conductive workpiece, the coil inductance can be estimated as:

…… (4.3) where l is the length of the coil and µ is the magnetic permeability of free space. Thus as 0 the number of turns in the coil increase, its inductance increases. The Peak current of a coil, Io at a particular energy level is related to its inductance L, by the Equation 4.2

124

C I =V ……. (4.4) 0 o L where C is the capacitance and Vo is the voltage of the circuit. Thus as the number of turns in the solenoid increase, the Peak current reduces because of an increase in the inductance of the coil. The Energy of electromagnetic launch E is given by Equation 4.3:

1 E = CV 2 ……. (4.5) 2

Where C is the capacitance and V is the voltage of the system. The rise time (time to peak) for the primary current is estimated [19] as

π t = LC …… (4.4) rise 2

Thus it is evident that as the number of turns in the coil increase, the rise time increases because of increase in its inductance.

A number of test runs were conducted for all sample – coil configurations as listed in

Table 4.3. Peak current and Rise time data was gathered for each case. The measured inductances of the 2, 4 and 10-turn coils were 0.36, 0.78 and 1.98 µH respectively. The measured average rise times for the two, four and ten turn coils were 16, 18 and 24 µs respectively. From Equations 4.2 and 4.4, it can be deduced that Io α √ E. This relation is clearly borne out by the experimental results. Figure 4.29 shows the relation between the peak current and the energy of launch for the experimental test runs for all coil – sample configurations. It is evident that the peak current increases with the energy of launch. For a particular energy level, highest peak currents are obtained with a two-turn coil.

125

200 175 150 125 2-turn coil rent (kA) 100 4-turn coil 75 10 - turn coil 50 Peak Cur 25 0 01234 (Energy (kJ)) 0.5

Figure 4.29: Variation of Peak primary current with root of energy for all samples tested with two, four and ten-turn coils.

4.10 DISCUSSION: COMBINED TRUE STRAIN DATA FOR

DIFFERENT SAMPLE LENGTH – COIL CONFIGURATIONS

Figure 4.30 shows the True strain FLD with strain data from various samples tested in different sample length – coil configurations. Pictures of samples from which the strain data has been taken, are also shown in the figure. With the help of different coil turns – sample length pairs, strain distribution for the entire FLD for high velocity deformation has been obtained. For each test case, a different symbol has been used. For each case, a solid symbol represents an unsafe strain while the corresponding open symbol represents safe strain in the sample. As is evident from the Figure 4.30, mostly a uniaxial strain distribution has been obtained with the different configurations, except the tall sample

126

(8.51 cm) – four-turn coil configuration. In this configuration, the reported unsafe strains are biaxial while the safe strains, which were taken from areas away from the maximum bulge ‘waistline’, are in plane strain. Along with the high strain forming limits, the quasi

– static FLD has also been shown. This FLD has been adapted from [8]. Quasi-static FLD for AA6063-T6 has not been found in the literature. This material is in full-hard condition and is thus of not much interest in formability studies. It is clear that substantial improvements in formability have been obtained for this material under high velocity conditions. AA6063-T6 has a ductility of about 8-12%. Comparison of the quasi-static

FLD and the high strain data also indicates the positive influence of velocity on formability

Major strains in the range of 10-55% have been shown which in some cases represents a nominal 400% increase in formability in some cases, under high velocity conditions. It should be kept in mind that the inertia of the material or the velocity of launch is the primary influencing factor here in the absence of a high velocity impact. Changes in constitutive behavior of the material could also be important and will be discussed in

Chapter 5.

127

configurations coil-sample length

Figure 4.30: True strain FLD for different 128

4.11 CONCLUSIONS

Electromagnetic tube expansion experiments were successfully implemented with different configurations of coil – sample length. Samples were deformed in different strain states, thereby generating data for both sides of the FLD. Thus a complete data set was created for a FLD under high velocity conditions.

This data was generated to understand the influence of inertia and constitutive parameters on formability. In the absence of a high velocity impact, these factors are the primary influencing factors on high velocity formability. It is clear from the experimental results outlined in Sections 4.4-4.6 that substantial improvements in formability can be obtained at high velocities even in the absence of a high velocity impact. This is true for all strain states on both sides of the FLD. Figure 4.30 shows the true strain data from most of the experiments along with the quasi-static FLD. True strains in the range of 10-55% have been achieved from AA6063 T6 which has a quasi-static ductility of 8-11%. This high increase in formability through high velocity free expansion, has been demonstrated for the first time, to the best of our knowledge. It can be the basis of important practical design guidelines.

This is also a useful study to understand the basic concepts in high velocity formability. It outlines the significance of inertia and constitutive parameters which are responsible for this tremendous improvement in formability from an aluminum alloy in full-hard condition. It opens new avenues for the use of materials with low quasi-static ductility 129

materials with other attractive properties, in the use of applications which require higher formability in it. In addition, this study provides a simple method of testing the high velocity formability of any material which can be successfully electromagnetically formed.

In addition to this, the use of our in-house Mathematica code, as discussed in Section 4.8, is a powerful and easy to use tool for predicting values and properties like the current traces, velocity, temperature rise, constitutive behavior of the material. This complements the experimental data set and also gives the trends for properties which we cannot or do not have the capability, to measure.

An important aspect of this study was also to generate an FLD under high velocity conditions to provide important design guidelines, as is the case under quasi-static case.

Successful creation of a complete data set has been achieved for strain and current data for sample runs, for both sides of the FLD. This data was used as important input parameters for simulation of high velocity FLD. The theory and results of these simulations will be discussed in Chapter 5.

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BIBLIOGRAPHY

[1] Balanethiram V.S., Hyperplasticity: Enhanced Formability of Sheet Metals at High Velocity, Ph.D. Thesis, The Ohio State University (1996).

[2] Balanethiram V.S. and Daehn Glenn S., Hyperplasticity: Increased Forming Limits at High Workpiece Velocity, Scripta Materialia, Vol. (30) (1994), pp.515-520.

[3] Seth Mala, High Velocity Formability of High Strength Steel Sheet, M.S. Thesis, The Ohio State University, (2003)

[4] Seth Mala, Vohnout V.J. and Daehn G.S., Formability of steel sheet in high velocity impact, J. of Materials Processing Technology, Vol. (168) (2005), pp. 390-400.

[5] Imbert J.M., Winkler S.L., Worswick M.J., Oliveira D.A. and Golovashchenko S., The effect of tool/sheet interaction on damage evolution in Electromagnetic Forming of Al alloy sheet, J. of Engg. Mat. Tech., Vol. (127) (2005), pp. 145-153

[6] Imbert J.M., Winkler S.L., Worswick M.J., Oliveira D.A. and Golovashchenko S., Formability and damage in electromagnetically formed AA5754 and AA6111, 1st International conference on High speed forming, Dortmund Germany (2004), p. 201

[7] Olieviera D.A., Worswick M.J., Finn M. and Newman D., Electromagnetic forming of aluminum alloy sheet: Free-form and cavity fill experiments and model, J. of Mat. Processing Tech., Vol. (170) (2005) pp. 350-362.

[8] Thomas J., Seth M, Daehn G., Bradley J. and Triantafyllidis N., Forming limits for electromagnetically expanded aluminum alloy tubes: theory & experiment, submitted for publication, Acta Met. (2006).

[9] http://www.suhm.net

[10] www.matweb.com

[11] http://www.lectroetch.com 131

[12] http://homepage.ntlworld.com/rocoil

[13] Shenoy V.B, Freund L.B, Necking bifurcations during high strain rate extension, Journal of the mechanics and physics of solids, Vol. (47) (1999), pp. 2209-2233.

[14] Mercier S. and Molinari A., Analysis of multiple necking in rings under rapid radial expansion, Int. J. of Impact Engg., Vol. (30) (2004), pp. 403-419

[15] http://www.ansoft.com/products/em/max2d

[16] Maxwell Manual, Ansoft Corporation.

[17] Gourdin W.H., Analysis and Assessment of Electromagnetic Ring Expansion as a High-Strain-Rate Test, J. Applied Physics, Vol. 65(2) (1989) pp. 411-422.

[18] Pon Wen Fu, A Model for Electromagnetic Ring Expansion and Its Application to Material Chacterization, PhD thesis, The Ohio State University (1997).

[19] Jablonski J. and Winkler R., Analysis of the Electromagnetic Forming Process, Int. J. Mech. Sci., Vol. 20 (5) (1978) pp. 315-325.

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CHAPTER 5

SIMULATION OF FORMING LIMITS FOR

ELECTROMAGNETIC TUBE EXPANSION

The experimental tube expansion work done, as described in Chapter 4, was sponsored by General Motors and was done in collaboration with the University of Michigan, Ann

Arbor. One of the goals of the experiments was to support analysis of the concept of

Forming Limit Diagrams (FLD’s) to model the ductility of electromagnetically formed specimens. The simulations for studying the ductility of freely expanding electromagnetically loaded aluminum tubes were done by J.D Thomas and Dr. N.

Triantafyllidis [1] at The University of Michigan. A description of the model and simulation results will be presented here followed by a critical discussion of the model.

5.1 BACKGROUND AND MOTIVATION

Experimental work involving EMF with impact with a die [2, 3] has shown dramatic increases (compared to conventional hydroforming) in the ductility of aluminum alloys 133

thus making them significantly more ductile than conventional mild steel. A theoretical explanation for this observed increase in formability based on fully coupled electromagnetic and thermomechanical modeling of EMF ring free expansion was provided by Triantafyllidis and Waldenmyer [4]. It was suggested that the strain rate sensitivity occurring in aluminum alloys at high EMF strain rates was the main mechanism responsible for the observed higher necking strains of the ring.

The goal in the present study was to apply the above-mentioned work in modeling the ring experiment [4] to more complicated EMF geometries, to allow comparison with experiments and further investigate formability mechanisms. However this fully coupled electromagnetic and mechanical model of the actuator and the workpiece would be accurate and best applied to specific EMF processes (with known part and actuator geometries). In order to help a potential designer of electromagnetically formed components, a model which involved simple and considerably more rapid calculations to give a reasonable estimate the ductility under EMF conditions was needed. With this requirement in mind, a general theory to calculate EMF-based FLD’s was proposed, in which the calculation of strains at the onset of necking in a sheet accounted for the presence of electric currents and the resulting ohmic heating effect. Details about this model and a comparison of the theoretical simulation results and the experimental data from EM tube expansion, as described in Chapter 4, will be presented next.

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5.2 PROBLEM FORMULATION

The ductility prediction of an electromagnetically free formed sheet (formed without die impact) is based on a forming limit diagram concept which is widely used in the analysis of conventional (i.e. purely mechanical) sheet metal forming processes. The formulation has been proposed by Thomas and Triantafyllidis [4] and starts with the “weak band” analysis for the localization of deformation in a biaxially stretched sheet subjected to electric currents. Complete details about the model can be found elsewhere [1].

5.2.1 Localization of deformation analysis

By ignoring curvature and inertia effects, the sheet was idealized as a thin plate under plane stress conditions, as depicted in Figure 5.1. It was assumed that deformation is localized in a narrow band B with normal N and tangent S in the reference configuration.

The band was distinguished from the sheet by the presence of an initial imperfection, which was implemented as a discontinuity in the reference configuration, either of a material or a geometric (thickness) parameter. The goal was to calculate the deformation gradient FB, stresses σB, currents jB, temperature θB and internal variable (plastic strain)

p ε B inside the band given the knowledge of the counterpart of those quantities outside the

A A A A p band F , σ , j , θ and ε A. A full Lagrangian (reference configuration) formulation of the problem was adopted. For simplicity only incompressible materials were considered.

Displacement continuity across the band gives:

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Figure 5.1: The weak band model showing band orientation in reference configuration

……….(5.1) while traction continuity across the band implies

…… (5.2) where Π is the first Piola-Kirchhoff (P-K) stress. For an incompressible solid the first P-

K stress is related to the Cauchy stress by

.…... (5.3)

Electrical principles require continuity of the current and of the tangential component of the electric field across the band. Current continuity gives

……. (5.4)

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where J is the current density vector in the reference configuration. Continuity of the tangential component of the electric field E across the band requires,

.…… (5.6)

Finally, from energy conservation, assuming that adiabatic heating takes place in the sheet both inside (A) and outside (B) the weak band, one has for an incompressible solid

…….. (5.7)

• where µ is the mass density, cp is the specific heat, θ is the rate of temperature

• • p p change, σ e ε is the plastic dissipation ( σ e equivalent stress, ε rate of plastic strain) and

χ is the plastic work conversion factor (0 < χ < 1).

5.2.2 Constitutive response of the metal sheet

During an EMF process the material experiences both high strain rates and high temperatures. Consequently a temperature-dependent viscoplastic constitutive law was used for modeling its stress-strain response. The accumulated plastic strain in the solid,

p ε , determines the size of the material’s current yield surface σe and is related to the solid’s quasistatic uniaxial response σ= g (εp , θ) by

… … ( 5 . 8 )

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⋅ p where m is the solid’s rate-sensitivity exponent and εo a material constant. Further, the sheet was assumed to be transversely isotropic, under plane stress and incompressible.

The rate-independent uniaxial response adopted in the calculations [5] was,

…… (5.9) where n is the hardening exponent, α the thermal sensitivity, σy the yield stress, εy = σy /E the yield stain, θm the melting temperature and θo the reference temperature. For isotropic materials that do not exhibit the Baushinger effect, i.e., materials that exhibit no difference between their tensile and compressive responses, the following yield surface criterion (based on results from Barlat et al. [6]) was used

……. (5.10) where β is an experimentally determined exponent and σi are the principal values of the

Cauchy stress tensor. With the help of these equations, the mechanical constitutive response of the material was determined. The material’s electric constitutive law, i.e., the relation between current j and electric field e, was taken to be an isotropic Ohm’s law

.…… (5.11) where the resistivity of the metal r is taken temperature-independent for the temperature range of interest. The electric constitutive equation is completed in a thin sheet by j3 = 0 since the current in the sheet can flow only in its plane. Thus all the governing equations

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those are required for the calculations of the field quantities inside the band, given the knowledge of their counterpart outside the band, have been established.

5.2.3 Selection of material constants

The experiments for electromagnetic expansion were done with AA6063-T6. However, the material constants required for the determination of the quasistatic FLD were obtained from experiments using flat sheet blanks. In addition, since an independent measurement of rate and thermal sensitivity parameters, at the strain rates and temperatures of interest, requires highly specialized equipment not available, the uniaxial quasistatic test measurements from AA6063-T6 (Section 4.2.3) were used to obtain some mechanical properties, as listed in Table 5.1. For the remaining material parameters, existing experimental data for AA6061-T6 [5], as listed in Tables 5.2 and 5.3 was used.

Table 5.1: Material parameters of AA6063-T6 tube obtained from experiments.

Table 5.2: Material parameters used from AA6061-T6 [5]

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Table 5.3: Density, specific heat, plastic work conversion factor and resistivity of Al.

Figure 5.2: AA6063-T6 uniaxial quasi-static true stress vs. true strain plot: experimental data and corresponding best fit

Figure 5.2 shows the uniaxial test data plot from Section 4.2.3, along with the best fit which was achieved, using the values in Table 5.1. The remaining parameters pertain to the characterization of the yield surface. The band was modeled by a discontinuity in the yield stress, using

……. (5.12)

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where ξ is the imperfection amplitude. The values of ξ and of the yield surface exponent

β (equation 5.10) were chosen to give the most reasonable agreement with the available quasistatic FLD experiments on AA6061-T6 by LeRoy and Embury [7] and are given in

Table 4. The parameters given in Tables 1 to 4 completely characterize the mechanical, thermal and electrical properties of the model used for the simulations of the electromagnetic free expansion of the aluminum alloy tubes.

Table 5.4: Material parameters for FLD of AA6061-T6

5.2.4 Strain, Strain rate and Current density profiles

When calculating FLD’s under quasi-static loading for rate-independent solids, the time dependence of strain is irrelevant. However, for the time-dependent viscoplastic response of the material in EMF processes, strain history influences the solid’s response and hence a strain profile ε1(t) is required. In addition, calculations of FLD’s are based on

• • the simplifying assumption of proportional strain paths. It is assumed that ε2/ε1 = ε2 / ε1

= ρ where −1/2 ≤ ρ ≤1 with the lower limit corresponding to uniaxial stress and the upper to equibiaxial plane stress. Determining the exact strain profile ε1(t) would require a solution of a coupled electromagnetic and thermomechanical problem of the tube plus its

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actuator coil, a feasible but complicated and time consuming task. Such a modeling approach would be the 2D version of the 1D ring calculations done by Triantafyllidis and

Waldenmyer [4]. In the interest of simplicity, the following sinusoidal-shape strain rate pulse was assumed since a pulse-like strain rate history was expected for the hoop strains at any height of the expanding tube:

…… (5.13)

where 4τ0 is the duration of the strain rate pulse and εmax is the maximum strain in the hoop direction. The electromagnetic nature of the problem also requires knowledge of the time-dependent current density. Again a sinusoidal pulse current density was assumed in the hoop direction

…… (5.14)

where Jmax is the maximum current density achieved in this process. Experimental observations, as well as fully coupled electromechanical calculations in the ring problem

[4] showed that the time duration of the first (and much larger) current pulse is approximately half the duration of the strain rate pulse, thus explaining the reason for the choice in equation (5.14). The characteristic time τ0, which is half of the measured duration of the main current pulse (which is the first half current pulse), and the maximum density Jmax are available from the experimental results described in Chapter 4 and are given in Table 5.6. These parameter values are obtained from electric current

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versus time traces taken during the tube expansion experiments for the four cases for which the results were described in details in Sections 4.4 and 4.5. Figure 5.3 shows a comparison between the experimental and simulated current traces obtained from equation 5.14. The experimental current trace shown here is a part of the complete current trace shown in Figure 4.11, for the test case a, in Table 5.5.

Table 5.5: Parameters from tube expansion experiments for four test cases

It should be noted that the current traces here are the induced current traces. One half of the first current pulse has been simulated for the model. Figure 5.3 also shows the dimensionless strain versus time trace for which the time duration is twice the current pulse. As outlined in Section 4.3, the samples were launched at incremental energies until a sample in which necking or failure was detected, was reached. The choice of maximum hoop strain εmax in equation (5.13) was made following the same trend. For each strain ratio ρ, a simulation was run with a certain value of εmax for which no necking was detected. A simple forward marching technique was used to gradually test larger values

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of εmax until necking was achieved at 0.99 εmax. For example, for case (a) the maximum simulation strain rate ranged from 4932 s−1 (plane strain) to 8793 s−1 (uniaxial)

Figure 5.3: Comparison of the experimentally determined current density profile with the simulated current density profile. Simulated dimensionless strain profile also shown. These plots correspond to a 31.7mm tube deformed with a 4-turn coil at 6.72 kJ of energy (case (a)).

5.2.5 Numerical algorithm

The governing equations for the principal solution and for the localization problem was

. cast as a system of first order Ordinary Differential Equations (ODE’s) x = f (x,t) . A fourth order Runge-Kutta algorithm was used for each case. For the localization problem along each path, a much larger system of nine ODEs had to be solved for each value of the reference localization angle Φ. For each load path (i.e. given ρ, τ0, Jmax, εmax), a value

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of the localization angle Φ was selected in the interval 0 ≤ Φ < π/2. Necking was detected when the plastic strain rate inside the band becomes unbounded i.e. necking was said to

. • p • p occur at tneck, the time when ε B /ε A > 10 and εneck ≡ ε1 (tneck). The choice of value 10, although arbitrarily selected, had negligible effect on the localization strain. The entire Φ range 0 ≤ Φ < π/2 was scanned using π/180 increments and the critical angle Φ was considered to be the one minimizing εneck(Φ). As in the quasistatic case, Φ ≠ 0 for ρ < 0

(Uiaxial) while Φ = 0 for ρ ≥ 0 (plane strain to equibiaxial). The typical time step used for the simulations was 0.0001 τ0.

5.3 COMPARISON OF EXPERIMENTAL AND SIMULATION

RESULTS

A comparison between the experimental true strain data presented in Sections 4.4 and

4.5, with the theoretical results from the model explained above, will be presented next.

The experimentally obtained true strain FLD’s for cases (a) through (d) (Table 5.5) plus the corresponding theoretical simulation results are presented in Figure 5.4 – 5.7. For comparison purposes the conventional quasistatic FLD’s for the same cases, were calculated in the absence of currents and using much larger pulse duration times τ0 are also plotted in these figures to show the ductility increase due to the EMF process.

The FLD results for the short tube/short coil combination (case (a) in Table 5, experimental results in Section 4.4) are presented in Figure 5.4. It is clear that the 145

experimental data are all for ρ < 0 and clustered about the uniaxial stress path (ρ = −1/2).

This is expected from the deformed sample profile which was discussed in Section 4.4.1 and shown in Figure 4.9, which shows that the localization and hence the place of measurement of strains, is at the top and bottom ends of the short tube. There is a reasonable (for FLD simulations) agreement between theory and experiment both in the numerical values of the critical strains as well as on the slope of the FLD curve

(increasing critical strains with decreasing strain ratio ρ).

Although the simulation overestimates the forming limits, both the measured and computed results show a significant increase in ductility in the electromagnetically expanded AA6063-T6 tube compared to the quasistatic curve. Moreover, from Figure

5.2, the uniaxial quasistatic AA6063-T6 necking and failure (approximately εneck = 0.11).

This corresponds with the ρ = −1/2 quasistatic forming limit in Figure 5.4 due to the use of rectangular low aspect ratio samples in the uniaxial quasistatic tests. This illustrates a purely experimental increase in free forming using EMF, which is captured well here.

Figure 5.5 shows the results for the shorter tube with the longer coil (case (b) in Table

5.5, experimental results in Section 4.5.2). It indicates the largest discrepancy between theory and experiments. This deviation could be because of the deformed sample profile with flaring at the top and bottom edges, as shown in Figure 4.18. Thus, the failed tube is highly distorted, while the assumptions adopted for the computation of the FLD were based on uniformly expanding tubes.

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Figure 5.4: Comparison of simulated and experimental forming limits for an AA6063-T6 tube of length 3.17 cm electromagnetically expanded with a 4-turn coil at 6.72 kJ of energy (case (a)).

Figure 5.6 shows the FLD corresponding to the long sample with short coil configuration, which is case (c) in Table 5. This experiment generated data in the ρ > 0 region, as expected from the centrally bulged sample profile shown in 4.12. It should be noted that in this case, the strain data was taken from a sample deformed at 7.52 kJ, but since current data could not be gathered in this case, it was taken from a similar sample run at

6.72 kJ, the details of which were given in Section 4.4.2. Experimental points on ρ = 0

(plain strain) showed agreement with theoretical predictions while experimental points for ρ > 0 (biaxial) showed large deviations from theoretical results. This discrepancy could be explained from the fact that yield surface parameters and anisotropy of sheet play a crucial role for the determination of the right hand side (ρ > 0) of the FLD.

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Figure 5.5: Comparison of simulated and experimental forming limits for an AA6063-T6 31.7 mm tube deformed with a 10-turn coil at 8 kJ of energy (case (b)).

Figure 5.6: Comparison of simulated and experimental forming limits for an AA6063-T6 85.1 mm tube deformed with a 4-turn coil at 7.52 kJ energy (case (c))

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In this simulation a simplified isotropic yield surface was used. It should be noted however, that despite overestimation of the forming limits, the experimental data, clearly shows an increase in formability. Figure 5.7 corresponds to the long tube/long coil combination (case (d) in Table 5.5 and experimental results in Section 4.5.1). As in case

(a), the experimental data points are clustered around the uniaxial stress path ρ = −1/2.

This is expected from the straight deformed sample profile shown in Figure 4.15 which shows initiation of failure near the top and bottom ends. This comparison shows the closest agreement between experiment and simulation, with the forming limits minimally overestimated. It should be noted that the theoretical predictions for all four experiments are predictably close to each other given the proximity of the values of the strain rates, current densities and characteristic times between the four different experiments.

Figure 5.7: Comparison of simulated and experimental forming limits for an AA6063-T6 85.1 mm tube deformed with a 10-turn coil at 13.92 kJ energy (case (d))

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Figure 5.8: The localization angle Φ as a function of strain ratio, for the simulation of the 31.7 mm tube deformed with a 4-turn coil at 6.72 kJ (experiment (a)).

Figure 5.8 shows the critical angle of the band, Φ versus the strain ratio ρ for the short tube/short coil configuration, which is case (a) in Table 5.5. Just like the quasistatic case,

Φ is a decreasing function of ρ for −1/2 ≤ ρ < 0 while Φ = 0 for ρ ≥ 0. Although localization angles are difficult to measure, for the few cases that such an angle could be estimated along the full length of the tube (case (d), corresponding to ρ = −1/2) Φ ~ 40o.

The temperatures at localization inside and outside the band were also calculated as a function of strain ratio for the configuration of case (a) in Table 5.1, The necking temperature was minimum for ρ = 0.

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5.4 DISCUSSION

A fully coupled electromagnetic and mechanical modeling of the actuator and workpiece would be the most accurate way to model the tube expansion experiments. However, the above model, with all its generalizations and assumptions has presented a relatively easy way to predict parameters and develop an FLD which can be an efficient tool in designing and understanding high rate forming processes. A few comments and proposed improvements for the model will now be proposed.

This model is based on the hypothesis that localized necking initiates from a pre-existing non-homogeneity and was first proposed by Marciniak and Kuczynski (M-K model) [8].

As has been discussed in Section 3.4.2, the success of the M-K model in accurately predicting FLD’s is sensitive and dependant on a number of factors. The choice of the yield criterion used has a significant influence on the forming limits. Studies [9-13] have shown that M-K theory can be used successfully to calculate FLD’s only if it is suitably coupled with an appropriate yield function. The level of the predicted FLD is also fairly sensitive to the assumed value of the initial imperfection f=(1-ξ), which is an adjustable fitting parameter in the theory. In the M-K analysis, a linear imperfection of infinite length is assumed which has been challenged and it has been shown that both the size and aspect ratio (length/width) of the imperfection (band) affect the localized necking process

[14] and this influence was found to be of comparable magnitude to the inhomogeneity factor f. By predicting lower forming limits for a given f value, the M-K curve represented too severe a forming condition.

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Some studies [15-18] for the determination of quasi-static FLD’s have been proposed as improvements to the M-K model by integrating the aspect of void growth, through

Gurson’s yield function, in the analysis. In order to obtain accurate analytical predictions, the geometric instability should be assumed to result from the combined effects of a thickness imperfection and damage accumulation. This also reduces the imperfection sensitivity of the M-K model. It has been shown [17, 18] that while the FLD curve without damage globally overestimated the limit strains, the curve with damage, is in better agreement with the experimental FLD. Thus the integration of damage parameters

(through Gurson's model) in the present model might yield better results in terms of agreement, quantitatively as well as qualitatively with the experimental data. The consideration of damage evolution produces flattening of the FLD for positive strains, which might give a match with the experimental data for the ρ > 0 strain ratios, i.e. for the configuration of a long tube with short coil, case (c) in Table 5.5.

Another basic assumption in this model is that inertia can be ignored, while changes in constitutive behavior, mainly, the increasing strain rate sensitivity of aluminum alloys, is the primary factor influencing high velocity formability. However, it has been noted by other studies that inertial effects and die-impact can have separate and important effects in improving formability. At high velocity, inertia becomes an important factor that stabilizes deformation. A number of numerical analyses [19-22], clearly depict that inertia aids formability enhancement and changes the necking pattern at high velocities.

A detailed description of these models will be provided in Section 7.1. Their broad outcome is that inertia slows down the rate at which an imperfection grows. 152

One study [19] showed that at high strain rates, localization in the inhomogeneity region

(band), B, is inhibited due to inertia such that the quasi-static failure criterion of ‘dєB /dєA

. • p • p → ∞’, (implemented as ε B /ε A > 10 , in the present model) is no longer applicable. The quasi-static prediction of the M-K model, that the strain in the uniform region, reaches a limiting value at which point, the local strain in the defective (band) region, suddenly becomes unstable, does not stand under high velocity conditions. This study also supported evidence that the M-K model should be applied to high strain rate conditions, only when inertial effects are included. Figure 5.9 depicts this postulation and shows the

B A variation of local strain εθ with the uniform strain εθ outside the band. Point B in the figure is the point of static load maximum, which shows the condition of dєB /dєA → ∞.

However as the strain rate is increased, this shooting up of the strain in the inhomogeneity region, does not happen and the growth of local strain is inhibited by increasing strain rate. This delay in necking due to inertia was observed regardless of whether the material was represented as a rate dependant or rate independent model.

Another study [21] analyzing the formation of necks in high velocity ring expansion revealed that as deformation progresses, the mean strain rates in necking sites are only slightly higher than the background rates due to inertial resistance, which again questions the premise of the present model of using the traditional M-K based analysis.

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Figure 5.9: Influence of inertia in imperfection growth. B is point of static load maximum (Considere’s Criterion). It is clear that growth of local strain is inhibited by increasing the strain rate.

The total elongation of the samples under high velocity conditions is increased primarily by inertia at high velocities due to an increase in the post-uniform elongation because significant acceleration is developed only after necking commences. The primary influence of inertia in improving formability at high velocities is provided by stabilizing deformation against neck growth and can as much as double the strains to failure [20]. In addition to this, work by Triantafyllidis [4] on electromagnetic ring expansion has shown that with increase in ring density, while all other ring properties are kept constant, the ductility of the rings increases. Thus it is clear that inertia plays a dominant role in improving high velocity formability and it should not be ignored.

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It has also been shown that necking pattern at high velocities is established by interplay between the destabilizing effect of geometrical softening (due to section reduction) and the stabilizing effect of inertia and rate sensitivity [22]. Thus strain rate sensitivity can be an important contributory factor in enhancing formability. However, it has also been shown that the tests that are typically used (Split Hopkinson Pressure Bar) for depicting enhanced strain rate sensitivity are prone to size effects and wave propagation effects [23,

24]. Different thickness samples, yield different stress strain curves. The sharp increase in rate sensitivity at high strain rates is more pronounced with thick specimens. Thus it has been suggested that the apparent strain rate sensitivity is an artifact and not inherent in the material. This could again be another indication of the importance of inertial effects and thicker specimens would have a higher inertial resistance.

In addition to inertia and changes in constitutive behavior, high velocity impact is a dominant mechanism that greatly enhances high velocity formability. The dramatic improvements in formability when high velocity impact is involved [1] point towards the importance of inertia in suppressing necking and ‘inertial ironing’.

Several simplifications and assumptions were made in the model, which could be improved upon. The assumptions about the strain and current path could differ substantially from the actual ones at the necked zones. An isotropic yield surface was used although it is known that anisotropy plays an important role in the accurate prediction of the right side of the FLD. Only the first half current pulse was solved for instead of the full first current pulse during which most of the forming takes place. The 155

characteristic time was taken to be half of the first current pulse, which is not completely accurate. Instead the rise time, which is the time needed for the first current pulse to peak, could be more appropriate. Although the temperature rise predicted in this model was close to that predicted by the Mathematica model described in Section 4.8, the basic assumption here is that plastic work dissipation is the major source of temperature rise.

A complete constitutive behavior data set which is necessary for an accurate constitutive description for the alloy that has strain and temperature sensitivity was missing. All the above factors count towards the predictions of the model not being in accurate correspondence with the experimental data.

5.5 CONCLUSIONS

The goal of the present work was the quantitative comparison between theoretical calculations for the onset of necking in sheets and the experimental results obtained from the free expansion of electromagnetically loaded aluminum alloy tubes. Although a fully coupled electromagnetic and thermomechanical analysis is required to accurately model the experiments, considerable insight can be gained by using this generalization of the

EMF FLD to study the ductility of sheets, as measured locally in the necked regions of the failed tubes. Overall, in spite of the inherent assumptions and simplistic approach here, the basic trends of improved formability are well captured by the model. The comparison between theory and experiments shows that the EMF-based FLD concept is a useful tool to predict ductility limits of metal sheet. However, most of the predicted FLD 156

curves from this study consistently overestimated forming limits from the experiments.

Also, a dominant factor that has been completely ignored is the incorporation of inertial effects. It has been shown through studies discussed above that inertia becomes very important at these high velocities and it could be even more significant than changes in constitutive behavior. There is ambiguity regarding the enhanced rate sensitivity at high strain rates and a complete experimentally obtained constitutive characterization of the material is needed for further improvements in predictions. Other improvements in assumptions regarding the circuit parameters like the rise time as suggested above are also important. However, even though a number of improvements can be proposed for the collected experimental data set and the model, the trends for improved formability even in free electromagnetic forming have been well established.

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BIBLIOGRAPHY

[1] Thomas J., Seth M, Daehn G., Bradley J. and Triantafyllidis N., Forming limits for electromagnetically expanded aluminum alloy tubes: theory & experiment, submitted for publication, Acta Met. (2006).

[2] Balanethiram V.S., Hyperplasticity: Enhanced Formability of Sheet Metals at High Velocity, Ph.D. thesis, The Ohio State University (1996).

[3] Balanethiram V.S. and Daehn Glenn S., Hyperplasticity: Increased Forming Limits at High Workpiece Velocity, Scripta Materialia, Vol. (30) (1994), pp.515-520.

[4] Triantafyllidis N. and Waldenmyer J., J Mech Phys Solids, Vol. (52) (2004), p.2127.

[5] Yadav S, Chichili D and Ramesh K., Acta Met Mater, Vol. (43) (1995), p. 4453.

[6] Barlat F, Maeda Y, Chung K, Yanagawa M, Brem J, Hayashida Y, Lege D, Matsui K, Murtha S, Hattori S, Becker R and Makosey S., J Mech Phys Solids Vol. (45) (1997), p. 1727.

[7] LeRoy G, Embury JD. The utilization of failure maps to compare the fracture modes occurring in aluminum alloys. In: Hecker SS, Ghosh AK, Gegel HL, editors. Formability Analysis, Modeling, and Experimentation. New York (NY): AIME, (1978). p.183.

[8] Marciniak Z. and Kuczynski K., Limit strains in processes of stretch forming sheet metals, Int. J. of Mech. Sci. Vol. 9 (1967), pp. 609–620.

[9] Chan, Forming limit diagrams: concepts, methods and applications, edited by RH Wagoner, KS Chan, SP Keeler, The Minerals and Materials society, 1989.

[10] Wonjib Choi, Peter P and Jones SE, Calculation of forming limit diagrams, Met. Trans A, Vol. (20A) (1989), pp. 1975-1987.

[11] Barlat F., mater. Sci. eng., Vol. (91), (1987), pp. 55-72.

[12] Butuc MC, Gracio JJ, Barata da Rocha A., A theoretical study on forming limit diagrams prediction, J. of mat. Proc. Tech., Vol. (142) (2003), pp. 714-724.

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[13] Burford DA and Wagoner RH, A more realistic method for predicting the forming limits of metal sheets, Forming limit diagrams: Concepts, methods and applications, edited by RH Wagoner, KS Chan, SP Keeler, The Minerals and Materials society, (1989).

[14] Narasimhan K and Wagoner RH, Finite Element modeling simulation of in-plane forming limit diagrams of sheets containing finite defects, metal. Trans. A, Vol. (22A) (1991), pp. 2655.

[15] Ragab A.R., Saleh Ch., Zaafarani N.N, Forming Limit diagrams for kinematically hardened voided sheet metals, J. of mat. Proc. Tech., Vol. (128) (2002), pp. 302- 312.

[16] Ragab A.R., Saleh Ch., Effect of void growth on predicting forming limit strains for planar isotropic sheet metals, Mechanics of materials, Vol. (32) (2000), pp. 71-84.

[17] Brunet M., Mguil S. and Morestin F., Analytical and experimental studies of necking in sheet metal forming processes, J. of materials Processing Tech., Vol. 80 (1998), pp. 40-46.

[18] Brunet M. and Morestin F., Experimental and analytical necking studies of anisotropic sheet metals, J. of materials Processing Tech., Vol. (112) (2001), pp. 214-226.

[19] Rajendran A.M. and Fyfe I.M., Inertia effects on the ductile failure of thin rings, J. of Applied Mechanics, Vol. 49 (1982), pp. 31-36.

[20] Hu Xiaoyu and Daehn Glenn S., Effect of velocity on flow localization in tension, Acta mater.Vol. (44), No.3 (1996), pp 1021-1033.

[21] Sorenson N.J. and Freund L.B., Unstable neck formation in a ductile ring subjected to impulsive radial loading, Int. J. of solids and structures, Vol. (37) (2000), pp. 2265-2283. [22] Fressengeas C., Molinari A., Fragmentation of rapidly stretching sheets, Eur. J. Mech. A/Solids, Vol. 13 (2) (1994), pp. 251-288.

[23] Oosterkamp L.D., Ivankovic A. and Venizelos G., High strain rate properties of selected aluminum alloys, Materials Science and Engineering Vol. A278 (2000), pp. 225-235.

[24] Gorham D.A., An effect of specimen size in the high-strain rate compression test, Journal De Physique III, Vol. 1 (1991), pp. 411- 418. 159

CHAPTER 6

TUBE IMPACT EXPERIMENTS

From tube free expansion experiments, a considerable increase in formability of an aluminum alloy was demonstrated. This was in contradiction to a few experimental studies [1-3] which had noted little or no improvement in formability in the absence of a high velocity impact. It has been demonstrated in earlier studies [4-7] that impact can be beneficial in further enhancement of formability, in terms of the measured strains that can be obtained from a sample forming into a die. It was desired to learn how impact would benefit formability of the same aluminum alloy using the same experimental setup used in Chapter 4 (Section 4.2). The goal here was not as much to see the amount of increased observable strains as to see if impact could benefit the necking and tearing in the samples.

6.1 BACKGROUND AND MOTIVATION

The impact of a sheet metal moving at high velocity, with a fixed massive die, quickly decelerates it to a stop and produces through-thickness compressive stresses that also

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cause lateral extension of the material. This can produce a forming process more like ironing than conventional sheet metal stretching. It is called ‘inertial ironing’ [4, 5] as ironing is used in conventional forming to generate large through-thickness stresses are between tools to laterally stretch out the sheet materials by squeezing it in the through- thickness direction. The through-thickness pressure can easily become on the order of the materials flow stress in electromagnetic forming, and then should be able to affect the deformation characteristics in forming.

A number of experimental and numerical studies have demonstrated the tremendous increase in high velocity formability when impact is involved. Balanethiram et al. [4, 5], performed electrohydraulic forming (EHF) experiments for IF iron, Armco steel OFHC copper and AA6061 – T4, involving high velocity sheet impact into a conical die. The biaxial formability of these alloys was investigated under quasi-static and high strain rates of up to 1000 /s. Figure 5.1 shows the limit strain plot for both cases with close to a

5-fold increase in formability. The high rate samples withstood major strains in excess of

100% without failure even under nearly plane strain conditions. It is clear from this figure that increasing strain rate has a potent effect on formability. However it should be noted that in these EHF experiments the pressure of the water in contact with the sheet material may be an important factor in enhancing formability as well.

Similarly experiments of EMF of cold rolled, high strength steel sheets [6, 7] involving high-speed impact with relatively massive punches have revealed almost a ten-fold 161

increase in formability for high strength steels with formability ranging from 1.3-25.6%.

A similar increase in formability was revealed for these steels and AA6111-T4, when they were formed against a hemispherical die, with a 9mm air rifle, with pellets of different geometries, at velocities ranging from 136-205 m/s [6].

Figure 6.1: FLD for AA6061 and added low and high rate forming data [4]

While inertia could be accountable for as much as doubling the limit strain, the observed improvements in formability from experimental studies, is far more dramatic. The rapid deceleration of a workpiece as it makes contact with a tool in high velocity forming produces through-thickness compressive stresses, which promote stretching of the material and inhibit localization. Thus a fairly unique aspect of high velocity forming is

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that when two solid bodies impact with significant velocity very high pressures are created [8]. Even at modest impact speeds, it is easy to develop pressures large enough to produce plastic deformation.

In order to further understand how a high velocity impact benefits formability, Imbert et al. [1, 2] studied the effect of tool/sheet interaction on damage evolution in the electromagnetic forming of aluminum alloy sheet. They had observed a very modest increase in formability over the conventional FLD in the case of forming into an open cavity (free forming) versus forming into a conical die involving die strike, which generated a higher increase in formability, as was shown in Figure 4.1. Metallographic analysis was conducted on the free formed and conical die impact formed samples, to quantify the amount of damage generated by the EMF operations. Microstructure of a necked and safe free formed sample revealed a 15 and 3 times increase respectively in porosity levels when compared with the undeformed sheet. This means that the damage in free-formed parts increases with deformation eventually leading to localization and failure in the necked region. Metallographic analysis of die-impacted samples revealed a reduction in damage in some areas while some areas showed no change in damage compared to the undeformed sheet. This indicated a damage suppression effect due to tool-sheet interaction.

By integrating the Gurson’s model [9] into their FEM code LS-DYNA, they studied the change in void volume fraction with time, as the sheet metal impacted the conical die.

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The predicted void volume fraction did not reach the void coalescence criteria, while this was not the case with the free formed sample. Their analysis revealed that the sheet undergoes bending and straightening as it confirms to the shape of the die that results in a complex stress state. After impact, as the sheet straightens, compressive bending stresses occur at top surface of the sheet facing the die, while the bottom surface of the sheet

(away from the die) is in tension. This causes reduction of damage in the surface of the sheet close to the die and increase in damage on the surface away from the die. Although none of the void volume fractions exceed the critical value to initiate void coalescence.

Figure 6.2 and 6.3 show the void volume fraction and hydrostatic stress history as a function of time for a free formed sheet and sheet with die impact respectively.

Figure 6.2: Predicted hydrostatic stress and void volume fraction history for failed elements in the free form simulations. The hydrostatic stress is normalized by yield stress. Both values keep increasing till failure [1].

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Figure 6.3: Hydrostatic stress and void volume fraction history for the top surface of a sheet undergoing die impact. Reversal in void volume fraction and hydrostatic stress after impact is evident [1].

The predicted hydrostatic stress history reveals that following impact and during straightening of the sheet, large compressive hydrostatic stresses develop on the surface of the sheet facing the die which result in reduction of void volume fraction while damage at the bottom surface of the sheet increases with deformation. In the case of a free-formed sample, the void volume fraction and the predicted hydrostatic

(positive/tensile) stress increase steadily as deformation progresses and no reduction of damage is observed. Thus damage suppression in die impact can be attributed to the difference in stress state created by the tool-sheet interaction. This reduction in damage associated with die impact could also contribute towards improving formability.

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Evidence from all these experimental and numerical studies points towards the beneficial influence of impact in increasing high velocity formability by ‘inertial ironing’ i.e. with the development of through thickness compressive stresses in the sheet thickness and literally squeezing the sheet; and also by reduction of damage due to change in stress state after impact. Keeping these facts in mind, experiments were designed with the existing tube expansion setup to take advantage of the benefits of impact in reducing necking and tears. The experimental setup was not meant to show improvements in the recorded strain in the deformed samples, and will be described next.

6.2 EXPERIMENTAL PROCEDURE

The experimental setup used in these experiments was almost the same as used for tube free expansion experiments. It consisted of a solenoidal actuator, with a tube workpiece around it, connected to a capacitor bank. A cylindrical die was placed around the tube workpiece with the help of spacers. When current was passes through the capacitor bank, a primary current ran through the actuator which induced a current in the opposite direction, in the workpiece. The electromagnetic repulsion between the two fields caused the workpiece to expand away from the actuator and hit the die at high velocities. Most of the components of the setup have already been described in Section 4.2 and will be only briefly mentioned here. A schematic of the experimental setup along with its components is shown in Figure 6.4.

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Figure 6.4: Schematic of the tube die impact experimental setup

6.2.1 Capacitor bank

A commercial Maxwell Magneform capacitor bank, with a total energy of 16kJ was used for these experiments (Section 4.2.1). The energy of the bank was stored in 8 capacitors, each with a capacitance of 53.25 µF. However during these experiments, one capacitor out of the eight capacitors that make the complete bank was taken out due to damage.

Hence the total available energy here was 14 kJ and not 16kJ.

6.2.2 Actuator

A four-turn solenoid made from ASTM B16 brass wire of 6.35mm diameter was used for the experiments. The coil had an outer diameter of 5.4 cm and a pitch (center to center distance between consecutive wire turns) of 9.4 mm. Further details are in Section 4.2.2 167

6.2.3 Workpiece

The workpiece used was a AA6063 T6 tube with an inner diameter of 5.7 cm and a wall thickness of 1.75 mm. The material had a quasi-static ductility of 8-11%. The samples were cut into lengths (or heights) of 3.17cm (1.25”). They were not electrolytically etched with a circle grid because the aim here was not to measure strains. Also, from previous die impact experiments, it was seen that the grid was ‘washed away’ during die impact. Further details about the workpiece material were given in Section 4.2.3.

6.2.4 Rogowski probes

Two Rogowski probes from Rocoil [10] were used to measure the primary and induced currents. The schematic for the probe setup for measuring primary and induced currents was shown in Figure 4.8. As shown in Section 4.2.4, one Rogowski Probe measured the primary current while the other probe measured the induced current in the tube in addition to the product of the number of coil turns and the primary current. The placement of the probes is also clear from Figure 6.4 which shows a schematic of the experimental setup. An error could be present in this reading though because of the second probe also measuring the induced current in the die. However, due to the big gap between the die and the workpiece and the low conductivity of the die, this current is expected to be very small and thus not a major source of error.

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6.2.5 Die Arrangement

The die used was a cylinder with an inner diameter of 7.82 cm and wall thickness of 9.95 mm. It was machined from a stainless steel tube to a very good surface finish. It was sturdy and massive to be able to withstand the high velocity impact. The inner diameter of the tube was roughly 33% bigger than the tube workpiece outer diameter. The die was placed on the actuator with the help of two spacers made from a non-conductive lexan sheet. The spacers were circular discs with a concentric circular disc of smaller diameter, machined out (removed) from them. The outer diameter of the spacers was equal to the inner diameter of the die and their inner diameter was equal to the outer diameter of the potted coil. The spacers were made to slide on to the potted actuator and then the die was fitted on top of the spacers. The arrangement is clear from Figure 6.4 which shows the die sitting on the actuator with the help of spacers. The tube sample was placed in between the coil and the die and is not visible in the picture.

6.2.6 Contact plates

In order to ensure good contact between the round wire of the coil and the flat connection in the capacitor bank, copper plate contacts were machined. Each contact consisted of two copper plates with hemispherical cavities in them. The distance between the plates was adjustable with the help of sunken screws, to form a snug fit around the coil wire.

Thus this setup ensured a better contact between the actuator and the bank and reduced the chances of sparks. These plates are shown in Figure 6.5.

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Spacers

Cu plates Potted, Die for better four-turn contact actuator

Figure 6.5: Die arrangement with steel die, spacers and actuator used in tube die impact experiments. Tube sample has not been shown in the picture.

6.3 METHODOLOGY

A similar methodology, as adopted in tube free expansion experiments, was adopted here.

The samples were electromagnetically launched at increasing energy levels to impact with the die. The goal here was to stop the expanding sample in motion, by die impact. It was essential, to let the sample freely expand away from the actuator for some time so that it accelerates to sufficiently high velocity levels. It was also imperative to stop the sample while it was still in motion, with some kinetic energy still left in it so that it hit the die with sufficient remnant velocity. The velocity vs. radius profile schematic for a sample in motion, with and without die impact is shown in Figure 6.6.

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The die inner diameter was made roughly 33% higher than the undeformed tube’s outer diameter. This figure was partly based on availability of the tube sections to machine out a die and partly on calculations and experience. From tube free expansion experiments, the radius of the sample in which tearing has just initiated along with the energy level and peak velocity estimate for the sample run were known to us.

Die wall

ty With impact loci

Ve Without impact

X X

Radius Tear in sample

Figure 6.6: Velocity vs. Radius profile schematic for an expanding tube specimen, with and without impact.

The aim was to stop the sample at a distance (or radius) smaller than the sample in which tearing had initiated, and thus the placement of the die wall. This enabled the use of higher energy levels and corresponding peak velocities which in turn ensured more benefits from inertial forces. Thus greater advantage of inertial stabilization of necks and inertial ironing can be obtained by stopping the moving sample with a die wall. As shown in Figure 6.6, in the case involving die impact, a higher velocity profile is used and the sample is stopped before tears can initiate.

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6.4 RESULTS AND DISCUSSION

As explained in Section 6.3, the 3.17cm long AA6063-T6 specimens were electromagnetically launched at increasing energy levels onto a cylindrical die. The objective was to stop the sample while it was still supposed to be in motion (according to free expansion studies). The samples were launched at increasing energy levels of 5.6,

7.28, 8.4, 9.8 and 11.2 kJ. The effect of die impact in each case was clear from the sample profile and the strain readings from the sample dimensions. Current traces for primary and induced currents were also captured for each run.

6.4.1 Sample profiles at increasing energy levels

The initial sample runs at energy levels of 5.6 and 7.28 kJ resulted in samples which were distorted near the edges because they were not able to completely fill out the die cavity.

Hence necking took place near the sample edges, just like it would in tube free expansion experiments. The next sample run at 8.4kJ, resulted in the sample expanding more, and almost filling out the die. However it was not able to completely fill out the die due to insufficient energy and thus necking bands which were seen in the free expansion experiments were also seen in this sample. These bands were seen criss-crossing around the sample and in some places of intersection, necking and tears were evident. These tears were seen close to the top sample edge which had not impacted the die because of the non-uniform field of the coil near this sample edge. This is attributed partly to the wide pitch of the coil and partly due to the sample not aligning perfectly with the coil length. 172

Figure 6.7 shows a picture of the sample impacted at 8.4 kJ. This sample measures

7.76cm in diameter at the central region and 7.54cm near the top edge, as shown in the figure. Thus the benefit of impact is only evident in the central and bottom regions of the sample. In the top portion, tears are seen due to portions of it, not impacting the die at high velocities. This sample had an average engineering strain in the thickness of -12.9%

(i.e. it became thinner) and a corresponding average engineering strain in the height, of -

12.8% (i.e. it became shorter). The strain in the circumferential direction was about

25.6% as measured from the sample circumference in different areas, with a scotch tape.

The sample that was impacted at 9.8 kJ showed no necking and was almost the same size as the die. Thus there was an interference fit between the sample and die and it was difficult to pull the deformed sample away from the die. The sample surface was also appeared shiny due to impact with the polished inner surface of the cylindrical die.

Similarly the sample that was impacted with the die at 11.2 kJ produced an interference fit with the die and completely filled out the die cavity. It impacted the die uniformly throughout the sample length at high velocity and hence was able to benefit from the impact. There was no necking or tearing evident along the sample length.

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7.54 cm

7.76 cm

Figure 6.7: Picture of sample of length 3.17cm, impacted with a cylindrical die at 8.4kJ. Necking and tearing due to the sample not completely filling the die in some areas is evident.

Figure 6.8 shows a picture of the sample deformed at 11.2kJ. It is clear that there are no big necks or tears near the edges, as was the case in the previous runs. The sample does have a slightly bumpy profile which is attributed to the relatively high pitch of this coil which resulted in a non-uniform field. This sample, impacted at 11.2kJ had an average engineering strain in the thickness direction of -17.7% and in the height direction, -

12.6%. This should be compared with the corresponding values in the previously discussed sample with 8.4kJ impact. That sample had an average engineering strain of -

12.9% and -12.8% in the thickness and height directions respectively. Thus this particular sample has almost the same reduction in height as the previous sample. The difference lies in the thickness strain due to a higher velocity impact.

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7.82 cm

Figure 6.8: Picture of sample of length 3.17cm, impacted with a cylindrical die at 11.2kJ. No necking or tearing is evident. Sample completely filled the die cavity.

This corroborates the theory of ‘inertial ironing’. This sample had higher through thickness compressive stresses, had a higher amount of squeezing in the thickness. Thus it took greater benefit from high velocity impact. This is clear form the absence of big tears which are seen in the absence of impact or with insufficient velocity impact. To put this in perspective, Figure 6.9 shows pictures of the two samples discussed above along with a free formed sample. In the free formed sample, tearing initiated at 6.92 kJ, with the big tear progressing from edge to edge. The diameter of this sample was about 8.02 cm.

The sample die impacted at 8.4 kJ did not completely fill the die cavity and had neck and tear formations near the sample edge. The diameter of this sample was 7.76 cm, which is slightly less as compared to the die (7.82 cm). In the sample impacted at 11.2 kJ, an interference fit with the sample diameter, equal to the die inner diameter, is produced. 175

This sample has no failures, had thinned more as compared to the previous sample and was able to achieve greater benefits of ‘inertial ironing’.

Without impact With impact With impact at 6.98 kJ at 8.4 kJ at 11.2 kJ

8.02 cm 7.76 cm 7.82 cm

Sample failed but Sample not completely No failures!! in one piece filled the die cavity Interference fit

Figure 6.9: Pictures and forming conditions for electromagnetically formed samples with and without impact.

6.4.2 Current traces

The primary and induced currents were measured for each sample run with the help of

Rogowski probes and a Fluke Scopemeter. Figure 6.10 shows a current time trace for the sample impacted at 11.2 kJ. As expected, the peak current was high. Here the peak primary current was 176 kA and the Rise time was 18 µs.

176

500 Primary 400 Induced 300 A)

k 200 ( 100 0 Current -100 0 100 200 300 400 -200 -300 Time (µs)

Figure 6.10: Current vs. Time trace for sample impacted at 11.2 kJ. Peak primary current = 176 kA, Rise time = 18µs.

6.5 TUBE FLANGING EXPERIMENTS

Flanging experiments, involving die impact were designed as a variation of the tube die experiments described above. The experimental setup and results will be briefly presented here. The same four-turn coil was used to electromagnetically launch a circle gridded AA6063-T6 tube of 3.17cm length, and impact it against an L-shaped die.

Pictures of the experimental setup and the flange die are shown in Figures 6.11 and 6.12.

The die had an entry radius of 4 mm and it sat snugly outside the workpiece tube which was placed on the four-turn actuator. The tubes were electromagnetically launched at increasing energy levels into the flange die. The tube initially expanded radially outward.

Then with further deformation, it bent-over the L-shaped die forming a flange. 177

Capacitor Bank

Flange die

Workpiece

Actuator

Figure 6.11: Picture of experimental setup for tube flanging

10.16 cm

6.2 cm (a) (b)

Figure 6.12: (a) Flange die side view and (b) top view.

The tube was positioned inside the die fixture in such a way that flange of length 1.5 cm was generated. Successful creation of the flange, without tears was obtained at particular 178

energy levels. However, it was observed that further increase in launch energy resulted in initiation of tears at the flange edge, progressing inward. The length of tears increased with further increase of launch energy. A few samples showing this trend are shown in

Figure 6.12.

Figure 6.13: Pictures of flanged samples at increasing energy levels.

) 40 % g g

n 7.2 kJ

e 30 ( 8 kJ n i

ra 9.6 kJ t 20

r s Quasi-static o j a

M 10

0 -20 -15 -10 -5 0 5 Minor strain (engg%)

Figure 6.14: Limit strains in tubes flanged at different energy levels. 179

Limit strains were also measured from as close to the edge of the flange as possible. As expected the strain distribution close to the edge was uniaxial and became closer to plane strain away from the edge. The strain readings from a few of these samples are given in

Figure 6.14. This material has a quasi-static transverse ductility of about 10%. However after high velocity impact with the flanging die, strains in the range of 25-40% were observed. The observed strains increased with energy of launch.

Thus these experiments were designed to demonstrate the possibility of making stretch flanges in tubes using EMF. In conventional forming processes, it is difficult to make flanges of length around 5mm. While in this case, flanges of length 1.5cm were successfully created from a material in full-hard condition. Noticeable improvements in formability were observed. The trend of initiation of tearing, beyond particular launch energy indicates could be due to too much thinning at the edges when they impact the die. Tensile stresses in the circumferential direction at the edge of the flange can be responsible for tearing. In addition to this, inhomogeneities along the machined sample edge could also be instrumental in initiating tears. It was also observed that at too low energies, the sample just tends to bulge out and not straighten out (perpendicular to the initial tube) to form the flange. Thus there seems to be a range of die impact velocities which result in successful creation of the stretch flange.

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6.6 CONCLUSIONS

Tube die impact experiments were designed to understand the effect of impact in reducing or eliminating tears in a sample that if free formed, would fail. The objective was to take advantage of a die impact arresting the same before it can fail. It was observed that at energy levels equal to and even higher than the energy levels which initiate tears in free formed samples, a high velocity impact reduces or eliminates tears by arresting localization in the sample. This work also stands to corroborate the theory by

Imbert et al. [1, 2], that a high velocity impact has the beneficial influence of damage suppression. Thus the reduction in tears and necking as observed for a high velocity impact, in addition to inertial ironing, shows that reduction in damage is a beneficial aspect of die impact.

In order to achieve the maximum benefits of impact and completely eliminate tears in a sample, it seems that it is important to strike it at sufficiently high energy levels or velocities. If the sample is not completely able to fill the die cavity with sufficient kinetic energy, the complete benefits of impact will not be achieved. This can sometimes be misconstrued as a lack of formability improvement under high velocity conditions, as was done in a recent study by Oliveira et al. [3]. In this study die impact experiments by electromagnetically launching sheets into flat bottomed and hemispherical bottomed die cavities. Although they did see increasing dome heights with increasing energy levels, they reported the strain levels from both the die configurations, to be well below the conventional FLD. They considered this to be ‘a lack of hyperplasticity’. However the 181

reason behind this lack of improvement in the observed strains was that they had increased the cavity depth too much in order to get rid of the arcing issues thus the sample did not impact the die with sufficient kinetic energy which is dissipated as the sample moves further away from the actuator.

From tube flanging experiments, it was shown that it is possible to make stretch flanges without tears in tubes, with EMF. However a proper energy level is needed. Too low energy could not make a flange while too high energy resulted in tearing. All these studies demonstrate that it is very important to design the forming system well to achieve maximum benefit from high velocity impact. There appears to be a threshold level of impact velocity /energy that is needed to get the improvements in formability associated with a high velocity impact. This level would depend on the forming setup. This forms an important practical guideline for designing high velocity manufacturing setups. It once again bears out the fact that boundary conditions play a very important role in enhancement of formability.

It is also important to have a uniform field from the actuator so that the sample launch can be uniform. In this regards, a coil like the uniform-pressure actuator, recently developed at The Ohio State University [11] might yield better results. The coil used in the present experiments was designed to be robust, made with a thick wire, with enough pitch to prevent arcing.

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BIBLIOGRAPHY

[1] Imbert J.M., Winkler S.L., Worswick M.J., Oliveira D.A. and Golovashchenko S., The effect of tool/sheet interaction on damage evolution in Electromagnetic Forming of Al alloy sheet, J. of Engg. Mat. Tech., Vol. (127) (2005), pp. 145-153

[2] Imbert J.M., Winkler S.L., Worswick M.J., Oliveira D.A. and Golovashchenko S., Formability and damage in electromagnetically formed AA5754 and AA6111, 1st International conference on High speed forming, Dortmund Germany (2004), p. 201

[3] Olieviera D.A., Worswick M.J., Finn M. and Newman D., Electromagnetic forming of aluminum alloy sheet: Free-form and cavity fill experiments and model, J. of Mat. Processing Tech., Vol. (170) (2005) pp. 350-362.

[4] Balanethiram V.S., Hyperplasticity: Enhanced Formability of Sheet Metals at High Velocity, Ph.D. thesis, The Ohio State University (1996).

[5] Balanethiram V.S. and Daehn Glenn S., Hyperplasticity: Increased Forming Limits at High Workpiece Velocity, Scripta Materialia, Vol. (30) (1994), pp.515-520.

[6] Seth Mala, High Velocity Formability of High Strength Steel Sheet, M.S. Thesis, The Ohio State University, (2003)

[7] Seth Mala, Vohnout V.J. and Daehn G.S., Formability of steel sheet in high velocity impact, J. of Materials Processing Technology, Vol. (168) (2005), pp. 390-400.

[8] Daehn G.S., High Velocity Metal Forming, submitted, ASM Handbook (2004).

[9] Gurson A.L, Continuum Theory of Ductile rupture by void nucleation and growth: part I-Yield criteria and flow rules for porous ductile media, J. of Eng. Mat. Tech., Vol. (99), No.2 (1977), pp. 2-15.

[10] http://homepage.ntlworld.com/rocoil

[11] Kamal Manish, A uniform pressure electromagnetic actuator for forming flat sheets, PhD Thesis, The Ohio State University, 2005. 183

CHAPTER 7

RING EXPANSION EXPERIMENTS

From tube free expansion and tube die impact experiments, a lot of fundamental concepts about high velocity formability like the significance of inertia, constitutive behavior and impact were established. It was desired to understand how workpiece size and aspect ratio affect its high velocity formability. This in turn is a good way of further understanding the influence of inertia on formability under high velocity conditions. To this effect, experiments were designed for the electromagnetic expansion of rings with different aspect ratios. The motivation for these experiments along with studies in the literature will be discussed followed by the experimental setup and results.

7.1 BACKGROUND

The aim in this section is to give a brief background about the ring expansion test and present the work done in the literature about improved ductility and changes in necking

184

pattern .i.e. multiple fragmentation. Thus the results of a few experimental studies [1-4] will be presented first which reveal the existing trends about high velocity. The main motivation for the present experimental study was to study the influence of aspect ratio on formability. Numerous numerical studies for understanding this influence have been found in the literature. These studies also try to understand the importance of inertia.

7.1.1 Advantages of ring geometry and trends about ductility

Ring expansion study has been a fairly popular high-strain rate test method. The main reason for this is the elimination of stress waves that are experienced near the mobile end of a high strain rate tensile test like split Hopkinson pressure bar. The ring geometry results in a state of nearly uniform uniaxial tensile stress without any inhomogeneities of deformation.

Grady and Benson [1] performed fragmentation studies on electromagnetically expanded

AA1100 and copper rings to maximum velocities of 200 m/s and corresponding strain rates of 104 /s at fragmentation. A continuous increase of dynamic fracture strains and number of fragments with velocity was observed for both the materials. A similar result was also obtained from an experimental study conducted by Altynova et. al. [2, 3] on

AA6061 and OFHC Cu rings. Within the expansion velocities studied (50-300 m/s), the experimental results showed that ductility of both the metals increased monotonically with increasing velocity and the sample strain at failure at high velocities was nearly twice as great as that in static condition. This improvement in ductility was attributed to 185

inertial effects. These trends can be seen in Figure 7.1a which shows the total elongation and average uniform elongation of the rings as an increasing function of velocity and strain rate. At high expansion velocities, multiple necks were formed, with some necks leading to fracture. A picture of the fragmented rings is shown in Figure 7.1b.

Maximum strain rate (/s)

1.0e+4 1.5e+4 2.0e+4 2.5e+4 100

80 Cu

60 (%) ave u 6061 Solutionized Al 40 or e T e 6061 T6 Al 20

0 0 100 200 300 400 v max (m/s) r (a) (b)

Figure 7.1: (a) Ductility as a function of launch velocity. Open symbols indicate the average uniform elongation and the solid ones represent the measured total elongation. (b) Photographs of the original ring geometry and fragmentation after a total strain of 45% [2, 3].

The ring expansion test is an excellent way of addressing the issue of wave propagation experienced in a dynamic tensile test as was conducted by W. W. Wood [4], by fixing one end of the sample and moving the other with some velocity. He studied several materials and consistently found that there were three distinct regions in the ductility- velocity plot. Below a first critical velocity strain to failure was independent of endpoint 186

velocity. Between a first and second critical velocity, strain to failure was significantly increased and beyond this second critical velocity, also known as von Karman velocity, strain to failure dropped to nearly zero. This is because above this velocity, the velocity imposed to the endpoint is greater than the effective plastic wave speed. Therefore significant deformation is deposited in the driven end of the sample before the stress is fully transmitted to the fixed end of the sample, resulting in the sample pinching off near the driven end. These results are shown in Figure 7.2a which shows a schematic of the tensile test and the conditions under which, the observed ductility increases to levels higher than the quasi-static ductility, then decreases with further increases of velocity, till the failure occurs near the driven end. The numerical simulation conducted by Hu and

Daehn [5, 6], for this test profile is shown in Figure 7.2b. The same trends are revealed.

A comparison of Figure 7.1a and 7.2b shows how the axisymmetric ring expansion test is preferred over a tensile test. Its launch profile does not have the non-uniform velocity distribution and stress waves prevalent in the tensile test profile. Thus varied sample launch conditions like sample geometry, could produce very different results. Formability enhancement with velocity occurs only in a particular range of velocities. Hence the establishment of a proper velocity field is essential.

7.1.2 Multiple fragmentation

The trend about multiple fragmentation at high velocities has been shown in the experimental studies of ring expansion listed above. This feature distinguishes dynamic

187

loading from quasi-static behavior wherein generally failure proceeds from the intense localization of one neck. In dynamic conditions, there is independent nucleation of necks at multiple sites ultimately leading to fragmentation. Thus one of the most important aspects of dynamic fracture is that at the end of the fracturing sequence, the body is divided into many parts.

Quasi Static

Peak Ductility

High Velocity

(a) (b)

Figure 7.2: (a) Schematic of the tensile test geometry and failure morphology. Right side of sample is driven and failure produced at driven end at high velocity beyond the second critical velocity. (b) Plot of experimentally observed strain to failure as a function of velocity with predicted behavior. [5].

The reason for fragmentation, from an energetic point of view is that dynamic loading provides a body with kinetic energy that is not available to it under quasi-static loading.

Independent nucleation of many necks takes place simultaneously with each neck at a different level of localization. As fracture in one region occurs, stress relief fronts

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advance and unload the neighboring areas, thereby leading to a larger number of arrested necks, than fragments. The competition between the necks, some of which thrive, while other arrest, is an interesting feature here. Relief waves cause the driving force for the extension of the weak necks to drop, with the result that the necks eventually arrest.

Multiple necking at high velocities is attributed to inertia which reduces strain localization, and the rate of growth of an existing neck. A number of fragmentation theories [7-9] have been formulated to calculate number of fragments and cracks, velocity of interface between the loaded and unloaded material etc.

From the experimental ring expansion studies performed by Grady and Benson [1], and

Altynova et al. [2, 3] it was established that the number of observed necks and fragments continuously increase with velocity or strain rate. This increasing trend of both the total and uniform elongations and the number of necks and fragments with increasing velocity is considered to be a result of inertia. As was explained in Section 3.7, inertia acts to diffuse deformation from one particular localized neck, thereby allowing additional elongation in the specimen before it ultimately fails after multiple necking. A number of perturbation analysis studies have tried to predict the number of necks formed in a specimen at particular velocities. A background of these models was given in Section 3.7.

A few specific studies which study multiple necking will be mentioned here.

Han and Tvergaard [10] performed a numerical simulation for dynamic ring expansion to understand the influence of inertia. Using FEM, the formation of necks in ring segments

189

with an imperfection in the form of a taper in their wall thickness was studied at different loading rates. They revealed that for perfect rings under axisymmetric loading, there was no wave propagation around its circumference i.e. no necking occurs in the specimen.

However, small initial imperfections change that. At low strain rates (<100 s-1), necking appeared at relatively low strains and in the thinnest cross-section, which is characteristic of quasi-static behavior. At high strain rates (>1000 s-1), necking was delayed and more than one neck appeared simultaneously, with additional necks appearing in regions without initial imperfections. These necking sites were seen to be uniformly spaced throughout the specimen and were promoted by the dynamic effect. They also found that imperfections have a significant influence on the number of dynamic necks.

Imperfections were found to be essential for triggering the onset of multiple neck formation. More and uniformly spaced necks were found to occur in the presence of a small imperfection. On the other hand, a bigger imperfection resulted in the necks concentrated around it. Under rapid loading, the bigger the imperfection, the earlier necking began. Also, the lower the loading speed, the earlier the necking began because of a lower inertial effect. They showed that at high loading rates, inertia is the dominant factor in delaying the occurrence of necking and increasing ductility.

Using a model, very similar to the above [10], Sorenson and Freund [11] studied the development of necks with time in a part of a ring specimen undergoing high velocity radial expansion, under plane strain conditions. The influence of imperfections in the form of non-uniform wall thickness and of variations on constitutive behavior was

190

studied. The computations showed that instead of long-wavelength imperfections (which would result in the development of one or very few necks and occur in quasi-static case, just after maximum load point has been reached), a critical nearly periodic mode with short wavelength (thus resulting in a large number of necks with small space between them), appears at rather large overall strains. Even in a dynamically loaded specimen, in the early stages of deformation, necks evolved only from the sites around the ring circumference at which the wall thickness is minimum. However, as deformation progresses, the mean strain rates in these sites are only slightly higher than the background rates (due to inertial resistance), thus indicating that the background velocity is too large to allow neck formation at these sites. In the later stages of deformation, the gradual thinning from imperfections is interrupted by the appearance of multiple necks with a more or less regular spacing. The location of the necks showed a weak or no correlation to the imperfection distribution. This reveals that in some way, imperfection sensitivity of materials is suppressed at high strain rates.

The sequence of neck development, as mentioned above is shown in Figure 7.4. It shows the evolution of necks as a function of time along a segment of the circumference of the ring indicated by the angle θ (0 < θ < 22.5o), in terms of a normalized strain rate, q. As shown in the figure, after some initial elastic oscillations, a non-localized zone of thinning appears and grows relatively slowly. Later in the deformation, a cascade of instabilities appears with a nearly constant spacing of localization sites around the circumference. Some of these result in fractures while others are arrested.

191

Figure 7.3: Neck development in terms of level curves of q, a normalized inelastic strain rate, over the θ-time plane [11]. First (left) a non-local zone of slow thinning appears (large parabola), which gives way to (right) a cascade to instabilities with different levels of strain. Some of them result in fracture.

Shenoy and Freund [12] studied the dynamic necking bifurcations during rapid plane strain extension of a block of material which is considered to be a part of a thin walled sheet deforming at high strain rate. They showed that at high velocities, an intermediate spacing of necks is observed because the rate of growth of very long and very short wavelength modes of non-uniform deformation are suppressed by inertia. With increasing velocity, the spacing between necks decreases in magnitude. The necking pattern was observed to be independent of the strain hardening exponent.

192

Gurduru and Freund [13] adapted the above analysis [12] to a homogenously deforming cylindrical rod in order to facilitate comparison with ring expansion experimental results

[1-3]. Overall there was good qualitative correspondence between the experimental and analytical results, in terms of number of necks at a particular extension velocity. There was some discrepancy in the calculated fracture strain, which was attributed to the uncertainty in constitutive data used in the analysis. Both these analyses neglected the curvature of the rings and other effects like strain rate sensitivity (which could have influenced the results) and attributed all the results to inertia.

Pandolfi and Ortiz [14] used a cohesive element based FEM model to simulate the experimental ring expansion results of Grady and Benson [1]. Good correspondence was achieved in the two studies. It was also revealed that there was variability in fragment sizes from samples. As was observed in experiments, the simulations predicted a statistical distribution of fragment sizes from a number of samples tested, with fragment mass from different samples, clustering around a particular mass value, with lighter and heavier fragments becoming increasingly infrequent.

Needleman [15] found that inertia slows down neck development and lowers stress triaxiality in the neck center. Long wavelength modes are suppressed at high velocities since their rate of formation is too slow because of inertial resistance. Mercier and

Molinari [16] developed a linear perturbation analysis for rapid radial expansion of rings.

Their analytical results compared well with the experimental results of Altynova et al. [2,

193

3] shown in Figure 7.3. They showed that for a viscoplastic material, the size of fragments is an increasing function of strain rate sensitivity and a decreasing function of applied velocity. Fressengeas and Molinari [17] studied the dynamic necking instability in rapidly stretching sheets formed with explosives. They found that the necking pattern was established by interplay between the destabilizing effect of geometrical softening

(due to section reduction) and the stabilizing effect of inertia and rate sensitivity. They found that viscosity effects were responsible for diminishing the growth of very short wavelength modes. Hutchinson and Neale [18] investigated the effect of strain rate sensitivity (neglecting material inertia) on necking in the uniaxial tensile test. They had found that rate sensitivity had a strong influence on post-uniform elongation. The analysis of Sorensen and Freund [11] which was explained above and was on the lines of the analysis of Han and Tvergaard [10], indicated that the details of material behavior, like void growth, rate sensitivity and thermal softening play only a minor role in determining the necking pattern on ring expansion.

Thus there is diverse opinion in the literature about which factor, the constitutive behavior or inertia is responsible for suppressing short wavelength non-uniform deformation modes. There is agreement however that the long wavelength modes are suppressed by inertia and ultimately an intermediate wavelength mode of deformation is observed. However it should be noted that from the experiments [1-3], minor influence of rate sensitivity was observed on the necking pattern. Similar trends for copper which is rate sensitive and aluminum which has minimal rate sensitivity were observed.

194

7.1.3 Effect of aspect ratio on formability

A number of numerical investigations [10-19] have been carried out to study the effect of workpiece aspect ratio on its formability. Aspect ratio is typically defined as the ratio of a sample’s width to its height or vice versa. The results about multiple necking from a few of these studies were discussed in the previous section. Some studies with their results about the effect of aspect ratio on necking or formability will be discussed next. These studies provided the main motivation for the present study.

Sample aspect ratio has been shown to influence the determination of ductility in tensile tests. In a tensile test, if the sample gauge length is much greater than the sample diameter (or width), the contribution of necking to total elongation is very small. Plastic deformation occurs in a non-homogenous manner along the gauge length. Plastic strain is localized and maximum in the neck while outside it little elongation takes place. On the other hand of the gauge length is much smaller than the sample diameter, necking elongation accounts for most of the elongation. Due to this a fixed aspect ratio of the sample is taken in tensile tests, according to ASTM standard (usually gauge length/diameter = 4).

In quasi-static tensile tests it has been shown that the aspect ratio of a sample can influence its fracture strain in a particular range of aspect ratios. This can be seen in

Figure 7.4 which shows specimen size effects on tensile tests performed with and without neutron irradiation [19].

195

Figure 7.4: Influence of sample aspect ratio of tensile sample on ultimate strain [19]

As shown in the figure, sample aspect ratio is an important factor in establishing quasi- static ductility with low aspect ratio (i.e. slender) samples having lower ductility. Few studies in the literature also establish the influence of aspect ratio on high velocity ductility and will be presented next.

An experimental study carried out by Tamhane et al. [20], with AA6061-T4 and annealed

OFHC copper rings of a fixed diameter (30mm) and wall thickness (1mm), but with varied heights, revealed the effect of sample size on high velocity formability. The heights of the rings were varied from 1 to 16mm with the same coil. Figure 7.5 shows the maximum strain without failure, as a function of ring height for both the materials. Very high extensions were seen to be available just by changing the geometry of the sample 196

being launched. They also noticed a higher resistance to thinning and a difference in the necking pattern for the 1 and 16mm rings. The apparent anisotropy of the samples, also the fact that as the heights of the tubes were increased the stress state was not strictly uniaxial anymore, points towards the fact that with this kind of a setup, the magnetic field can induce axial pressure on the tube and also the constraints imposed by inertia may effect the apparent r-value in tube expansion [21]. There is a strong influence of the strain state on accessible material extension. Thus a better way is needed to study the effect of sample aspect ratio on its formability.

70 60 Cu and static ductility 50 Extension (%) 40

30 Al and static ductility 20

Circumferential 10 height 0 124816 Ring Height (mm)

Figure 7.5: Circumferential extension without failure in free form electromagnetic expansion of AA6061-T4 and OFHC Cu samples vs. Ring height [20].

Shenoy and Freund [12] studied the dynamic necking during rapid plane strain extension of a block of strain hardening material, at a constant end velocity. The material was

197

assumed to have a rate independent constitutive behavior. They studied the effect of aspect ratio of a block, which was defined as a ratio of the block’s height or thickness with its length. The analysis indicated that for small aspect ratio blocks, the number of necks per unit length is an increasing function of deformation velocity and decreases with the aspect ratio of the block and predicted an empirical relation showing this dependence.

Nilsson [22] studied the effect of inertia on the necking pattern for tensile bards under dynamic loading. It was observed that two bars which had the same aspect ratio, with one bar ten times wider and longer than the other, had overall the same necking pattern and necking sites. However when studying bars of different aspect ratios, the bar with a larger aspect ratio (thicker or wider for the same length) had a larger level of overall logarithmic strain at which necking initiated and localized, and also a smaller number of necks, thus a higher ductility. Thus the aspect ratio of the sample was more important than its size. Han and Tvergaard [10] also found the dependence of number of necking sites on the aspect ratio of the ring, with a larger aspect ratio sample having fewer necks. From the point of view of the onset of necking, a larger aspect ratio ring tended to delay necking. The neck spacing was found to be effected more by the aspect ratio than by the level of strain hardening in the material.

The prediction of dependence of ductility on aspect ratio seems to be a common outcome in all these studies, despite different model setups and assumptions. There is some diversity regarding whether or not rate sensitivity is a dominant factor in high velocity

198

formability. However the importance of inertia in influencing the necking pattern and ductility is clear and is also corroborated from the influence of aspect ratio on formability. As the aspect ratio and hence mass increases, inertial forces increase. Hence, a larger aspect ratio sample also has higher inertial forces.

There is a lack of experimental data to support and analyze the results of these studies. It has been explicitly stated in some of these analyses [12, 13] that these predictions about the effect of aspect ratio, number of necks etc., are incomplete till they are complemented by experiments which either support or refute them. The existing experimental results are limited and do not investigate the effect of aspect ratio on formability. Thus with this motivation, electromagnetic forming experiments with square cross-section rings with different aspect ratios (varying wall thicknesses and heights for a fixed inner diameter) were designed and will be explained next.

7.2 EXPERIMENTAL PROCEDURE

Electromagnetic free-form expansion experiments were performed with AA5754 rings with square cross-section and different aspect ratios. The setup for ring expansion experiments was designed primarily to understand and quantify the effect of sample cross-section and size on its formability, as outlined in Section 7.1. Ring specimens were radially accelerated by imparting electromagnetic energy to them. A capacitor bank, a solenoidal coil, probes for measuring primary and induced currents, and a high speed digital array camera for obtaining velocity estimates for the expanding ring, were

199

essential components of the experimental setup. Figure 7.6 and 7.7 show a schematic and a picture of the setup which shows the ring specimen placed around a solenoidal coil connected to the capacitor bank. The current in the coil induces an eddy current in the ring in a direction opposite to the primary current in the coil. This causes mutual repulsion between them resulting in the radial expansion of the ring.

7.2.1 Capacitor bank

A Maxwell Magneform capacitor bank with a total energy of 16 kJ was used for the experiments. The energy of the bank was stored in 8 capacitors, each with a capacitance of 53.25 µF. The system had a maximum working voltage of 8.66 kV. Both the number of capacitors and charging voltage could be adjusted to control the discharge energy and voltage. For the duration of the experiments, only four out of the available eight capacitors were used. Hence the total energy of the bank was 8kJ.

7.2.2 Actuator

The actuator used was a closely wound five-turn solenoid made from ASTM B16 brass wire which had a 4.7mm x 4.7 mm square cross-section. The outer diameter of the coil was 5.93 cm and its pitch of 6mm. It was fabricated by commercial spring manufacturer

[23] and was wound as tightly as possible without consecutive turns touching each other.

It was designed to result in an axisymmetric radial launch of the ring specimen. The wire in the actuator was covered with heat shrink wrap tubing and then potted in urethane to give structural support to the coil. 200

Monitor Pearson probe Rogowski probe Ring

Capacitor Bank High Speed Coil Camera

Oscilloscope

Figure 7.6: Schematic of experimental ring expansion setup with camera and oscilloscope.

Coil Ring Capacitor bank

Figure 7.7: Electromagnetic ring expansion experimental setup picture with capacitor bank, actuator and ring specimen.

201

RR==11 9.5.1 mm 6 mm 3. 3 c m

5.93 cm OD RR==11 9.5.1 mm Le g Le R=4 mm ngth = 12.5 c m R=11.1 mm

(a) (b)

Figure 7.8: Schematic of five-turn brass wire actuator used for EM ring expansion experiments. (a) Front view, (b) Top view

Figure 7.9: Picture of insulation covered five-turn ASTM B16 brass wire actuator

202

The chemical and mechanical properties of the ASTM B16 brass wire were provided in

Tables 4.1 and 4.2 respectively. Figure 7.8 shows a schematic of the front and top view of a four-turn coil along with all the measurements. Figure 7.9 shows a picture of the five- turn coil, covered with shrink wrap tubing which was potted in urethane before using it.

7.2.3 Workpiece

The workpieces in these experiments were rings made from a AA5754-O alloy tube with an OD of 6.985 cm (2.75”) and a wall thickness of 4 mm. Three different square cross- section rings were machined from this tube such that all three types have approximately the same inner diameter which is almost equal to the outer diameter of the potted coil.

The ring cross-sections and diameter details are given in Table 7.1 while Figure 7.10 shows pictures of the rings. The three different types of rings are referred to as 4x4, 2x2 and 1x1 according to the approximate dimensions of their cross-sections (height x thickness). The ring types will be referred to with this convention throughout.

Thickness Height Inner diameter Type Aspect Ratio (mm) (mm) (cm) 4x4 3.98 3.91 6.21 0.02

2x2 1.95 1.95 6.33 0.01

1x1 0.99 1 6.37 0.005

Table 7.1: Dimensions of the three different cross-section rings

203

Figure 7.10: Picture of 1x1, 2x2 and 4x4 AA5754-O ring specimens

In Table 7.1, the aspect ratio of the rings was calculated as the ratio of the ring thickness or height to its average circumference. This is the same convention adopted in some numerical studies mentioned in Section 7.1.3, specifically Shenoy and Freund [12].

In order to determine the quasi-static tensile properties of the 5754 alloy tube, sections were cut from the tube in longitudinal and transverse directions and then were carefully straightened. The tensile samples were water jet cut from these sections according to

ASTM standard with a gauge length of 2.54cm (1”) and width of 0.63 cm. The samples were then tested in an MTS machine at a strain rate of 3.3x10-3. An average of four sample runs was conducted for each orientation. The results for two samples in each orientation are shown in Figure 7.11 which shows the engineering stress-strain plot for the 5754 aluminum alloy tube in the as received condition. In the plot, ‘L’ stands for longitudinal section and ‘T’ stands for transverse section. Table 7.2 lists the average values mechanical properties of the alloy as deduced from the tensile tests in the

204

longitudinal and transverse directions. It is clear from the plot that the material is slightly more ductile in the transverse direction. The difference is not acute and can be due to material anisotropy. It could also have been influenced by the straightening process.

150

125 )

pa 100 M (

s 75 s e r

St 50 5754-L2 5754-L3 25 5754-T2 5754-T3 0 0 3 6 9 12151821242730 Strain (engg%)

Figure 7.11: Engineering Stress vs. Strain plot for longitudinal and transverse sections of the AA5754-O tube used for making the rings

Ultimate Tensile Strength Yield Strength % Elongation (MPa) (MPa)

154 95 24.8

Table 7.2: Mechanical properties of the AA5754-O tube, used for making ring specimens. 205

7.2.4 Rogowski and Pearson Probes

In order to measure the primary and induced currents in the coil and ring specimen, a

Pearson probe (Model 4418) [24] and a Rogowski probe [25] were used respectively. The setup and details about the Rogowski probe were provided in Section 4.2.4. It had a peak range of 150, 000 Amperes/Volt. For these experiments, the setting of 50,000

Amperes/Volt was used. The Pearson Probe had a maximum output of 200V and a calibration of 1000 Ampere/Volt. Thus the maximum measured peak current was 200 kA and the usable rise time was 200 ns. The Pearson probe measured the primary current while the Rogowski probe measured the induced current in the ring along with five times the primary current. The setup of both the probes along with the actuator and ring assembly is shown in Figure 7.12. The outputs were transferred to a two-channel digital storage scope (Fluke Scopemeter 99 Series II).

7.2.5 High Speed Camera

A high speed digital array camera (Cooks Flashcam [26]) was used for estimating the position of the sample vs. time as the ring expanded. The coil and ring assembly was covered with a safety cage which was a large diameter PVC pipe. The pipe had a lexan sheet transparent cover so that the camera could capture the motion of the ring. The inside of the cage/pipe was layered with soft black modeling clay to collect the ring fragments without damaging them. The camera circuit included a flash that triggered along with the camera, a Fresnel light and some studio lights and a monitor Figure 7.13 shows a picture of the experimental setup with the camera and light assembly.

206

Capacitor Bank

Rogowski probe Ring

Actuator Pearson probe To Oscilloscope

Figure 7.12: Picture of experimental setup with the actuator-ring assembly and the Pearson and Rogowski probes used for measuring primary and induced currents.

High speed camera Fresnel connected light

Monitor Oscilloscope

Safety cage

Figure 7.13: Picture of experimental setup with high speed camera and light assembly. 207

In order to account for the difference in scales for the images on the monitor and their actual size, a scale factor was calculated by marking a ring of 5cm diameter on the front face (facing the camera) of the actuator. In order to easily differentiate the edge of the ring specimen as it was expanding, marks of a known pattern were made on the ring. The marks also helped in placing back the ring fragments after electromagnetic expansion.

The camera had adjustable exposure and delay times ranging from one microsecond to one millisecond. After the preset delay, the trigger started an exposure for the preset

‘width’ (exposure) setting. This block of delay + width was repeated for the number of preset cycles (which in turn ranged from 1 to 10 times). In these experiments, exposure intervals (delay + width) of 11-23 microseconds and number of cycles of 4-7, were used to capture the images. By knowing the exact exposure time of the camera between successive images, and calculating the distance traveled by the ring edge in that time taking into account the scale factor, its velocity was estimated. Different settings were required for each ring cross-section type to capture the different velocity profiles. In addition to that the light intensity had to be adjusted if it was expected that the ring would fragment a bright flash was produced as the ring fragmented and the circuit was broken.

7.3 METHODOLOGY

The basic approach used in these experiments was to expand the ring workpieces with electromagnetic forces generated between the solenoid actuator and the workpiece, with

208

the help of the capacitor bank. For each ring cross-section, the samples were launched at incremental launch velocities (by varying the energy discharged from the capacitor bank).

A few sample runs were conducted at low energies with the ring expanded but still in one piece. Then, an effort was made to expand the sample at an energy level at which only one failure took place and the sample was still in one piece. Successive launches at higher energy levels resulted in ring fragmentation. For each experimental run, the current traces and the edge of the sample in motion were captured. Uniform and total elongation, number of necks etc. were also measured from each sample. Prior to using the machined rings, trials were also conducted with saw cut rings, to adjust the camera parameters viable for capturing the maximum number of cycles possible.

In the entire document, 4x4, 2x2 and 1x1 rings have been represented respectively by the symbols,

7.4 RESULTS AND DISCUSSION- POSITION AND VELOCITY

PROFILES

As mentioned in Section 7.3, experimental runs at incremental energy levels were conducted for the 4x4, 2x2 and the 1x1 rings. Experimental data collected included high velocity images from which the velocity and position profiles were deduced; primary and induced current traces; uniform and total true strains; and number of necks and fragments. The results for the different aspect ratio rings - velocity and position profiles; true strains; number of necks and fractures; and current traces, and will be presented next. 209

7.4.1 Position and Velocity profiles for 4x4 rings

With the help of the high speed camera, images of the ring edge in motion were captured.

A few high velocity images along with the calculated velocity and position vs. time profiles for a series of 4x4 rings will be presented. Figure 7.14 shows an experimental data set for a 4x4 ring electromagnetically expanded at 2.96 kJ. Figure 7.14a shows the picture of the expanded ring with no failure in it. The high velocity image of this ring in motion is shown in Figure 7.14b. This picture shows five superimposed images of the ring in motion, captured by the camera. The last two images are partially overlapping as the ring has slowed down. Knowing the camera parameters of the exposure time of 22µs

(Width = 19µs and Delay = 3µs) the distance between successive camera exposures can be calculated to obtain an estimate of the position of the sample as a function of time.

This profile is shown in Figure 7.14c. By knowing the distance traveled by the sample in a fixed time interval, its velocity profile was calculated and is shown in Figure 7.14d.

The velocity of the ring peaks to a particular level as it accelerates away from the ring due to electromagnetic repulsion and then due to its own inertia. The ring then slows down and ultimately comes to a stop. It should be noted the sophisticated equipment like a VISAR can determine the velocity data quite precisely. However in the absence of such equipment, the present method can estimate the motion of the sample. The error bars in

Figure 7.14c indicate the degree of accuracy of the position of the sample which was

±0.5mm and ±3µs. Figures 7.15-7.17 show a sequence of high velocity images, position vs. time plots and velocity vs. time plots respectively for 4x4 rings at increasing energy

210

levels. All these pictures indicate the axisymmetric launch of the ring and that the sample launched at both low and high energy levels, maintain this circular profile up until fragmentation. Figure 7.15 shows a few high velocity images of 4x4 rings launched at increasing energy levels. There pictures have been shown to scale.

9.5cm

(b) (a)

4.8

Measured terminal ring )

m 4.5 outer radius c ( 4.2 dius a

r 3.9 r e 3.6

ng out Measured initial i 3.3

R ring outer radius 3 0 25 50 75 100 125 Time (µs)

175 150 )

/s 125 m

( 100 y t i

c 75 o l e 50 V 25 0 0 25 50 75 100 125 Time ( µs) (d)

Figure 7.14: Experimental results for a 4x4 ring electromagnetically expanded at 2.96 kJ. (a) Pictures of expanded ring, (b) High velocity image with camera parameters: Width = 19µs, Delay = 3µs and number of cycles = 5, (c) Position vs. time plot, (d) Velocity profile

It should be noted that in Figure 7.15 d, e and f, the big flash produced as the sample fragments in evident. It tends to wash out the image. To control this, the light levels were kept low, thereby sacrificing some clarity. Also the camera was set to shut itself off after a few (4-6 cycles) so that when the big flash was produced, the camera would not wash off the rest of the cycle images. However, the main idea here is to capture the peak velocity of the ring in motion and to get an estimate of it’s velocity profile. Figures 7.16 and 7.17 show the position and velocity plots for the samples corresponding to the high velocity images in Figure 7.15. As expected, as energy of electromagnetic launch is increased, the sample expands to higher strain levels indicated by the increased position coordinate with increase in energy as shown in Figure 7.16.

212

(a) (b)

(f) (e)

Figure 7.15: High velocity images shown to scale for 4x4 rings electromagnetically expanded at (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ , (e) 4 kJ and (f) 5.6 kJ. 213

The corresponding increase in peak velocities with increase in energy of launch is also evident in Figure 7.17. As energy in increased, there is increase in peak velocity in the ring along with a raise in level of the entire plot. The overall shape of the curve remains the same even at high energy levels, except for the evident reduction in slope of the deceleration curve after the peak velocity. This is to be expected as the ring that accelerated to higher energy level also tends to decelerate slowly and maintains its velocity path for a longer time.

) 5.5 ) 5.5 m m c c ( ( 5 5 Measured terminal Measured terminal dius ring outer radius dius a a 4.5 4.5 ring outer radius r

r r e e 4 4 ng out

ng out 3.5 3.5 i i Measured initial Measured initial R R ring outer radius ring outer radius 3 3 02550751001250255075100125 Time(µs) Time(µs)

(a) (b)

) 5.5 5.5

m Measured terminal ) c Measured terminal m

( ring outer radius c s 5

ring outer radius ( 5 u di us a di

r 4.5

4.5 a r r e r t

4 e (c) 4 (d) ou out ng

i 3.5

Measured initial ng i R 3.5 ring outer radius R Measured initial 3 ring outer radius 3 0255075100125 0 25 50 75 100 125 Time (µs) Time (µs) (c) (d)

214

5.5 ) 5.5 Measured Measured terminal ) m ring outer radius m terminal ring c c (

5 ( 5 outer radius dius dius

a 4.5 4.5 a r r r

r e 4 e 4

ng out 3.5 ng out i 3.5 Measured initial ring i Measured initial R outer radius R ring oute r radius Figure3 7.16 3 0 25 50 75 100 125 0 25 50 75 100 125 Time(µs) Time(µs)

(e) (f)

Figure 7.16: Position vs. Time plots for 4x4 rings measured from high velocity images for ring expanded at energy of (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ , (e) 4 kJ and (f) 5.6 kJ.

250 250 ) ) s

/ 200 200 /s m (m 150 (

y 150 ty t

i i c o loc l 100 100 e V Ve 50 50

0 0 0 25 50 75 100 125 0 25 50 75 100 125 Time (µs) Time (µs) (a) (b)

250 250 )

) 200 200 /s /s m

m (

( 150 150 y y t t i i

loc 100 loc 100 e e V V 50 50 0 0 0 25 50 75 100 125 0255075100125 Time ( µs) Time ( µs) (c) (d) 215

250 250 ) ) s

/ 200 /s 200 m (m 150 (

y 150 ty t

i i c 100 lo

loc 100

2.2 e Ve 50 V 50

0 0 0 25 50 75 100 125 0 25 50 75 100 125 Time (µs) Time ( µs) (e) (f)

Figure 7.17: Velocity vs. Time profile for 4x4 rings measured from high velocity images for ring expanded at energy of (a) 2.16 kJ, (b) 2.56 kJ, (c) 2.96 kJ, (d) 3.36 kJ

, (e) 4 kJ and (f) 5.6 kJ.

7.4.2 Position and Velocity profiles for 2x2 rings

A similar set of data was obtained for 2x2 rings. Due to a lower mass, these rings needed lower energy levels for expansion at high velocities. The same trends of increasing expansion and velocity with increased energy of launch were obtained, as expected.

Figure 7.18a shows a picture of a ring specimen with three fragments, expanded at

1.04kJ. Figure 7.18b shows the high velocity image obtained from the camera. Parts of it have been washed out due to the flash generated on fragmentation. Knowing the camera parameters of the exposure time of 21µs (Width = 17µs and Delay = 4µs), the distance between successive camera exposures was calculated to obtain an estimate of the position of the sample and its velocity profile as a function of time. They are shown in Figure

7.18c and 7.18d respectively.

216

(a) (b)

4.5 ) m 4.25 c

( Measured

s terminal ring u 4 i

d outer radius a

r 3.75 r e t 3.5 ou ng i 3.25 Measured initial R ring outer radius 3 0 20406080100120 Time (µs) (c)

217

175 150 )

/s 125

(m 100 y t i

c 75 lo

e 50 V 25 0 0 20 40 60 80 100

Time (µs) (d)

Figure 7.18: Experimental results for a 2x2 ring electromagnetically expanded at 1.04 kJ. (a) Picture of fragmented ring, (b) High velocity image with camera parameters: Width = 17µs, Delay = 4µs and number of cycles = 5, (c) Position vs. time plot, (d) Velocity profile.

7.4.3 Position and Velocity profiles for 1x1 rings

Incremental energy levels were used to launch the 1x1 rings as well. Similar trends for velocity and position profiles were obtained. As expected the low mass rings needed low energy levels to fragment and to accelerate to high velocities. Figure 7.19 shows an experimental data set for a 1x1 ring fragmented at 0.56 kJ. Figure 7.19a shows the two ring fragments and Figure 7.19b shows the high velocity image. As the ring was relatively thin, it was more difficult to distinguish the superimposed images. Also since the 1x1 rings, internal diameter was larger than the other rings (Table 7.1), this ring sat loosely on the actuator. Due to this it appears farther away from the actuator in parts

218

(bottom-right) of the image. Since these rings were relatively slender, they distorted

(which is evident in Figure 7.19a.) after expansion, when they hit the soft lining of the safety cage, Figures 7.19 c and d show the position and velocity profile respectively.

(a) (b)

4 Measured terminal m)

c 3.8 ring outer radius (

dius 3.6 a r r e

t 3.4 ou

g 3.2 Measured initial n i ring outer radius R 3 020406080 Time (µs)

(c)

219

120 100 ) s / 80 (m ty

i 60 c o l 40 Ve 20 0 020406080 Time ( µs) (d)

Figure 7.19: Experimental results for a 1x1 ring electromagnetically expanded at 0.56 kJ. (a) Picture of fragmented ring, (b) High velocity image with camera parameters: Width = 14µs, Delay = 4µs and number of cycles = 4, (c) Position vs. time plot, (d) Velocity profile.

7.4.4 Peak velocities for all three ring types

The high velocity images were used for estimating peak ring velocities. However during a few sample runs, the high velocity camera did not trigger and the peak velocities were estimated using Figure 7.20. The experimentally measured peak velocities were plotted against the energy of electromagnetic launch, to obtain linear trend lines. For each aspect ratio ring, a linear trend line was obtained. As expected, the peak sample velocity monotonically increases with the energy of launch. The extrapolation of these lines can also be used to estimate sample velocities outside the experimental range. As expected, the lower mass 1x1 rings have the highest peak velocity for a particular energy of launch.

220

300 4x4 )

/s 250 2x2 m ( 200 1x1 y t i c 150 lo e

v 100 k a

e 50 P 0 012345678

Energy (kJ)

Figure 7.20: Experimentally measured peak velocity vs. energy of launch plot for

4x4, 2x2 and 1x1 rings.

7.5 RESULTS AND DISCUSSION – TRUE STRAINS

For each electromagnetically expanded ring, true uniform and total strains were measured. The true uniform strain is the reduction in cross-section of the uniform (i.e. away from necks) regions. It was determined by taking an average of five readings of the thickest cross-sections in the deformed sample. These thickest regions were regions away from the necked region and hence were used to measure the true strain. The total elongation is the circumferential strain in the samples. In the cases in which the rings were fragmented, the total elongation was calculated by putting together the ring fragments and measuring the circumference of the ring (or length of the fragments).

While in the cases involving no fragmentation, the total elongation was denoted by taking

221

an average of five measurements of the diameter of the expanded ring. True strains were calculated from the measured engineering strains in all cases. Figures 7.21 – 7.23 show the true uniform and total strains in a 4x4, 2x2 and 1x1 rings respectively, as a function of the measured peak velocity attained by them. Each plot shows the strain measurements from a number of rings launched at incremental energy levels. For all the different ring cross-sections, the observed strains increase with the peak velocity. Initially strain increases significantly with increase in velocity. However with further increase in velocity, the increase of strain is lower. The increase in strain with velocity is expected due to a higher energy of launch and thus a higher kinetic energy in the sample. This translates into a higher circumferential extension. In all the plots the increase in true total strain with velocity is higher than the corresponding uniform strain.

60 4x4-uniform 50 4x4-total Don’t Fail Fail )

(% 40

ra 30 t

s Quasi-static tot. e

u 20 r T 10 Quasi-static uni.

0 25 75 125 175 225 275 325 Peak velocity (m/s)

Figure 7.21: True uniform and total strains for a number of 4x4 rings electromagnetically launched at increasing energy levels.

222

60 2x2-uniform 50 2x2-total )

% 40 ( Don’t Fail Fail n i a r

t 30

Quasi-static tot. ue s 20 Tr Quasi-static uni. 10

0 25 75 125 175 225 275 325 Peak velocity (m/s)

Figure 7.22: True uniform and total strains for a number of 2x2 rings electromagnetically launched at increasing energy levels.

60 1x1-uniform 50 1x1-total )

% 40 ( n i Don’t Fail Fail a r

t 30

Quasi-static tot. ue s 20 Tr Quasi-static uni. 10

0 25 75 125 175 225 275 325 Peak velocity (m/s)

Figure 7.23: True uniform and total strains for a number of 1x1 rings electromagnetically launched at increasing energy levels.

223

In all the plots, the quasi-static uniform and total true strains, as determined by the quasi- static tensile tests have been shown with dotted and solid lines at values of 14.98% and

24.3% respectively. From Figure 7.21, it is evident that for the 4x4 rings (aspect ratio

=0.02) launched at relatively low peak velocities of up to 135 m/s, no improvement in total elongation in comparison with the quasi-static elongation, is seen, although there is some improvement in the uniform elongation. However in the samples with higher peak velocities, an incremental improvement in both total and uniform elongation is seen. The observed true total elongation from the 4x4 rings at higher peak velocities (290 m/s) is almost 50% which represents a 100% improvement from the quasi-static value. This substantial improvement in total elongation for the 4x4 rings is also accompanied by a corresponding doubling of the uniform elongation at the highest peak velocities. Figure

7.22 shows the true strain vs. peak velocity plot for the 2x2 rings (aspect ratio =0.01).

Once again it is seen that at relatively low peak velocities of up to 125 m/s, no improvements in high velocity total and uniform elongations is seen over their quasi- static values. However, with further increase in peak velocity, there is a steady increase in ductility of the rings. At the highest peak velocity tested (285 m/s) a 50% and 25% improvement in total and uniform elongation over the quasi-static values respectively is observed. Thus a noticeable increase in high velocity formability of the 2x2 rings and its favorable response to increase in velocity is clearly evident.

Figure 7.23 shows the true strain vs. peak velocity plot for a series of 1x1 rings (aspect ratio=0 .005) electromagnetically launched at increasing energy levels. In this plot, the 224

high velocity total and uniform seem to increase slightly with increase in peak velocity.

However, the high velocity values never increase beyond their quasi-static counterparts.

So much so that even at the highest tested peak velocity (165 m/s), the high velocity uniform and total elongation are 48% and 30% lower than their respective quasi-static values. Thus it is clear that this aspect ratio ring generated high velocity true strains lower than the quasi-static values and although a slight increase in formability, with increase in peak velocity was seen, no improvement in formability was observed.

Thus the 4x4 rings showed a substantial improvement in formability with almost a 100% improvement in both total and uniform elongations over the quasi-static values. The 2x2 rings also showed a big improvement in formability although the improvement was not as much as the 4x4 rings. On the other hand, the 1x1 rings showed no improvement in formability under high velocity conditions and recorded true total and uniform elongations lower than their respective quasi-static values. These observations are shown in Figure 7.24 which is a combined plot of true stress vs. strain for all the aspect ratio rings. It shows the influence of aspect ratio on high velocity formability. The highest aspect ratio rings also got the most benefit in formability with velocity while a decreasing improvement or even in worsening effect of velocity on high velocity formability was seen with decreasing aspect ratio. This observation links to the results of numerical studies in the literature, mentioned in Section 7.1.3. Dependence of formability on aspect ratio was an outcome of a number of numerical studies.

225

60 4x4-uni. 50 4x4-tot. 2x2-uni. 2x2- tot.

) 40 1x1- uni. %

( 1x1- tot. n

i 30 a r Quasi-static tot.

St 20

ue Quasi-static uni.

Tr 10

0 25 75 125 175 225 275 325 Peak velocity (m/s)

Figure 7.24: Combined true strain vs. peak velocity plot for all aspect ratio rings.

Thus the plot in Figure 7.24 corroborates the outcome of numerical studies of the dependence of the improvement in formability with velocity. This in turn points to the importance of inertial factors in high velocity formability. Even though, all the aspect ratio rings were expanded at high velocities, larger improvements were seen for larger aspect ratio rings which had higher inertial resistance to necking and failure. The higher mass of the larger aspect ratio rings provided larger inertial stiffness and a corresponding stabilizing influence to them thereby allowing them to keep extending and avoiding failure. This also points towards the fact that if only constitutive behavior, specifically the strain rate sensitivity was the governing factor, then all aspect ratio rings should have had improvements in formability at these high strain rates.

226

Another trend observed in all the above plots is of a larger increase in total elongation with velocity, as compared to an increase in uniform strain with velocity. This was more the case for the rings whose formability actually benefited from velocity. In 4x4 rings, the improvement in total elongation in the experiment span of peak velocities from 125 – 290 m/s was 85% higher than the corresponding improvement in uniform elongation. In 2x2 rings, the improvement in total elongation in the experiment span of peak velocities from

100-285 m/s was 60% more than the corresponding improvement in uniform elongation.

On the other hand in the case of 1x1 rings this figure was just 7%. Thus the rings whose formability was most improved with velocity had a larger improvement in total elongation as compared to the uniform elongation. On the other hand the 1x1 rings, whose formability did not benefit at all from velocity, had similar improvements in total and uniform elongation.

This observation also points towards the influence of inertial effects. Xu et al. [5, 6] with the help of a 1-D model to study inertial effect, had shown that the total elongation of samples is increased by inertia at high velocities primarily due to an increase in the post- uniform elongation because significant acceleration is developed only after necking commences. The primary influence of inertia in improving formability at high velocities is by stabilizing deformation against neck growth. Post-uniform elongation when necking commences, higher acceleration of the material into a growing neck results in higher inertial forces which in turn have a stabilizing influence.

227

7.6 RESULTS AND DISCUSSION – NECKS AND FRACTURES

Electromagnetic expansion of the rings of all three aspect ratios resulted in an increasing formation of necks and fragments, with increase in velocity. The number of fractures was counted for the fragmented rings. The number of necks was counted in each case. The absolute number of necks for a particular sample could vary depending on the criterion used to designate it as a neck. The highly localized necks were easy to distinguish however the diffuse necks were more difficult to ascertain. Thus a criterion was adopted.

A neck was considered to be a region in which there was a steady increase in wall thickness of the ring, in moving away from it, on either side. Thus a region was not considered to be a neck only if it had a slightly lower wall thickness compared to an adjoining area. This could also be an inhomogeneity. The region was only considered to be a neck if the wall thickness of the adjoining areas steadily increased in moving away from it. Thus it was tapering out away from the neck. Readings were taken for the ring wall thickness carefully with digital calipers throughout the ring circumference.

Figure 7.25 shows the results of the measurements of the number of necks measured by the criterion stated above, as a function of peak velocity attained for all the three different aspect ratio rings. The same symbols used to designate the 4x4, 2x2 and the 1x1 rings as used in earlier plots have been used. Open symbols have been used to indicate the number of necks while corresponding solid symbols indicate the number of fractures.

228

For all the different aspect ratio rings, the number of necks and fractures increase with increase in velocity. The number of fractures initially increases more with increase in velocity. However the flattening of the fracture curves in the figure indicate a less increase in number of fractures with further increase in velocity. The number of necks steadily increases with velocity in all three aspect ratio rings. However at any given velocity, highest number of necks is observed for the 1x1 rings and the lowest number of necks is observed for the 4x4 rings. This indicates an increase in the number of necks with reduction of aspect ratio for a particular peak velocity. The same trend is also seen for the number of fractures which for a particular velocity are highest for the 1x1 which are the smallest aspect ratio rings.

35 s e

r 30 4x4- Necks u t Necks ac 25 4x4- Fractures r f 20 2x2- Necks

s and 15 2x2- Fractures ck

ne 1x1- Necks 10 of . 1x1- Fractures o 5 Fractures N

0 25 75 125 175 225 275 325 Peak velocity (m/s)

Figure 7.25: Number of necks and fractures for rings of all three aspect ratios, as a function of the peak velocity attained by them during electromagnetic expansion.

229

The increasing number of necks with velocity corroborate the previous experimental observations by Altynova et al [2, 3] and Grady and Benson [1]. This was shown in

Figure 7.3 in Section 7.1.2. Multiple necking at high velocities is attributed to inertia which reduces strain localization, and the rate of growth of an existing neck. As is clear from Figures 7.24 and 7.25, both the observed strains and the number of necks and fragments increase with increasing velocity. Thus inertial forces act to diffuse deformation from particular localized necks there allowing the region outside of a growing neck to strain, thereby resulting in increase of total elongation before fracture.

The numerical studies discussed in Section 7.1.2 had predicted the influence of aspect ratio on the number of observed necks. The present study supports the outcome that as the aspect ratio of a sample increases, the number of necks at a particular velocity level reduce. It can also be deduced that inertial forces are responsible for suppressing the long and short wavelength modes as an intermediate spacing between the necks was observed.

A uniform distribution was observed throughout the rings with some necks ultimately leading to fragmentation.

7.7 RESULTS AND DISCUSSION- SAMPLE PICTURES

As noted earlier, rings of the three aspect ratios were electromagnetically expanded at increasing energy levels to get samples with one failure followed by samples with increasing number of fragments with increasing energy levels.

230

rmability.

) a ( 6.3cm ce of aspect ratio on fo uen

) b ( to indicate the infl ng. (a) 1x1 rings launched at 0.56 and 0.96 kJ, (b) 2x2 ched at 3.36 and 6.8 kJ. 6.3cm ng energy levels

) c ( 6: Pictures of rings at increasi 6.3cm Figure 7.2 The innermost ring in each case is the undeformed ri launched at 0.96 and 2.4 kJ, (c) 4x4 rings laun

231

In order to visually show the difference in the total elongations for the different aspect ratio rings, Figure 7.26 shows pictures of the rings. The innermost ring in each picture is the original undeformed ring followed by rings launched at increasing energy levels.

Typically the middle ring is the ring launched at an energy level such that only one failure occurred in the specimen. Thus a ring launched at energy lower than this level resulted in a ring with no fractures. A further increase in energy level resulted in multiple fragmentation.

However, for the 1x1 ring in Figure 7.26a, it was not possible to obtain a ring with only one fracture; hence it was substituted with a ring with two fractures. The outermost ring in each picture is the ring launched at the highest energy level in the experimental range.

This sample had the highest number of fragments amongst all the samples tested for that ring type. All the pictures have been shown to scale and have the same magnification. As shown in Figure 7.26, there is a noticeable difference in the total elongations obtained from rings of the different aspect ratios. The innermost ring in each picture has roughly the same inner diameter. However there is a big difference in the diameters of the subsequent rings. For the set of 1x1 rings shown in Figure 7.26a, there is hardly any difference in the diameters of the center, middle and the outermost rings which was also seen in the total elongation data presented earlier. However for the 2x2 and 4x4 rings, in

Figures 7.26b and c respectively, there is considerable difference in the diameters of the rings at increasing energies with the 4x4 rings having the highest diameter of the expanded ring. The outermost rings in Figures 7.26a,b and c for the 1x1, 2x2 and 4x4

232

rings respectively have 6, 12 and 8 fractures, and 29, 28 and 22 necks. Although these rings don’t have the same peak velocities, the higher number of necks in the smallest aspect ratio rings relates to the data shown in Section 7.6.

The nearly uniform spacing between necks can be seen from the outermost rings in

Figure 7.26b and c which show the 2x2 and 4x4 rings respectively. This distance appears to be an intermediate distance. The fragments though are all not of equal length due to relief waves stopping the localization of adjacent necks in some cases.

7.8 RESULTS AND DISCUSSION – CURRENT TRACES

As outlined in Section 7.2.6, Pearson and Rogowski probes were used to measure the primary current in the actuator and the induced current in the ring. Figures 7.27-7.29 show a typical experimentally measured current vs. time plots. The rise time i.e. the time needed for the primary current to reach its peak, for this actuator was 22µs. All the current traces have been presented for rings with close measured peak velocities of around 140 m/s, in order to show a comparison of the induced currents. Figure 7.27 shows a current time trace for a 1x1 ring electromagnetically launched at 0.8 kJ. In this plot the peak primary current was 32 kA and while the peak induced current was 18 kA which was roughly half the peak primary current. Figure 7.28 shows a similar plot for a

2x2 ring launched at 1.04 kJ.

233

150 Primary 100 Induced ) A 50 t (k n e r

r 0 u 0 50 100 150 200 250 300 350 400 C -50

-100 Time (µs)

Figure 7.27: Current vs. Time trace for a 1x1 ring electromagnetically launched at

0.8 kJ. Peak primary current = 32 kA, Peak induced current = 18kA.

150 Primary 100 Induced ) A 50 (k t n

rre 0 u

C 0 50 100 150 200 250 300 350 400 -50

-100

Time (µs)

Figure 7.28: Current vs. Time trace for a 2x2 ring electromagnetically launched at

1.04 kJ. Peak primary current =39.12 kA, Peak Induced current = 56 kA.

234

150 Primary Induced 100 ) 50 t (kA en r r 0

Cu 0 50 100 150 200 250 300 350 400 -50

-100 Time (µs)

Figure 7.29: Current vs. Time trace for a 4x4 ring electromagnetically launched at

2.16 kJ. Peak primary current = 55.2 kA, Peak Induced current = 106 kA.

Here, the primary peak current is 39.12 kA and the induced peak current is 56 kA which is roughly 1.5 times the peak primary current. Figure 7.29 shows trace for a 4x4 ring launched at 2.16 kJ. The peak primary current here is 55.2 kA and the peak induced current in the 4x4 ring is 106 kA which is roughly twice the peak primary current. As expected, the 4x4 rings have a better coupling with the actuator than the smaller rings.

7.9 RESULTS AND DISCUSSION – COMPARISON OF

EXPERIMENTAL AND NUMERICAL RESULTS

Since one of the motivations of this study was to provide experimental data to support or refute the results of numerous numerical studies in the literature, a comparison will be

235

made for a few results. Specifically, it is possible to compare the results with the study by

Shenoy and Freund [12], which was briefly discussed in Section 7.1.3. In this study, the ring experiment was modeled by performing linear stability analysis for plane strain extension of a block of material. They provided an equation for calculating the number of necks, which was dependant on the velocity of extension and aspect ratio of the block of material. Due to this simple equation, it is possible to compare the trends from their study with the present experimental study. In most other studies, a direct comparison of our data with their results was not possible due to lack of an empirical outcome. A few details about the study will be given next.

Shenoy and Freund [12], through linear stability of a block in plane strain extension, developed an empirical relation to calculate the number of necks within the length 2l1 of the block, given by qmax, the dominant necking mode in the block

……… (7.1)

Where K is a proportionality constant (=1 here), Vo is the block’s extension velocity, Vp is

k a constant, known as the material characteristic velocity given by V = , where k is p ρ the strength coefficient and ρ is the density of the material. α is the aspect ratio of the block given by l2/l1. Nc is the critical necking rate whereas N is the rate as which a perturbation in velocity grows relative to the background uniform strain rate. A given mode q, satisfies a failure criterion when N = Nc which has been chosen to be 50,100, 236

500 or 1000. Thus according to this study, the number of necks is universally related to the extension velocity and aspect ratio via a power law. However this expression was shown to be valid only for low values of Nc and α.

Thus dynamic formability is influenced by both the aspect ratio of the block and the deformation velocity. These relations are shown in Figures 7.30. As shown in Figure

7.30a, for a particular extension velocity, a larger aspect ratio block (i.e. larger thickness or height for a fixed length), produces a smaller number of necks. In all cases, the number of necks continuously increases with the extension velocity. Thus aspect ratio seems to have a sizeable influence on the number of necks. A larger aspect ratio block, offers higher inertial resistance and thus a higher resistance to localization or necking.

With increasing velocities, an increase in ductility is shown by an increase in the dynamic stress or strain a material can support. Figure 7.30b shows the variation of critical stress, sc, with the non dimensional velocity. sc is the dynamic stress a block of material could support for which the failure criterion is met and the necking rate is equal to the critical necking rate Nc. Thus it is a measure of the stress and thus the strain before failure, and thus a parameter that can represent ductility. As shown in the figure, the critical stress increases monotonically with velocity which corroborates the stabilizing influence of the homogenous background velocity in improving ductility.

237

The figure also shows the dependence of sc on the aspect ratio of the block. A larger aspect ratio block has a higher critical stress parameter and hence ductility for a particular velocity. The increase in sc with velocity is attributed to ‘inertial stiffness’ caused by background velocity. Once again, a large aspect ratio block offers higher inertial pressure and provides higher ‘inertial stiffness’, which is in the form of a hydrostatic pressure gradient which increases with the square of the aspect ratio leading to higher ductility.

In order to compare the results of our study with that of Shenoy and Freund [12], we calculated the material characteristic velocity for the AA5754 material to be 275 m/s. The aspect ratio α of the 4x4, 2x2 and 1x1 rings was determined to be 0.02, 0.01 and 0.005 respectively as shown in Table 7.1. The value of the critical necking parameter Nc, which is an adjustable parameter in the study, was set to 50 as better agreement was obtained with it. The Proportionality constant K was set to 1, just like in the numerical study.

Using equation 7.1, the number of necks was calculated for the non-dimensional velocity parameter Vmax / Vp , where Vmax is the peak velocity generated by experimental measurements. It should be noted that in the original numerical study, Vo was taken to be a constant extension velocity of the block for simplification. However, we have substituted that with the peak velocity in the velocity profile. Then the experimental number of necks and the calculated number of necks using equation 7.1 was plotted together in the same format as in Figure 7.30a.

238

(a)

(b)

Figure 7.30: (a) Number of necks qmax , and (b) critical stress or extension sc, as a function of extension velocity Vo / VP for different aspect ratios α [12].

239

Figure 7.31 shows the experimental and numerical results of the number of necks as a function of non-dimensional velocity. The numerical trends were calculated for velocity ranges beyond the experimental one. As shown in the figure, excellent agreement is obtained between the experimentally measured and the empirically calculated number of necks. This result points towards the validity of the equation and the entire study. The number of necks being related inversely to the square root of the aspect ratio seems to be a valid outcome of the study as shown in Figure 7.31. The highest aspect ratio rings (4x4) have the least number of necks at a particular velocity level and the plots of all three aspect ratios agree excellently with the experimental results.

60 Experimental 4x4 Numerical 4x4 50 Experimental 2x2 Numerical 2x2

s 40 Experimental 1x1 1x1, α=0.005 Numerical 1x1 neck 30 2x2, α = 0.01 . of

o 4x4, α = 0.02

N 20

0 0 0.2 0.4 0.6 0.8 1

Vmax / Vp

Figure 7.31: Comparison of experimentally measured and numerically [12] calculated number of necks as a function of non-dimensional velocity.

240

Another comparison that can be made is the experimentally measured fracture strain or the true total elongation and the critical stress parameter sc in the study. This parameter is a measure of the fracture stress and hence the strain and ductility of the material. Due to lack of an empirical equation from the study for this parameter, our results cannot be expressed in the same format. Nevertheless, trends of the experimental results for the dependence of fracture strain on aspect ratio and velocity can be compared. Figure 7.32 shows the fracture strain or the experimentally measured true total elongation of the three aspect ratio rings as a function of the non-dimensional velocity.

35 4x4 30 2x2 (%) n i 25 1x1 4x4, α = 0.02 ra t

s 20 e 2x2, α = 0.01

tru 15 re

tu 10 c 1x1, α = 0.005 a r 5 F 0 0 0.2 0.4 0.6 0.8 1

Vmax/Vp

Figure 7.32: Fracture strain vs. non-dimensional velocity for all three aspect ratio rings for comparison with numerical results.

241

This figure when compared to the numerical prediction shown in Figure 7.30b indicates a good agreement for the trends for the different aspect ratio rings. The trends about the increasing fracture strain (critical stress parameter [12]), with increase in velocity and also the influence of aspect ratio is in full agreement. The highest aspect ratio rings (4x4) have the highest dynamic critical stress or fracture strain before failure.

Excellent correspondence between the experimental results of this study and the numerical predictions of Shenoy and Freund [12] supports the validity of their study.

Prior to this study, no experimental results were present which could validate the influence of aspect ratio of formability. However, now the dominating influence of specimen aspect ratio of the necking pattern and ductility is clear. This also means that the assumptions made in the numerical study are validated by the positive correlation with the experimental results. This study had considered a rate independent material and also showed the insensitivity of the necking pattern to the strain hardening parameter. It showed that the rates of both very long and very short wavelength modes are suppressed by inertia, thus promoting a necking pattern at an intermediate wavelength. Their theoretical calculations showed that changing the strain hardening exponent from 0.1 to

0.4, made no difference on the necking pattern, the dynamic critical stress or the number of fragments. The basic outcome of the study was that inertial effects alone establish the necking pattern. Thus the results of this experimental study also validate the assumptions of the numerical study about the dominating influence of inertial effects.

242

7.10 CONCLUSIONS

Electromagnetic ring expansion experiments were conducted with rings of three different aspect ratios made from AA5754-O. Using high velocity images, position and velocity profiles as a function of time were measured for the rings. Peak velocities of 45-290 m/s were recorded for the rings launched at incremental energy levels subsequently leading to fragmentation. For all ring types, the number of necks and fragments increased with increase in velocity. The uniform and total elongations were also seen to increase with increase in velocity for all the rings. Primary and induced currents in the rings were also measured for each sample launch.

The experiments with rings of three aspect ratios established the important influence of sample aspect ratio on its high velocity formability. This influence of aspect ratio was seen on the sample’s total and uniform elongation and the necking pattern. A higher aspect ratio ring displayed a higher total and uniform elongation as compared to a low aspect ratio ring at a particular peak velocity. The high aspect ratio ring (4x4) was seen to benefit more from an increase in velocity as compared to a lower aspect ratio ring (2x2).

The lowest aspect ratio ring (1x1) actually showed a worsening effect of velocity on its formability with its total and uniform elongations lower than the corresponding quasi- static values. The higher aspect ratio rings also showed a larger increase in total elongation than uniform elongation over the velocity range tested, indicating a higher benefit to post-uniform elongation due to inertial effects.

243

A study of the necking pattern in the different rings also indicated an influence of aspect ratio. For the same velocity, a higher aspect ratio ring showed lower number of necks and as compared to a low aspect ratio ring. The 4x4 rings recorded the smallest number of necks while the 1x1 rings showed the highest number of necks. Necks with an intermediate spacing formed in the rings, with a few of them localizing and resulting in fragmentation. A range of lengths for the fragments was observed in all cases.

The experimental results were compared with the theoretical results from Shenoy and

Freund [12]. Very good correspondence was observed between the experimentally observed and calculated number of necks (using an empirical relation) from the study.

This corroborated the empirical relation which shows the inverse dependence of number of necks on the square root of aspect ratio and direct dependence on the velocity of launch. The experimental trends of increasing fracture strain with velocity and higher fracture strain for a higher aspect ratio block were also in good correspondence with the theoretical results. This numerical study was conducted for a rate independent block of material and also showed that the constitutive parameters like strain hardening coefficient were not primary influencing parameters on formability. Direct correspondence between the experimental results and this study in turn reveals the significance of inertial parameters which play a dominating influence in determining the necking pattern and also increasing the observed total elongations. Experimental results from this study suggest a more dominating role played by the sample aspect ratio and inertial effects than constitutive behavior, on high velocity formability.

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BIBLIOGRAPHY

[1] Grady D.E and Benson D.A, Fragmentation of metal rings by electromagnetic loading, Experimental mechanics, Vol. 23 (4) (1983), 393-400.

[2] Altynova MM, The improved ductility of aluminum and copper rings by electromagnetic forming technique, Masters Thesis, The Ohio State University (1995).

[3] Altynova M, Hu XY, Daehn GS, Increased ductility in electromagnetic ring expansion, Metall Mater Trans A 27 (7) Jul 1996, pp. 1837-1844.

[4] Wood W.W, Experimental Mechanics, Vol. (19) (1967), p. 441.

[5] Hu Xiaoyu and Daehn Glenn S., Effect of velocity on flow localization in tension, Acta mater.Vol (44), No.3 (1996), pp 1021-1033.

[6] Hu Xiaoyu, Wagoner Robert H., Daehn Glenn S. and Ghosh Somnath, Metal. Trans. A, Vol. (25A) (1994), pp. 2723-2735.

[7] Mott N.F, Proc. Roy. Soc., London, Vol. 300 (1947).

[8] Louro L.H.L and Meyers M.A., Stress wave induced damage in Alumina, Proc. DYMAT 88. J.Phys. Vol. (49) (1988) C3-333.

[9] Grady D.E. and Kipp M.E., Int. J. Rock Mech. Min. Sci., Vol. (17) (1980) p.147.

[10] Han Jiang-Bo and Tvergaard V., Effect of inertia on the necking behavior of ring specimens under rapid radial expansion, Eur. J. Mech. A/Solids, Vol. (14), No.2 (1995), pp. 287-307.

[11] Sorenson N.J. and Freund L.B., Unstable neck formation in a ductile ring subjected to impulsive radial loading, Int. J. of solids and structures, Vol. (37) (2000), pp. 2265-2283.

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[12] Shenoy V.B, Freund L.B, Necking bifurcations during high strain rate extension, Journal of the mechanics and physics of solids, Vol. (47) (1999), pp. 2209-2233.

[13] Gurduru P.R., Freund L.B., The dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials, Int. J. of solids and structures, Vol. (39) (2002), pp 5615-5632.

[14] Pandolfi A., Krysl P. and Ortiz M., Finite element simulation of ring expansion and fragmentation: The capturing of length and time scales through cohesive models of fracture, Int. J. of fracture, Vol. 95 (1999), pp. 279-297. [15] Needleman (1991)

[16] Mercier S. and Molinari A., Analysis of multiple necking in rings under rapid radial expansion, Int. J. of Impact Engg., Vol. (30) (2004), pp. 403-419.

[17] Fressengeas C., Molinari A., Fragmentation of rapidly stretching sheets, Eur. J. Mech. A/Solids, Vol. (13) (2) (1994), pp. 251-288.

[18] Hutchingston and Neale (1978)

[19] Kohno Yukata , Akira Kohyama , Hamilton Margaret L., Hirose Takanori, Yutai Katoh Yutai and Garner Frank A., Specimen size effects on the tensile properties of JPCA and JFMS, Journal of Nuclear Materials 283-287 (2000) pp. 1014-1017

[20] Tamhane Amit, Altynova Marina M. and Daehn Glenn S., Effect of sample size on ductility in electromagnetic expansion, Scripta Materialia, Vol. (34), No. 8 (1996), pp 1345-1350.

[21] Fenton Gregg K. and Glenn Daehn S., Modeling of Electromagnetically Formed Sheet Metal, Journal of Materials Processing Technology, (75) (1998), pp.6-16.

[22] Nilsson Kristina, Effects of inertia on dynamic neck formations in tensile bars, Eur. J. Mech. A/Solids, Vol. 20 (2001), pp. 713-729.

[23] www.suhm.net

[24] www.pearsonelectronics.com

[25] http://homepage.ntlworld.com/rocoil

[26] Cookes Flashcam manual.

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CHAPTER 8

RING DIE IMPACT EXPERIMENTS

8.1 BACKGROUND AND MOTIVATION

Ring free expansion experiments with rings of different aspect ratios revealed the importance of aspect ratio and inertial effects on formability. It was revealed that ring true and total elongation and number of necks and fragments increase with velocity and are affected by the ring aspect ratio. It was desired to learn how a high velocity impact would benefit the formability of a ring specimen. The influence of a high velocity impact on the formability of a tube specimen was discussed in Chapter 6. It was shown that a high velocity impact is beneficial in improving the formability of the tube specimen by reducing or eliminating the number of tears in the specimen, once a proper velocity field is established. If an axisymmetric sample launch with a uniform electromagnetic field was created, a high velocity impact at very high energy levels could eliminate tearing altogether. However the setup of the tube die-impact experiments was such that the influence of impact on formability could not be quantified. It was hard to measure the exact influence of impact in terms of the number and length of the tears because of the

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relatively long geometry of the tube specimen. In addition to that since the electromagnetic field of the actuator used in those experiments was not very uniform due to the high pitch of the coil, it was difficult to obtain a straight profile of the impacting specimen. This could also influence the tearing in the sample.

In order to overcome the shortcomings of the tube die-impact experiments (Chapter 6) and also obtain the beneficial influence of impact revealed through those experiments, an experimental setup was designed for impacting the ring specimens. As in the tube die impact experiments, the goal here was not to measure an improvement in formability in terms of higher strains from the expanded rings. The goal was to see if a high velocity impact can reduce the number of fractures in the rings mainly by arresting an expanding ring with a die wall. Since the number of fractures in electromagnetically free expanded ring specimens was quantified earlier, a direct measure of the influence of impact could be obtained from the number of fractures in the die impacted rings. The studies in the literature for understanding the influence of die impact on formability were described in

Section 6.1 and will not be repeated.

8.2 EXPERIMENTAL PROCEDURE

Most of the components of the experimental setup used in these experiments were the same as used for ring free expansion experiments described in Chapter 7. It consisted of a solenoidal actuator connected to a capacitor bank. The ring workpiece was placed around the actuator with a cylindrical die placed on it with the help of spacers. When current was 248

passed through the capacitor bank, a primary current ran through the actuator which induced a current in the opposite direction, in the ring specimen. The electromagnetic repulsion between the two fields caused the workpiece to expand away from the actuator and strike the die at high velocities. Most of the components of the setup have already been described in Section 7.2 and will be only briefly mentioned here. A schematic of the experimental setup along with its components is shown in Figure 8.1. Figure 8.2 shows an actual picture of the setup.

8.2.1 Capacitor bank

Figure 8.1: Schematic of ring die-impact experimental setup 249

Capacitor bank Die

Ring

Rogowski probe (R1)

Coil

Rogowski probe (R2)

Figure 8.2: Picture of ring die-impact experimental setup

8.2.2 Actuator

A closely wound five-turn solenoid made from ASTM B16 brass wire with a 4.7mm x

4.7 mm square cross-section, was used as the actuator. The outer diameter of the bare coil was 5.93 cm and it had a pitch (center to center distance between consecutive turns), of

6mm. The wire in the actuator was covered with heat shrink wrap tubing and then potted in urethane to give structural support to the coil. The chemical and mechanical properties of the wire and the pictures of the actuator were given in Section 7.2.2.

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8.2.3 Workpiece

The workpieces in these experiments were rings made from a AA5754-O alloy tube with an OD of 6.985 cm (2.75”) and a wall thickness of 4 mm. Square cross-section rings with a cross-section of 4x4 mm and 2x2mm were machined from this tube such their inner diameter was approximately the same as the outer diameter of the potted coil. The details about the ring dimensions and mechanical properties were given in Tables 7.1 and

7.2 respectively. This material had an average total elongation of 24.8%

8.2.4 Rogowski probes

Two Rogowski probes were used to measure the primary and induced currents. As shown in Figures 8.1 and 8.2, one Rogowski Probe (R1) measured the primary current while the other probe (R2) measured the induced current in the tube in addition to the product of the number of coil turns and the primary current. Further details can be seen in Section

7.2. An error could be present in the reading of the induced current because of the second probe could also pick up the induced current in the die. However, due to the big gap between the die and the workpiece and the low conductivity of the die, this current is expected to be very small and thus not a major source of error.

8.2.5 Die arrangement

The die used was a cylinder with an inner diameter of 10.4 cm and wall thickness of 4.8 mm. It was machined from a stainless steel tube to a very good surface finish. It was 251

sturdy and massive to be able to withstand the high velocity impact. The inner diameter of the tube was roughly 50% bigger than the outer diameter of the ring workpiece. The die was placed on the actuator with the help of two spacers made from a non-conductive

G-10 sheet. The spacers were circular discs of thickness 6.35mm with a concentric circular disc of smaller diameter, machined out (removed) from them. The outer diameter of the spacers was equal to the inner diameter of the die and their inner diameter was equal to the outer diameter of the potted coil. The spacers were attached to the die and the workpiece with the help of long screws which went through the die and spacers to connect with the potted coil. Three screws offset by 120o angles were staggered around the circumference of each of the two spacers. The arrangement of the front spacers, screws and die is shown in Figure 8.3. The tightening of the screws enabled correct alignment and tight fitting of the die onto the coil. This was an improvement over the tube die-impact setup which was prone to spacers flying off, after high velocity impact.

Die Spacer

Coil

Figure 8.3: Front view of cylindrical die placed on the coil with the help of spacers and screws. 252

8.3 METHODOLOGY

The ring samples were electromagnetically launched at increasing energy levels to impact with the die. The goal here was to arrest the expanding ring in motion, by die impact, before it can fragment into pieces, (which was the case with ring free expansion). It was essential, to let the sample freely expand away from the actuator for some time so that it accelerates to sufficiently high velocity levels. It was also imperative to stop the sample while it was still in motion, with some kinetic energy still left in it so that it hit the die with sufficient velocity. The velocity vs. radius profile schematic for a sample in motion, with and without die impact is shown in Figure 8.4. As shown in the figure, in the case involving die impact, a higher velocity profile can be used to stop the sample before tears can initiate in it. This enables the use of higher energy levels and corresponding peak velocities which in turn ensure more benefits from inertial forces. From ring free expansion experiments, the outer diameter of the fragmented rings at particular energy levels was known to us. The objective here was to use a die which had an internal diameter capable of stopping the expanding ring before it reached the diameter at which it had fragmented. This is different from traditional die impact experiments in which advantage of inertial ironing is taken to improve the achievable strains.

8.4 RESULTS AND DISCUSSION

As mentioned above rings of 2x2 and 4x4 cross-section were impacted against the die at increasing energy levels. Individual results will be discussed next.

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Die wall ty i With impact c o l

Ve Without impact

X X RingX Fragmentation Radius

Figure 8.4: Velocity vs. Radius profile schematic for an expanding ring specimen, with and without impact.

8.4.1 Die impact of 2x2 rings

AA5754-O rings of 2x2 cross-section were electromagnetically impacted against the die at increasing energy levels of 3.6, 4.8 and 6kJ. The die impact of the sample launched at

3.6 kJ resulted in complete arresting of localized necks such that a sample without any failure was obtained. The ring was significantly flattened against the die such that it thinned and became taller as it stretched against the die. This sample had an average thickness and height of 1.2 and 3.1cm respectively. Thus it had an average engineering strain of -39.5% and 56.2% respectively along the thickness and height.

This sample can be compared with free formed rings. When freely expanded without die impact, these rings resulted in one failure when launched at 0.96 kJ, and eleven fragments at an energy level of 1.76 kJ. The latter sample had an outer diameter of 9.6cm which is

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very close to the diameter of the die impacted sample (10.4 cm). Figure 8.5 shows pictures of these three rings along with an original undeformed ring in the center. Moving outward are the free formed rings launched at energies of 0.96 and 1.76 kJ with one and eleven fragments respectively. The outermost ring is the die impacted ring at 3.6 kJ with no failure. Thus this figure compares a free-formed (1.76 kJ) and die impacted ring at almost the same strain levels. The high velocity impact arrested all localized necks in the sample also resulting in significant inertial ironing due to high compressive stresses generated in the thickness of the ring.

Figure 8.5: Picture of 2x2mm rings. Innermost ring is undeformed ring, followed by free formed rings launched at 0.96 kJ (one failure, circumferential strain = 22.9%) and 1.76 kJ (eleven failures, circumferential strain = 34.8%). The outermost ring is the die impacted ring (no failures, circumferential strain = 40.3%).

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When the rings were launched at higher energy levels than that needed for arresting all the neck, failures took place in the ring due to excessive thinning. The ring launched at

4.8 kJ resulted in one failure while its average thickness was 1.15 mm and its height varied from 3.08 to 3.80mm along the circumference. That makes an average engineering strain of -42% along its thickness and 68% extension along its height. Thus there was too much thinning which was responsible for the failure in the ring. The sample launched at even higher energy level of 6kJ had two fragments with an average compressive strain in the thickness direction, of -53% and an extension along its height of 115%.

8.4.2 Die impact of 4x4 rings

Rings of cross-section 4x4mm were launched at increasing energy levels of 3.68, 3.76,

5.6, 6 and 7.6 kJ against the cylindrical die. The launch of a ring specimen at 3.76 kJ resulted in its fragmentation into four pieces. The outer diameter of the deformed ring was 10.1 cm which is close to the inner diameter of the die (10.4cm). It struck the die at an approximate velocity of 135 m/s as determined from a free expansion experiment with the help of position vs. time and velocity vs. time plot.

Further the ring specimen was electromagnetically launched at a higher energy level of

5.6 kJ. This sample hit the die at very high velocity of almost 225 m/s as determined from the position vs. time and velocity vs. time plots from free ring expansion experiments (Chapter 7). This impact velocity was sufficient to completely arrest all the necks resulting in a ring without any fractures, with outer diameter same as the inner 256

diameter of the die cavity. The velocity of strike was so high that it caused flattening out of the ring and it became taller and thinner. This sample can be compared with a freely expanded ring launched at 3.36kJ. It had almost the same outer diameter as this die impacted sample (10.3cm), had one failure and is shown as the middle ring in Figure 8.6.

The outermost ring here is the die impacted ring launched at 5.6kJ. It should be noted that when free ring expansion was carried out at 5.6 kJ, the resulting sample had nine fractures and an outer diameter of 11.2cm.

Figure 8.6: Picture of 4x4mm rings. Innermost ring is undeformed ring. The middle ring is a freely expanded ring launched at 3.36 kJ (one failure, circumferential strain = 39.2%). Outermost ring is die impacted ring at 5.6kJ (no failure, circumferential strain = 39.5%).

Launch of successive samples at higher energy levels, resulted in similar samples with no failures, but higher thinning and stretching along their height. The rings were laterally 257

stretched out against the die cavity and also took benefit from inertial ironing at these high velocities. The 4x4mm ring launched at 6kJ had average thickness and height of 2.9 and 4.2mm respectively. The ring launched at higher energy of 7.6 kJ had these readings as 2.5 and 5.5mm indicating an even higher level of flattening of the ring against the die.

Thus from die impact experiments with 4x4 rings, it has been shown that impact can reduce the number of fractures significantly if sufficient impact velocity is established.

However for further comparison with free-formed samples, a larger die is needed so that free-formed fragmented rings and die impacted rings can be compared at the same strain levels, in accordance with Figure 8.4. With the present results, comparison can only be made between samples launched at similar launch energies. In the case of samples launched freely and with die impact at 5.6 kJ, it has been shown that the number of fractures can be reduced from nine to zero. Also launch of these 4x4 rings, at much higher levels will probably cause the same extent of thinning observed in 2x2 rings which resulted in increasing number of fragments with increasing launch velocities, beyond the required impact velocity to arrest all necks.

Details about different studies in the literature were provided in Section 7.1. Imbert et. al. (2004) had demonstrated the reduction in damage parameters due to compressive hydrostatic stresses developing after die impact as the sheet straightens as it confirms to the shape of the die. The reduction of intense neck localizations and number of fractures, with die strike corroborates the outcome of their numerical study. As the sample is 258

arrested with the die wall, significant inertial stabilization of necks and inertial ironing phenomena also occur resulting in flattening out of the ring.

8.5 CONCLUSIONS

Ring expansion experiments were successfully conducted which involved impact of 2x2 and 4x4mm cross-section AA5754-O rings with a cylindrical die at increasing energy levels. The beneficial influence of high velocity impact was seen when the ring struck the die at high energy levels. The results shown here are similar to the ones obtained from tube die impact experiments. However in the present study, the shortcomings in the design of the experimental setup of tube die impact were improved upon and the results obtained were quantifiable. In both the experiments, it was revealed that if the sample does not completely fill out the die cavity or strikes it at insufficient velocities, then the benefits of impact are minimal and the benefit due to inertial forces cannot be realized to the full extent. If the die impact occurs at an appropriate velocity level, the number of fractures can be significantly reduced. Comparison with freely expanded rings at the same strain level shows a dramatic reduction in the number of fractures. From the 2x2 rings, it was also seen that when too high velocity levels are used, sample thinning dominates and results in failure. Thus there appears to be a particular range of velocities for a forming setup, in which arrest of the sample with die impact is beneficial.

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CHAPTER 9

MODES OF FAILURE IN HIGH VELOCITY FORMING

Numerous studies exist in the literature for understanding the microstructure of samples failed in the shock regime which is typically at strain rates > 104 s-1 and impact speeds >

750 m/s. However, little has been done to analyze modes of failure in high velocity forming. The present study aims at briefly studying the observed modes of failure in samples formed at high velocities. Based on visual macroscopic and microscopic examination the modes can be classified as - necking, shearing and ‘spall-like dynamic rupture with little macro plasticity’. The latter mode is a new observed type of failure and the characteristics of this mode were analyzed on the basis of optical microscopy and

SEM. Figure 9.1 shows the schematics of the three broad classifications.

9.1 NECKING

Necking is a classic mode of failure in which diffused necks form, followed by localization and fracture. In ductile metals, it takes place by void nucleation, growth and coalescence and the details of this mode were given in Section 3.2.5 260

(a) (b) (c)

Figure 9.1: Three broad classifications of failure modes in high velocity forming. (a) Necking, (b) Shearing and (c) Spall-like dynamic rupture with little macro plasticity

Figure 9.2: Few pictures of necked samples, formed at high velocities

At high velocities, necking has been shown to be influenced significantly by inertia and size effects as discussed in details throughout this document. This mode has been the focus of research in high velocity forming so far. Figure 9.2 shows pictures of a few samples which are believed to have failed with this classic mode. The different strain states including uniaxial, plane strain and biaxial in which samples fail are represented by the forming limit curve in an FLD. 261

9.2 SHEARING

Shearing is the cutting of a workpiece between two die components. In shearing a narrow strip of metal is severally plastically deformed beyond its ultimate strength, such that it fractures at the surfaces in contact with the blades, with the fracture then propagating inwards. In conventional shearing, the controlling factors are the clearance between punch and die (typically between 2 and 10% of the sheet thickness), punch velocity, metal thickness, composition and ductility. The quality of the sheared edge is measured with the height of the burr (protrusion). A few conventional press operations based on shearing are blanking, punching and trimming [1].

Shearing at high velocities has not been widely studied until recently. Ongoing research at The Ohio State University [2] involves shearing of sheet metal by launching it at high velocities by electromagnetic forces, against a sharp cornered punch or die. Figure 9.3 shows pictures of few such samples formed with EMF. The controlling factors here are the energy of launch and the standoff distance.

Figure 9.3: Pictures of few samples sheared at high velocities.

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9.3 SPALL-LIKE DYNAMIC RUPTURE

A few samples formed with high velocity impact were observed to have different fracture characteristics from the conventional modes listed previously. These samples were formed with die impact at high velocities; involved hardly any macro plasticity; a number of cracks were seen on the surface of the sheet, in the area adjoining the fracture surface; sometimes the fracture surface had an angle; void linkages along specific planes were observed from cross-sections, SEM micrographs showed a dimpled surface like ductile fracture etc. Thus this mode of failure had some features which distinguished it from the other dynamic modes of failure and is being categorized here as ‘spall-like dynamic rupture’ as it has some features which are seen in spalling. A background and studies in the literature about spalling will be presented here to establish this connection.

9.3.1 What is spall?

Spallation is the planar separation of material parallel to the wave front as a result of dynamic stress components perpendicular to this plane [3, 4]. It is a material failure produced by the action of tensile stresses developed in the interior of a body when two decompression waves collide [5]. A dynamic deformation system involving projectile impact against a target is shown in Figure 9.4 in which spalling is evidenced in the side of the target opposing the impact surface. The material is pulled out of the back and complete separation occurs when the stress pulse amplitude and duration are sufficient. It also shows that dynamic deformation is experienced by the material at the impact surface.

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Spalling is typically characterized by planar separation of material along a spall plane where maximum tensile stress is experienced. Spallation in ductile materials is controlled by localized plastic deformation around small voids which grow and coalesce to form a spall plane [5]. Spallation in brittle materials takes place by dynamic crack propagation without noticeable plastic deformation. In this chapter, details about only spall in ductile solids will be presented.

Spalling differs from quasi-static fracture in that it does not involve the propagation of a single crack or wave front. Rather there is independent nucleation and growth of the individual micro-failure regions (these can be either voids or cracks) the growth of which occurs without interference from the distant ones. Thus ductile metals fail, both quasi- statically and dynamically, by void coalescence. However, in spalling, strong tensile stress in a spall region, which reduces fracture strain and little plastic deformation, takes place. Thus, fracture strain in spalling is invariably low. In dynamic conditions, additional considerations are that the heat generated by plastic deformation cannot dissipate itself due to high rate of deformation; the inertial effect associated with the displacement of the material adjoining the void walls is important; also wave interactions have a bearing on the final configuration [3].

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a projectile, y ent areas of a target impacted b

Figure 9.4: A dynamic deformation system with differ experiencing dynamic fracture [3].

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9.3.2 Fundamentals about spalling

The occurrence of spalling can be understood by from Figure 9.5 which shows the main configurations under which spalling is seen and experimentally studied. Figure 9.5A shows a standard plate impact setup in which a flat impactor of thickness δ traveling at velocity v, striking a stationary target of thickness > δ, spalling is produced in the target

1 at a distance of δ from the free end. A compressive wave of amplitude ρUv and 2 duration ~2 δ/U is generated in the target where U is wave velocity (elastic wave speed) in the target and impactor. Interaction of this compressive wave with free surface

1 produces tensile stress − ρUv at a distance δ from the free surface. If the pulse is of 2 sufficient magnitude and duration, spallation occurs. In this case, both the peak tensile stress and spall position are known a priori [5]. Plate impact tests typically have strain rates of the order of 106 s-1 [6].

In the explosive loading setup shown in Figure 9.5B, spalling occurs at distances of δ1 and δ2 from the free surface, when tensile stresses of magnitude –p1 and –p2 develop as the triangular shaped stress wave is reflected at the free surface. In this case, the spall position and amplitude are not uniquely determined a priori and there can be multiple spall layers with different values of peak tensile stresses. The mechanics of spalling become clearer from Figure 9.6 which shows how reflections of stress waves can occur at free surfaces, fixed ends or at discontinuities within the solid. As a compressive shock wave approaches the free surface of a plate (Figure 9.6a), it is reflected as a tensile wave. 266

Figure 9.5: Dynamic fracture for A, plate impact; B, explosive loading and C, expanding ring [5].

After summing the incident and reflected waves, a net tensile wave results as shown in

Figure 9.3b. As shown in Figure 9.3c, the rapid buildup of tensile stress can result in spall if the amplitude of the tensile stress is high enough.

Figure 9.6: Reflection of a shock wave from a free surface and development of a spall fracture [1].

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Spalling is dependant on the amplitude and duration of the reflected tensile pulse. The extent of spall damage is dependant on shock-wave parameters (stress, duration of stress, stress gradients, strain rate etc.) and material parameters. There is a critical value of the nominal tensile stress required to produce spalling and this value is a characteristic of the material [3]. The determined spall strength for aluminum alloys varies little with temperature but it can decrease sharply when temperature approaches Melting point. The spall strength also depends on the projectile thickness and loading time used [14].

In experimental studies, a VISAR can be used to observe the velocity histories for the free surface of the target away from die impact, to determine whether or not spall is occurring [3, 6]. When no spall is observed, the free surface velocity returns to zero after the passage of the shock pulse. The formation of a spall on the other hand generates a release wave that produces a hump (spall cusp) behind the shock wave. Both these velocity profiles are shown in Figure 9.7. For multiple spall layers, multiple humps can be seen in the velocity profile [5]. Figure 9.7b shows calculated damage as a function of position at various times. It is evident that this function peaks at a particular value (which is the same as the thickness of the projectile) which is referred to as the ‘spall plane’.

Nucleation of voids can take place by fracture of inclusions, debonding or fracture at grain boundaries (heterogeneous nucleation); or homogenously at dislocation tangles and fine impurity or precipitate particles [8]. Both types can concurrently occur if the stress is high enough [5]. In some samples voids and spalling have been observed even though no obvious heterogeneity is seen in them by optical microscopy. 268

Figure 9.7: (a) Calculated and observed velocities of free surface of target indicating spall and no spall conditions, (b) Void volume distribution in the central region of a 6.5mm thick target [6].

9.3.3 Microstructural aspects of spall

There are essentially two modes of dynamic fracturing – brittle and ductile, as shown in

Figures 9.8 and 9.9. Brittle fracture is characterized by sharp tips, with little plastic deformation. SEM micrographs indicate the cleavage surfaces as seen in quasi-static conditions. Ductile fracture is characterized by voids and SEM micrographs indicate a classical dimpled appearance. BCC and HCP metals are prone to spall by brittle mode and FCC metals exhibit higher ductility at high strain rates and spall in a ductile manner.

Observation of fracture in nickel under Quasi-static and dynamic conditions show the same dimpled appearance. However in the former case the dimples as observed with

SEM, are deeper. Dimple depth is indicative of ductility and thus this signifies a certain

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loss of ductility at high strain rates. This could also be because dynamic mode is more uniform with a number of voids growing simultaneously and independently [3].

Figure 9.8: (a) Spalling by brittle fracture in low-carbon steel (b) Spalling by ductile void formation in nickel [3].

Figure 9.9: Micrographs of spall in AA6061-T6. (a)Void growth indicative of ductile spall, (b) Cracks indicative of brittle spall [8, 9].

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Experimental studies [10] with aluminum alloys AA2024 and AA6061 under different stages of age hardening indicate that aging conditions influence the extent of spalling due to differences in the second phase particles (nucleation sites). Steels have different spall extents based on the presence of large inclusions (refined and low alloy steels are better)

[13]. Figure 9.10 shows the optical and SEM micrographs of spalled 2024Al which shows the ductile fracture of the aluminum matrix along with the fractured particles.

Figure 9.10: (a) Cracked second phase particles and voids in spalled AA2024. (b) SEM micrograph showing ductile fracture of matrix [10].

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Figure 9.11 shows an SEM micrograph of the spall surface of a copper specimen containing second phase particles. It has the classic dimpled appearance with second- phase particles inside the dimples [3, 11].

Figure 9.11: SEM micrograph of spall in Cu containing second-phase particles [11].

In a specimen in which physical separation along is plane, is evident the spall is referred to as ‘complete spall’ as shown in Figure 9.12a. On the other hand, even though macroscopically a specimen does not appear to have spalled, microscopically, there can be evidence of spall with a number of microflaws formed at the region of maximum tensile stress. This is referred to as ‘incipient spall’ and is shown in Figure 9.12b [3].

There can be quite a large difference between tensile stresses required to achieve incipient spall to that required for complete spallation [14]. A completely spalled sample can have macrocracks while an incipient spall is characterized by microvoids or microcracks dispersed throughout [15]. If tensile stresses lower than the spall strength of

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the material are present, partial spallation occurs. Through experiments with 2024-T351 specimens with spall strength of 1.67 GPa it was found that maximum tensile stresses of

1.6 GPa result in incipient spall while stresses of 1.4 GPa don’t even result in incipient spall and no microscopic damage was seen nor a reverberation signal was observed [14].

(a)

Figure 9.12: (a) Complete spalling of AISI 1008 steel and nickel plate. (b) Incipient spall damage in copper [3].

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Spall fracture depends on numerous factors like loading time, local stress, initial temperature and microstructure. Several numerical studies [3-5, 14-17] predict spall fracture based on different approaches. These have been typically developed to reproduce damage levels, spall location, growth rates etc. as observed in individual spall experiments. Some of these quantitative studies are outlined in [4].

9.4 EXPERIMENTAL STUDIES FOR SPALL-LIKE FAILURE

A few samples formed with high velocity impact from previous studies, were examined metallographically to understand the observed new mode of failure and compare them to the micrographs in the literature. The motivation of this study came from a few samples that were formed at high velocities with impact with a die. They had a peculiar fracture surface with separation along an angle, with hardly any macro plasticity. A few such examples will be discussed next with pictures of samples, their forming conditions, optical and/or SEM micrographs. For all cases, samples were sectioned along the thickness of the sheets and these surfaces were prepared by grinding followed by polishing with 1 micron diamond paste and/or colloidal silica gel.

9.4.1 Sample1- AA2024 disc formed with EHF

The first sample studied was a 2024-T3 sheet electrohydraulically impacted against a truncated conical die at 9.4 kJ. Further details about the experiments can be found in [18].

Figure 9.13a shows the experimental setup with a conical die, and the truncated cone die

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geometry used for forming the sample. Figure 9.13b shows the picture of the failed sample after impact. Figure 9.13c shows the small disc that popped out of the central portion of the sample in Figure 9.13b. This disc had a slanting fracture surface which was peculiar. There was also the formation of cracks visible in the surface of the sample which as shown in the figure is the surface away from die impact.

Bridge wire (a)

(b) 275

Surface away from Die impact

(c)

Figure 9.13: (a) Electrohydraulic forming setup and truncated conical die geometry [18], (b) Picture of sample formed at 9.4 kJ, (c) Picture of disc (with slanting fracture surface), which popped out from the central region of the sample.

Figure 9.14 shows optical micrographs of the sections along the line shown in picture of the disc shown in Figure 9.13c. These micrographs at various magnifications show the slanting fracture surface (top left); a macrocrack originating on the surface and progressing inwards (top right); voids linking up along with second phase particles inside them (bottom). These voids were seen linking along various planes but were more prominent near the surface which was away from die impact. This is characteristic of results found in the literature about existence of various spall planes.

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Figure 9.14: Optical micrographs of sections of disc sample at various magnifications.

Figure 9.15 shows SEM micrographs of the fracture surface. The high volume of voids near the surface away from die impact which is the bottom surface in all these pictures is clearly evident from the (bottom) figures. A dimpled appearance of the fracture surface, 277

characteristic of ductile fracture, along with a second phase particle inside a void, can be seen in the top left image. The top right image individual cracks on the surface which can eventually linkup. Figure 9.16 shows SEM micrographs from another disc sample which had a similar appearance as the previous sample.

Figure 9.15: SEM micrographs of the fracture surface of the disc sample

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Figure 9.16: SEM micrographs from another disc sample show a crack.

9.4.2 Sample 2- High strength steel formed with EMF

A high strength, cold rolled steel with thickness of 0.15mm, tensile strength of 633MPa and 3.2% elongation, was impacted against a wedge shaped punch with EMF at 4.8 kJ energy [19], as shown in Figures 9.17 a and b. The impacted sample was sectioned along the line shown in Figure 9.17c and the thickness surface was microscopically examined.

Punch Work piece ` with driver sheet below

Coil

To capacitor bank

(a) (b) 279

(c) Figure 9.17: (a) Experimental setup, (b) Wedge shaped punch used for impact, (c) Picture of high strength steel sample formed with impact against the punch.

Figure 9.18 shows optical micrographs for the cross-section of Sample 2, at various magnifications. Considerable void linkups are seen throughout the thickness of the sample. However the one near the surface away from die impact shown in bottom figure is prominent. Figure 9.19 shows SEM micrographs of the fracture surface of the sample.

From the pictures it appears that the fracture appearance is dimpled like classic ductile fracture. The bottom left image is of a tear in the sample elsewhere, while the rest of them are of the fractured edge. These do not appear to be different from a quasi-static ductile fracture or a ductile spalled sample.

280

Figure 9.18: Optical micrographs at various magnifications for Sample 2.

9.4.3 Sample 3 – Copper sheet impacted with EMF

Copper sheets, 0.37 mm thick was impacted against a flat plate with the help of a 3-bar coil at increasing energy levels. Figure 9.17a shows the experimental setup schematic.

The sheet was deformed in accordance with the pressure of the coil. 281

Figure 9.19: SEM micrographs of the fracture surface of Sample 2.

A standoff of 5 mm was maintained between the sheet and the plate die so that the sample could accelerate to a high velocity before impacting the die. At the highest energy launch, the sheet stuck/was welded against the flat plat die. A sample launched at 10.4 kJ energy, which is shown in Figure 9.20b, was sectioned along the line shown. Figure 9.21 shows the optical micrographs of the cross-section of the sample. Once again distinct void linkup is seen along planes to a higher extent than even the previous examples. 282

(b) (a)

Figure 9.20: (a) Experimental setup schematic, (b) Picture of copper sheet impacted with a flat plate at 10.4 kJ.

9.4.4 Sample 4 – Electromagnetic welding of AA6061 rod with tube.

Recently experiments were carried out at EWI by Zhang et al.[20], which involved electromagnetic welding of a AA6061 annealed tube with Inner diameter = 4.9 mm and wall thickness = 0.09mm onto a AA6061 rod of 4.55 mm and thickness 2mm using a concentric round coil. Figure 9.21 shows a mounted cross-section of the tube welded to the rod at launch energy of 45 kJ. As seen in the picture, physical separation took place in the thick rod when the thin tube impacted it at high energy. Classical spall took place as the material was pulled out at the back surface, away from tube impact. Physical separation along a plane is evidence that classic spall can take place with electromagnetic forces at very high velocities. Figure 9.22 shows optical micrographs of the cross-section

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of the AA6061 rod, which shows distinct void /crack linkages along a spall plane. These micrographs confirm that a classic spall fracture took place in this sample. However it should be noted that the energy level used for launch in electromagnetic welding is noticeably higher than that typically used for electromagnetic forming.

Figure 9.21: Optical micrographs of cross-sections of copper sheet at various magnifications. 284

Figure 9.22: Picture of mounted sample showing spall in AA6061 rod welded with AA6061 tube [20].

Figure 9.23: Optical cross-sections of the spalled AA6061 rod.

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9.4.5 Comparison of cross-sections of undeformed, free-formed and die impacted AA5754-O ring samples

Cross-sections of rings of AA5754-O, with a 4mm x 4mm cross-section, which were used for ring expansion experiments, with and without die impact, as described in

Chapters 7 and 8, were metallographically examined. Figure 9.23 shows optical micrographs of an undeformed, free formed necked and die impacted samples. The latter sample was electromagnetically launched at 5.6 kJ against a cylindrical die such that the die arrested all localized necks and the sample was flattened against the die wall. As seen from Figure 9.23a, the undeformed sample has some inclusions elongated in the direction of extrusion of the tube from which the rings were made. The free formed necked sample in Figure 9.23b shows big void formations, which is expected from this ductile material.

Figure 9.23c shows the sample which was impacted against the die at high velocity. In this sample, it can be seen that the inclusions which were already present in the undeformed sample have broken up in some places giving rise to voids. There is some ambiguity here as the lined up inclusions were already present in the material.

(a) (b) 286

(c)

Figure 9.24: Optical micrographs of cross-sections of AA5754-O rings. (a) Undeformed, (b) Free formed at 3.6 kJ and necked, and (c) Die impacted at 5.6 kJ

However in the die impacted sample, the voids along these inclusions do give the appearance of spall planes. In any case, it is clear that after die impact, changes take place in the microstructure giving rise to additional damage.

9.5 DISCUSSION AND CONCLUSIONS

The work presented here is by no means a complete study to categorize the different types of failures. This is an ongoing study and has been motivated by an observation of a failure mode which appears to be different from the conventional modes of necking and shearing. This mode ‘spall-like dynamic rupture’ was studied by doing optical and SEM

287

micrography on a few samples whose mode of failure seemed uncharacteristic. Thus a few studies in the literature about spall damage were examined.

Spall occurs when two unloading stress waves (from opposing free surfaces) converge to generate a planar zone of intense triaxiality. Although it cannot be said with certainty that the damage observed in the samples in this study is ‘spall’, and the exact mechanics of the same have not been established, there are some similarities with it, as established from the literature. Distinct void linkages have been seen in the cross-sections and are characteristic of ‘incipient spalled’ samples in which visual separation of the material has not occurred and thus macroscopically might not be considered to be spalled.

Comparison of micrographs of undeformed, free formed and die impacted samples suggest that changes in the microstructure in the form of increased damage along inclusions, can take place after die impact. It is known that a sheet goes through stress reversal as it bends and unbends as it confirms to a die and wave propagation effects are characteristic of die impact. It is possible that the conjugation of release waves in the sheet results in regions of high tensile stress, and subsequent spall planes. The extent of damage can depend on the material, its thickness and die impact velocity. The example of classically spalled sample with electromagnetic welding showed that it is possible to cause spall fracture in the velocity regime encountered in EMF, although the velocities used in that case were higher than that typically used in electromagnetic forming.

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Overall the characteristics of this mode ‘spall-like rupture’ are – it is brittle like with

‘hardly any macro plasticity’ and very small crack opening; often a series of cracks are seen on the surface of the sheet, adjacent to the fracture surface; fracture surface is often at an angle, void linkages are observed from cross-sections. Further work needs to be done to develop further understanding.

It should be noted that all the studies in the literature about spall, typically look at the thick target plate while the impacting projectile is expendable and is not studied. To our knowledge, there are no works in the literature which study damage in the thin moving part (or impactor) which is the moving sheet in our case. Thus the present study can be considered to be a ‘beginning’.

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BIBLIOGRAPHY

[1] Dieter G.E., Mechanical Metallurgy, McGraw-Hill Book Company, (2001) [2] Unpublished research by Dr. G.S. Daehn, Scott Golowin, Yuan Zhang and Manish Kamal at Ohio State University (2005).

[3] Meyers M.A and Aimone C.T., Dynamic fracture (spalling) of metals, Progress in Materials Science, Vol. 28 (1) (1983).

[4] Meyers M.A., Dynamic behavior of materials, John Wiley and Sons (1994)

[5] Johnson J.N., Dynamic fracture and spallation in ductile solids, J. of Applied Physics, Vol. 52 (4) (1981), pp. 2812-2825.

[6] Davison L, Stevens A.L. and Kipp M.E., J. Mech. Phys., Vol. 25 (1977).

[7] Curran D.R., Seaman L. and Shockey D.A., Shock waves and high strain rate phenomena in metals, eds. Meyers and Murr, Plenum NY (1981) p.132

[8] Grady DE, The spall strength of condensed matter, Journal of mechanics and solids, Vol. 36 (3) (1988) p 353.

[9] Christman (1971).

[10] Jones W.B., Thesis U. of Washington (1973).

[11] Christy S., Pak H and Meyers M.A., Metallurgical applications of shock-wave and high strain-rate phenomena, New York eds. L>E Murr (1986) p. 835.

[12] Shockey D.A., Dao K.C. and Jones R.L., Mechanisms of deformation and Fractures, (ed. K.E. Easterling) Pergamon Press, Oxford (1977) p. 77.

[13] Morton M.E., Woodward P.L. and Yellup J.M., Fourth Tewksburg Symposium, Melbourne (1979), p. 11.1.

[14] Rosenberg Z., Luttwak G., Yeshurun Y. and Partom Y., Spall studies of differently treated 2024A1 specimens, J. of Applied Physics, Vol. 54 (5) (1983) pp. 2147-2152. 290

[15] Rinehart J.S. and Pearson J., Behavior of metals under impulsive loads, American Society for metals, Cleveland (1954).

[16] Chevrier P. and Klepaczko, Spall fracture: mechanical and microstructural aspects, Engineering Fracture Mechanics, Vol. 63 (1999) pp.273-294

[17] Kaniel G.I., Razorenov S.V., Bogatch A., Utkin A.V. and Grady D.E., Simulation of spall fracture of aluminum and magnesium over a wide range of load duration and temperature, Int. J. of Impact Engg., Vol. 20 (1997), pp. 467-478.

[18] Balanethiram V.S., Hyperplasticity: Enhanced Formability of Sheet Metals at High Velocity, Ph.D. thesis (1996).

[19] Seth Mala, High Velocity Formability of High Strength Steel Sheet, MS Thesis, The Ohio State University (2002).

[20] Unpublished research at EWI by Dr. Glenn Daehn, Yuan Zhang and Dr. Peihui Zhang.

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CHAPTER 10

CONCLUSIONS AND FUTURE WORK

The overall goal of this study was to establish the factors that are important in determining formability of materials at high forming velocities and to understand basic concepts about it which can be inculcated into practical guidelines for designing high velocity forming (HVF) techniques. The influence of parameters like inertia, impact, sample dimensions and changes in constitutive behavior at high velocities were studied to understand their effect on formability. Several experiments were designed with this intent as described in detail earlier. The key issues, previous understanding about them, knowledge gained from the present studies and the open questions will be presented here.

10.1 ISSUE: Role of constitutive behavior in enhancing formability at high velocities

• Previous understanding: Constitutive behavior of materials, mainly its rate

sensitivity changes at high strain rates. The observed flow stress and strain rate

sensitivity of most metals increase significantly at strain rates from 103 – 104 s-1. 292

However the measurements for the same are not clear and the methods used for

evaluating constitutive behavior at high strain rates have been shown to be prone to

wave propagation effects. They show an influence of sample size on the results

with thicker samples showing higher rate sensitivity. Also it has been suggested

that this increase in rate sensitivity could be due to strain hardening.

• New Insights from present studies: Using present results and solving the inverse

problem to reveal constitutive behavior, LS-DYNA predicted higher flow stress at

higher strain rates in ring expansion experiments. Since both inertia and rate

sensitivity were considered in this model, it indicates that changes in constitutive

behavior may be existent as well. In addition to this, using strain and circuit

parameter data from electromagnetic tube expansion experiments, a theoretical

study was undertaken for developing an EMF-based FLD using a weak-band

analysis for localization in a biaxially stretched sheet subjected to electric currents.

The study primarily conducted by J.D. Thomas and Dr. N. Triantafyllidis at The

University of Michigan, used experimental data from our tube expansion

experiments, as input and neglected inertial forces and considered only changes in

constitutive behavior at high strain rates. Although qualitative trends matched the

data, the FLD curve over-predicted the forming limits. Simplistic assumptions

about neglecting inertia, an incomplete data set for constitutive behavior, using only

the first half current pulse etc. can be challenged. Overall, this study shows that

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changes in constitutive behavior can be important in high velocity formability but it

is insufficient in isolating this from the significance of inertia.

• Future work: Studies involving optimization with tools like LS-OPT are needed

to further study the influence of strain rate on its constitutive behavior. Solving

inverse problems to reveal constitutive behavior is an important method of further

developing an understanding of this aspect. In addition to this, it is important to

isolate the role of changes in constitutive behavior aside from inertia.

10.2 ISSUE: Influence of Inertia in High velocity formability and size effects in fragmentation and necking

• Previous understanding: Inertia aids formability by stabilizing neck localization

and plays a role in establishing the necking pattern and multiple fragmentation.

Experimental studies showing enhanced formability often involved high speed

impact which in itself maybe a significant contributing factor to high velocity

formability. Formability data over a wide range of strain states was lacking. A

number of theoretical studies indicated the influence of sample’s dimensions or

slenderness in rings on formability with a slender rings or smaller samples having

lower formability. However experimental proof for the same was missing.

• New Insights from present studies: Electromagnetic tube free expansion (without

die impact) experiments were designed with AA6063-T6 tubes of various lengths

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which were expanded by solenoid actuators of various lengths. These combinations created various strain states. The results indicated significant improvements in formability even in the absence of a die impact, which was contrary to a few studies in the literature. Strains in the range of 10-55% were obtained in some cases from a material with quasi-static ductility of 8-10%. However these increases in formability could be due to inertial enhancement and/or changes in constitutive behavior. To understand this issue further, electromagnetic ring expansion were designed with rings of different (slenderness) aspect ratios. Size effects are a direct manifestation of inertia as a less slender sample has higher inertial resistance to necking. From these experiments it was seen that the sample’s slenderness has a big influence on its formability. A less slender sample had higher uniform and total elongation and also fewer necks at a particular velocity level, as compared to a slender sample. The most slender samples in this study did not show any improvement in formability at high velocities, and had formability less than or equal to the quasi-static ductility of the material. The results of this study were an excellent match with the model of Shenoy and Freund (1999), which established that the number of necks can be related to sample’s aspect ratio and velocity by a power law. Since this numerical study considered only inertial forces and neglected strain rate sensitivity, and the measured number of necks in our experimental study had an excellent correspondence with this model, it can be deduced that inertia is a dominant factor in enhancing formability at high velocities. It was shown that number of necks and fragments, and formability increase with velocity but is highly

295

influenced by the slenderness of the samples. An interesting mode of failure was

observed in tall tubes formed in tube expansion experiments. They had repeating

band instabilities criss-crossing across the sample length sometimes with

perforations forming at their intersection. These are proof of periodic instabilities

represented in the literature as failure modes with a constant wavelength.

• Open questions: Although this study in itself is complete in establishing the

significance of size and inertia on formability, further experiments with sheets of

different thicknesses, formed into open cavities will further establish the results. A

thickness effect in sheets was previously observed in EHF experiments by

Balanethiram (1995), in which thicker sheets were easier to form, and also in Bullet

impact experiments by Seth (2003), in which it was observed that while it was

possible to form thicker sheets into hemispherical cavities, its was impossible to do

so with the thin sheets which always failed. In addition to this, a useful FLD model

can be developed based on these experiments.

10.3 ISSUE: Influence of high velocity impact on formability

• Previous understanding: Impact has been shown to be beneficial in a number of

experimental studies which have shown substantial improvements in formability

after a high velocity impact. It leads to through-thickness compressive stresses

which radially stretch out the material and the accompanied ‘inertial ironing’ also

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enhances formability. There is a theoretical study by Imbert (2004) which predicts

damage reduction on impact but quantified experimental evidence does not exist.

• New insights from present studies: Previous experimental studies involving a high

velocity impact like Balanethiram (1995) had shown a five times increase in

formability; Seth (2003) had shown an almost forty time increase in formability

after impact. These studies took advantage of ‘inertia ironing’. Another method of

taking advantage of a high velocity impact is by arresting a sample by impacting it

with the die wall, just before it is supposed to fragment. This further allows use of

higher velocity of launch and thus higher inertial benefits too. Thus tube and ring

die impact experiments were performed. Comparison of samples expanded freely

with samples impacting a die, showed that a high velocity impact can be highly

beneficial in improving formability by reducing the number of tears or fractures in

a sample which gives it the benefit of deforming to higher strains. However it was

hard to quantify the influence of impact from this study. From ring die impact

experiments it was seen that the number of fractures in the samples can be

drastically reduced when impact is involved. Comparisons of freely formed and die

impacted samples at the same strains revealed a big difference in the number of

fractures. Tube flanging experiments were also done with successful creation of

flanges of 1.5 cm length which was almost impossible to form quasi-statically.

From all the experiments involving impact, it was observed, that it is important to

have sufficient energy and velocity in the sample, for it to be able to benefit from

impact. Impact at insufficient velocities resulted in no or less enhancement in 297

formability. At the same time, too high velocities can result in failures in the

material due to excessive thinning during flattening of the material against the die

or due to new failure modes like spall. Thus there is an appropriate velocity range

within which the substantial improvements in formability can be obtained.

• Future work: Further experiments with 4x4 rings, involving impact with a larger

cylindrical die will help in comparing further comparison of samples at the same

strain level with and without impact. FEM simulations in LSDYNA can further

help understand the mechanics of die impact. Failure models can be developed

using such dynamic codes.

10.4 ISSUE: Dynamic failure modes

• Previous understanding: Very few studies exist which try to understand the failure

of samples formed in the EMF regime, while considerable studies about fracture

under shock conditions with strain rates > 104 s-1 and velocities > 750 m/s exists.

These are typically projectile penetration or Taylor impact studies which do not

always apply to the typical strain rates (1000-5000 s-1) encountered in HVF.

• New insights from present studies: Different failure modes were classified on the

basis of samples formed with a high velocity impact with EMF or EHF. Necking,

shearing and ‘spall-like dynamic rupture’ were the broad categories. The latter

mode was a new mode observed in a few samples which failed in a different

298

manner than the classically observed failures. Studies in the literature about ‘spall’

were explored and it was revealed that spall occurs when two unloading stress

waves (from opposing free surfaces) converge to generate a planar zone of intense

triaxiality. While all the elements of a typical spall failure were not completely

established, it is possible for regions of dynamic hydrostatic tension to develop and

cause damage interaction to form a fracture surface with unique characteristics.

Thus this mode of failure was referred to as ‘spall-like dynamic rupture’ as it had

some similarities with spall. The basic characteristics of this observed new mode

were – it was brittle like with hardly any macroscopic plasticity; there was hardly

any crack opening; often a series of cracks were seen near a big failure; the fracture

surface often had an angle; damage in the form of void linkages was also observed

metallographically. All these observations were made from a series of samples of

AA2024, AA6061, high strength steel and copper sheets, which were formed with a

high velocity impact. Optical micrographs of the cross-sections showed void

linkages along specific planes. SEM micrographs of the fracture surface showed

classical dimpled appearance. Micrographs of a classically spalled tube during

electromagnetic welding were also presented to show that it is possible to spall

specimens with EMF. The stress reversal of the sheet while it confirms to the die

and wave propagation effects maybe responsible for this different mode of failure.

• Future work: The present study is incomplete in a lot of ways and just marks the

way for further detailed studies about failure in dynamic conditions. A lot of things

299

like the exact mechanics of the failure, and the conditions under which it happens

are still not clear. However a few characteristics of this mode involving dynamic

rupture with little macro plasticity have been established. In order to fully

understand it, further analysis needs to be done. Samples formed at conventional

forming velocities to the same strain levels, sample with classical necking and

shearing, with and without impact, materials of different thicknesses etc. need to be

examined optically to see if voids link up in the same manner as in the samples in

this study. It is essential to have studies of modeling involving failure models,

coupled with experiments.

10.5 ISSUE: Appropriateness of a Forming Limit Diagram (FLD) for depicting high velocity formability

• Previous understanding: FLDs have proven to be very useful for conventional

forming operations in predicting failures in parts and providing practical

engineering guidelines. Such FLDs do not exist for HVF conditions.

• New insights from present studies: A theoretical study for predicting an FLD

based on EMF was proposed using experimental data and the MK model. This

study ignored inertia and considered only changes in constitutive behavior like

increased rate sensitivity. However, from previous and present studies, it can be

concluded that an FLD under high velocity conditions is not as useful and practical

300

a tool as it is in conventional forming due to the following differences that exist in

the HVF regime as compared to the conventional one:

o Boundary conditions like presence of impact, magnetic field, punch or tool

geometry play a much more significant role in HVF than in conventional

forming. In the HVF regime, these are more pronounced effects than the

quasi-static formability of the material. Thus each forming operation is

uniquely defined by its forming boundary conditions so that formability

cannot be predicted on the basis of an FLD.

o Conventional FLDs do not include inertial effects (which have been proven to

significantly influence formability in HVF) and thus cannot represent high

velocity formability.

o Constitutive behavior of materials changes at high velocities which can

influence formability and cannot be included in the prediction by an FLD.

o Failure modes change in high velocity conditions and failure does not

typically occur by the intense localization of one neck. Instead multiple necks

form with inertial factors establishing the necking pattern and ultimately

leading to multiple fragmentation at much higher strains. During a high

velocity impact, tool-sheet interaction can lead to newer failure modes like

spall-like dynamic rupture, which do not affect conventional forming.

Thus FLDs might not be practical tools for designing HVF processes. Instead formability is uniquely controlled by boundary conditions during each forming operation. 301

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