METAL FORMING: PROCESSES AND ANALYSIS
B. Avitzur, Lehigh University
PREFACE vii
NOMENCLATURE xvii
PART l BASlC CONCEPTS
CHAPTER 1 STATE OF STRESS 3
1.1 Introduction 3
1.2 Components of Stress 5
1.3 Stress Vector on an Inclined Plane 7
1.4 Differential Equations of Equilibrium 9
1.5 Symmetry of the Stress Tensor 12
1.6 Principal Stresses and Principal Axes of Stress 13
1.7 Mean Stress and Stress Deviator 15
1.8 Maximum Shear 16
PROBLEMS 17
CHAPTER 2 YIELD CRITERIA 21
2.1 Introduction 21
2.2 Von Mises' Yield Criterion 22
2.3 Tresca's Yield Criterion 24
2.4 Comparison of Yield Criteria 24
2.5 Yield Surface 26
PROBLEMS 28
CHAPTER 3 STRAIN AND STRAIN RATES 29
3.1 Strain Rates and Velocities 29
3.2 Principal Strain Rates and Their Axes 34
3.3 Incompressibility 37
3.4 Infinitesimal Strains and Displacements 37
3.5 Principal Finite Strains 39
PROBLEMS 41
CHAPTER 4 STRESS-STRAIN AND STRESS-STRAIN RATE LAWS 43
4.1 Stress-Strain Relations for Solids 43
4.2 Tensile Test 45
4 3 Von Mises' Stress-Strain Rate Law 49
4.4 Viscous Flow 50
PROBLEMS 51
CHAPTER 5 UPPER BOUND ON POWER 52
5.1 Strain Energy 52
5.1.1 Power 52
5.1.2 Work 54
5.1.3 Ideal Power and Work of Deformation 56
5.1.3.1 Rod Forming 5.1.3.2 Strip Forming
5.2 Kinematically Admissible Velocity Field and Velocity Discontinuities 58
5.3 Friction and Friction Losses 60
5.3.1 Coulomb Coefficient of Friction 60
5.3.2 Constant-friction Factor 61
5.3.3 Hydrodynamic Lubrication 61
5.4 Upper-bound Theorem 63
PROBLEMS 64
CHAPTER 6 LOWER BOUND ON POWER 67
6.1 Statically Admissible Stress Field and Stress Field Discontinuities 67
6.2 Lower-bound Theorem 69
6.3 Exact Solution 69
REFERENCES FOR PART ONE: 71
PART II MANUFACTURING PROCESSES
A) AXISYMMETRIC STATE
CHAPTER 7 FORGING OF DISKS 77
7.1 Introduction 77
7.2 Strain Rates and Equilibrium Equations in Cylindrical Coordinates 77
7.3 Solid Disk-Radial Flow without Bulging 78
7.4 Hollow Disk 81
7.4.1 Assumed Pattern of Deformation 82
7.4.2 Powers 84
7.4.3 Determination of Neutral Radius Rn 87
7.4.4 Results 89
7.4.5 Limit Cases, Ri/Ro ® 1 and m Ro/T ® 8 90
7.4.6 Typical Dimensional Changes during Pressing 93
7.4.7 Kudo's Approach to Coulomb Coefficient of Friction 97
7.4.8 Experimental Determination of Neutral Radius 102
7.5 Solid Disk with Bulging Taken into Consideration 109
7.6 Solid Disk-Free-body Equilibrium Approach 111
7.6.1 Preface 111
7.6.2 Velocity Field, Strain Rate Components, and Stresses 111
7.6.3 Free-body Equilibrium 112
7.6.4 Constant Friction Factor 113
7.6.5 Coulomb Coefficient of Friction 114
7.6.5.1 Sliding 7.6.5.2 Sticking 7.6.5.3 Combined Sliding and
Sticking
7.7 Power and Energy 121
7.8 Continuous Regions Separated by Surfaces of Velocity Discontinuity 121
7.8.1 Conical Surfaces of Velocity Discontinuity 123
7.9 Variable Friction 132
7.10 Lower-bound Solution for a Solid Disk 134
7.11 Appendix 145
7.11.1 Combined Sliding and Sticking 145
7.11.2 Lower-bound Solution 146
REFERENCES 150
PROBLEMS 151
CHAPTER 8 FLOW THROUGH CONICAL CONVERGING DIES 153
8.1 Introduction 153
8.2 Strain Rates and Equilibrium Equations in Spherical Coordinates 154
8.3 Upper-bound Approach 155
8.3.1 Velocity Field and the Theorem 155
8.3.2 Internal Power of Deformation 158
8.3.3 Velocity Discontinuities and Friction Losses 160
8.3.4 Applied Powers 162
8.3.5 Applied Stresses 162
8.3.6 Optimal Cone Angle 164
8.3.7 Dead-zone Formation 166
8.3.8 Shaving 169
8.3.9 Central Burst or Chevroning 172
8.4 Equilibrium Approach 176
8.4.1 Zero-friction Case 176
8.4.2 Friction with Constant Shear Factor 178
8.5 Free-body Equilibrium Approach with Coulomb Friction 178
8.6 Energy Approach with Coulomb Friction 181
8.7 Energy Approach with Hydrodynamic Lubrication 184
8.8 Distorted Grid Pattern 188
8.8.1 On the Completed Product 188
8.8.2 Intermediate Uncompleted Distortion 193
8.9 Strain Rates and Strains 195
8.10 Strain-hardening Materials 201
8.11 Strain Rate Sensitivity 201
8.12 Discussion 202
8 12.1 Characteristics of Drawing and Extrusion Stresses 202
8 12.2 Defects and Irregularities 205
8.13 Appendix 213
REFERENCES 211
PROBLEMS 216
CHAPTER 9 WIRE AND ROD DRAWING AND OPEN-DIE EXTRUSION 218
9.1 Introduction 218
9.2 Drawing Stress 221
9.2 1 Recapitulation of Results of the Analysis 221
9.2 2 Wistreich's Study 223
9.2.3 Concluding Remarks 226
9.3 Optimal Cone Angle 226
9.4 Maximum Reduction 227
9.5 Open-die Extrusion 229
9.6 Measurement of Friction 234
9.6.1 Review 234
9.6.2 Derivation of Basic Equations 23S
9.6.3 Experimental Procedure 236
9.7 Discussion 238
9.7.1 Shaving 238
9.7.2 Defects 240
9.7.3 1Iydrodynamic Lubrication 241
REFERENCES 246
PROBLEMS 248
CHAPTER 10 EXTRUSION 250
10.1 Introduction 250
10.2 Spherical Velocity Field 254
10.3 Unit Cylindrical Deforming Regions 257
10.4 Direct Extrusion (Steady State) 258
10.4 1 Upper Bound with Spherical Velocity Field 258
10.4 2 Upper Bound by Unit Cylindrical Deforming Regions 260
10.4.3 Free-body Equilibrium Approach 261
10.4.4 Approximation Combining the Spherical Velocity Field and the Free-
body Equilibrium Approaches 262
10.5 Indirect Extrusion (Steady State) 266
10.5.1 Upper Bound with Spherical Velocity Field 266
10.5.2 Upper Bound by Unit Cylindrical Deforming Regions 268
10.5.3 Spherical Velocity Field with Coulomb Friction 268
10.6 Direct and Indirect Extrusion, End of the Stroke, and Summary 269
10.7 Piercing Extrusion 274
10.7.1 Early Stage, Steady State 274
10.7.2 End of the Stroke 277
10.8 Appendix 286
10.8.1 End of the Stroke 286
10.8.2 Surface of Velocity Discontinuity 287
REFERENCES 291
PROBLEMS 292
CHAPTER 11 HYDROSTATIC EXTRUSION 295
11.1 Introduction 295
11.2 Analyses 298
11.3 Discussion 305
11.3.1 Required Pressure 305
11.3.2 Speed Control and the Peak Phenomenon 311
11.3.3 Central Burst and Shaving 3]3
11.3.4 Chamber Design 313
11.3.5 Die Design 316
11.4 Metal Forming under Pressure 317
11.5 The Effect of Hydrostatic Pressure during Forming on Subsequent Strength and Ductility
320
REFERENCES 324
PROBLEMS 326
CHAPTER 12 TUBE SINKING AND EXPANDING 327
12.1 Introduction 327
12.2 Spherical Velocity Field 330
12.2.1 Tube Sinking 331
12.2.2 Tube Expanding 339
12.3 Thickness Study 341
12.3.1 Internal Power of Deformation 345
12.3.2 Shear Power 345
12.3.3 Power Balance 347
12.4 Floating Plug 352
12.5 Appendix 353
REFERENCES 355
PROB1EMS 356
B) STATE OF P1ANE STRAIN
CHAPTER 13 FORGING OF STRIP 359
13.1 Introduction 359
13.2 Upper Bound 360
13.2.1 Parallel Velocity Field without Bulge 360
13.2.1.1 Constant Friction 13.2.1.2 Coulomb Friction
13.2.2 Forging with Bulge, Constant Friction 363
13.2.3 Triangular Field 369
13.3 Lower-bound Solution 377
13.4 Discussion and Conclusions 386
13.4.1 Average Pressure and the Friction Hill 386
13.4.2 Friction Characteristics 388
13.4.3 No-slip Region 391
13.4.4 Bulge 391
13.5 Appendix 392
REFERENCES 395
PROB1EMS 396
CHAPTER 14 FLOW THROUGH INCLINED PLANES 398
14.1 Introduction 398
14.2 Velocity Field, Velocity Discontinuities, and Strain Rates 399
14.3 Upper-bound Approach 402
14.3.1 Required Drawing and Extrusion Stresses 402
14.3.2 Optimal Angle 406
14.3.3 Dead-zone Formation 407
14.3.4 Chip Formation 409
14.4 Flow Characteristics 412
14.4.1 Distorted Grid 412
14.4.1.1 The Completed Product 14.4.1.2 The Intermediate Stage
14.4.2 Effective Strain Rate and Its Average 418
14.4.3 Effective Strain and Its Average 421
14.5 Discussion 424
14.5.1 Upper Bound on Power 424
14.5.2 Distorted Grid Shape 426
14.5.3 Strain Rates and Strains 427
14.5.4 Characteristics of the Drawing (or Extrusion) Stress and the Mode of
Flow 429
14.6 Appendix 429
14.6.1 The Elliptic Integral 429
14.6.2 Chip Formation 430
REFERENCES 434
PROB1EMS 435
CHAPTER 15 STRIP ROLLING 436
15.1 Introduction 436
15.2 Pressure Distribution and Roll Separation Force 437
15.3 Limiting Thickness and Limiting Reduction 445
15.4 Power Requirement and Roll Torque 448
15.4.1 Assumed Velocity Field 448
15.4.2 Power Balance 450
15.4.3 Minimum Required Power, Ideal Work, and Efficiency 457
15.4.4 Effect of Roll Flattening on Power Requirements 458
15.4.5 Coulomb Friction 459
15.4.6 Discussion 460
15.4.7 Hydrodynamic Lubrication 463
15.5 Forward Slip and the Position of the Neutral Point 468
15.6 Maximum Possible and Minimum Required Reductions 469
15.7 Minimum Required Friction and the Measurement of Friction Values 475
15.8 Drawing through Idling Rolls 478
15.9 Gauge Control and Mill Prestressing 480
15.10 Appendix 483
REFERENCES 491
PROBLEMS 492
INDEX 493
