Fourth Edition METAL FORMING MECHANICS AND METALLURGY William F.Hosford Robert M.Caddell CAMBRIDGE www.cambridge.org/9781107004528
Contents Preface to the Fourth Edition page xii 1 Stress and Strain.··.... 1 1.1 Stress 1 1.2 Stress transformation 2 1.3 Principal stresses 4 1.4 Mohr's circle equations 1.5 Strain 7 1.6 Small strains 9 1.7 The strain tensor 10 1.8 Isotropic elasticity 10 1.9 Strain energy 11 1.10 Force and moment balances 12 1.11 Boundary conditions 13 NOTES OF INTEREST.......·· 14 REFERENCES.·.······· 15 APPENDIX-EQUILIBRIUM EQUATIONS. 15 PROBLEMS.·············· 15 2 Plasticity........·· 17 2.1 Yield criteria 17 2.2 Tresca criterion 18 2.3 Von Mises criterion 20 2.4 Effective stress 21 2.5 Effective strain 22 2.6 Flow rules 23 2.7 Normality principle 24 2.8 Derivation of the von Mises effective strain 26 NOTES OF INTEREST.............................................. 27
Contents Preface to the Fourth Edition page xii 1 Stress and Strain ................................... 1 1.1 Stress 1 1.2 Stress transformation 2 1.3 Principal stresses 4 1.4 Mohr’s circle equations 5 1.5 Strain 7 1.6 Small strains 9 1.7 The strain tensor 10 1.8 Isotropic elasticity 10 1.9 Strain energy 11 1.10 Force and moment balances 12 1.11 Boundary conditions 13 NOTES OF INTEREST .............................................. 14 REFERENCES .................................................... 15 APPENDIX – EQUILIBRIUM EQUATIONS ............................... 15 PROBLEMS ..................................................... 15 2 Plasticity ............................................. 17 2.1 Yield criteria 17 2.2 Tresca criterion 18 2.3 Von Mises criterion 20 2.4 Effective stress 21 2.5 Effective strain 22 2.6 Flow rules 23 2.7 Normality principle 24 2.8 Derivation of the von Mises effective strain 26 NOTES OF INTEREST .............................................. 27 v
vi CONTENTS REFERENCES.………………… 28 PROBLEMS 28 3 Strain Hardening.·· 30 3.1 The tension test 30 3.2 Elastic-plastic transition 32 3.3 Engineering vs.true stress and strain 32 3.4 Power-law expression 34 3.5 Other strain-hardening approximations 36 3.6 Behavior during necking 36 3.7 Compression testing 38 3.8 Bulge testing 38 3.9 Plane-strain compression 39 3.10 Torsion testing 40 NOTE OF INTEREST ......... 40 REFERENCES.···· 41 PROBLEMS 41 4 Plastic Instability.·········· 43 4.1 Uniaxial tension 43 4.2 Effect of inhomogeneities 44 4.3 Balanced biaxial tension 45 4.4 Pressurized thin-wall sphere 47 4.5 Significance of instability 48 N0 TE OF INTEREST,……………………·· 49 REFERENCES.………· 49 49 5 Temperature and Strain-Rate Dependence.·..··..··...··..···· 52 5.1 Strain rate 52 5.2 Superplasticity 55 5.3 Effect of inhomogeneities 58 5.4 Combined strain and strain-rate effects 62 5.5 Alternative description of strain-rate dependence 63 5.6 Temperature dependence of flow stress 65 5.7 Deformation mechanism maps 69 5.8 Hot working 69 5.9 Temperature rise during deformation 71 NOTES OF INTEREST...········ 72 REFERENCES.....··· 73 PROBLEMS............ 73 6 Vork Balance.······ 76 6.1 Ideal work 76
vi CONTENTS REFERENCES .................................................... 28 PROBLEMS ..................................................... 28 3 Strain Hardening ....................................... 30 3.1 The tension test 30 3.2 Elastic-plastic transition 32 3.3 Engineering vs. true stress and strain 32 3.4 Power-law expression 34 3.5 Other strain-hardening approximations 36 3.6 Behavior during necking 36 3.7 Compression testing 38 3.8 Bulge testing 38 3.9 Plane-strain compression 39 3.10 Torsion testing 40 NOTE OF INTEREST .............................................. 40 REFERENCES .................................................... 41 PROBLEMS ..................................................... 41 4 Plastic Instability ....................................... 43 4.1 Uniaxial tension 43 4.2 Effect of inhomogeneities 44 4.3 Balanced biaxial tension 45 4.4 Pressurized thin-wall sphere 47 4.5 Significance of instability 48 NOTE OF INTEREST .............................................. 49 REFERENCES .................................................... 49 PROBLEMS ..................................................... 49 5 Temperature and Strain-Rate Dependence ..................... 52 5.1 Strain rate 52 5.2 Superplasticity 55 5.3 Effect of inhomogeneities 58 5.4 Combined strain and strain-rate effects 62 5.5 Alternative description of strain-rate dependence 63 5.6 Temperature dependence of flow stress 65 5.7 Deformation mechanism maps 69 5.8 Hot working 69 5.9 Temperature rise during deformation 71 NOTES OF INTEREST .............................................. 72 REFERENCES .................................................... 73 PROBLEMS ..................................................... 73 6 Work Balance ......................................... 76 6.1 Ideal work 76
CONTENTS vii 6.2 Extrusion and drawing 77 6.3 Deformation efficiency 18 6.4 Maximum drawing reduction 6.5 Effects of die angle and reduction 80 6.6 Swaging 81 REFERENCES..... 82 PROBLEMS 82 7 Slab Analysis.......... 。。。 85 7.1 Sheet drawing 85 7.2 Wire and rod drawing 87 7.3 Friction in plane-strain compression 88 7.4 Sticking friction 90 7.5 Mixed sticking-sliding conditions 90 7.6 Constant shear stress interface 91 7.7 Axially symmetric compression 92 7.8 Sand-pile analogy 93 7.9 Flat rolling 93 7.10 Roll flattening 95 7.11 Roll bending 99 7.12 Coining 101 7.13 Reducing the area of contact 101 NOTES OF INTEREST....·...··.·· .102 REFERENCES..............,... ...102 PROBLEMS ....102 8 Friction and Lubrication....... ·····..106 8.1 General 106 8.2 Experimental findings 109 8.3 Ring friction test 110 8.4 Galling 111 8.5 Ultrasonics 111 NOTE OF INTEREST ....... ...111 REFERENCES....................................................112 PR0 BLEMS...........112 9 Upper-Bound Analysis.············· ….113 9.1 Upper bounds 113 9.2 Energy dissipation on plane of shear 114 9.3 Plane-strain frictionless extrusion 115 9.4 Plane-strain frictionless indentation 119 9.5 Plane-strain compression 119 9.6 Another approach to upper bounds 122 9.7 A combined upper-bound analysis 123 9.8 Plane-strain drawing 124
CONTENTS vii 6.2 Extrusion and drawing 77 6.3 Deformation efficiency 78 6.4 Maximum drawing reduction 79 6.5 Effects of die angle and reduction 80 6.6 Swaging 81 REFERENCES .................................................... 82 PROBLEMS ..................................................... 82 7 Slab Analysis .......................................... 85 7.1 Sheet drawing 85 7.2 Wire and rod drawing 87 7.3 Friction in plane-strain compression 88 7.4 Sticking friction 90 7.5 Mixed sticking-sliding conditions 90 7.6 Constant shear stress interface 91 7.7 Axially symmetric compression 92 7.8 Sand-pile analogy 93 7.9 Flat rolling 93 7.10 Roll flattening 95 7.11 Roll bending 99 7.12 Coining 101 7.13 Reducing the area of contact 101 NOTES OF INTEREST ..............................................102 REFERENCES ....................................................102 PROBLEMS ..................................................... 102 8 Friction and Lubrication .................................. 106 8.1 General 106 8.2 Experimental findings 109 8.3 Ring friction test 110 8.4 Galling 111 8.5 Ultrasonics 111 NOTE OF INTEREST .............................................. 111 REFERENCES ....................................................112 PROBLEMS ..................................................... 112 9 Upper-Bound Analysis ................................... 113 9.1 Upper bounds 113 9.2 Energy dissipation on plane of shear 114 9.3 Plane-strain frictionless extrusion 115 9.4 Plane-strain frictionless indentation 119 9.5 Plane-strain compression 119 9.6 Another approach to upper bounds 122 9.7 A combined upper-bound analysis 123 9.8 Plane-strain drawing 124
vili CONTENTS 9.9 Axisymmetric drawing 124 REFERENCES..…·· .126 PROBLEMS.,,...,,.······ 126 10 Slip-Line Field Analysis ...... ...132 10.1 Introduction 132 10.2 Governing stress equations 132 10.3 Boundary conditions 136 10.4 Plane-strain indentation 137 10.5 Hodographs for slip-line fields 138 10.6 Plane-strain extrusion 139 10.7 Energy dissipation in a slip-line field 141 10.8 Metal distortion 141 10.9 Indentation of thick slabs 142 10.10 Plane-strain drawing 146 10.11 Constant shear stress interfaces 150 10.12 Pipe formation 151 NOTES OF INTEREST................. REFERENCES..·· ..154 APPENDIX.......··· ....154 PROBLEMS··· .157 11 Deformation-Zone Geometry...............................167 ll.lThe△parameter 167 11.2 Friction 168 11.3 Redundant deformation 168 11.4 Inhomogeneity 170 11.5 Internal damage 175 11.6 Residual stresses 179 11.7 Comparison of plane-strain and axisymmetric deformation 182 NOTE OF INTEREST ....... 184 REFERENCES ...184 PROBLEMS.····· .184 12 Formability........ ···..186 12.1 Ductility I86 12.2 Metallurgy 186 12.3 Ductile fracture 190 12.4 Hydrostatic stress 191 12.5 Bulk formability tests 195 12.6 Formability in hot working 196 NOTE OF INTEREST.··· ..197 REFERENCES.·.··· ...197 PROBLEMS 197
viii CONTENTS 9.9 Axisymmetric drawing 124 REFERENCES ....................................................126 PROBLEMS ..................................................... 126 10 Slip-Line Field Analysis .................................. 132 10.1 Introduction 132 10.2 Governing stress equations 132 10.3 Boundary conditions 136 10.4 Plane-strain indentation 137 10.5 Hodographs for slip-line fields 138 10.6 Plane-strain extrusion 139 10.7 Energy dissipation in a slip-line field 141 10.8 Metal distortion 141 10.9 Indentation of thick slabs 142 10.10 Plane-strain drawing 146 10.11 Constant shear stress interfaces 150 10.12 Pipe formation 151 NOTES OF INTEREST ..............................................152 REFERENCES ....................................................154 APPENDIX ...................................................... 154 PROBLEMS ..................................................... 157 11 Deformation-Zone Geometry ............................... 167 11.1 The parameter 167 11.2 Friction 168 11.3 Redundant deformation 168 11.4 Inhomogeneity 170 11.5 Internal damage 175 11.6 Residual stresses 179 11.7 Comparison of plane-strain and axisymmetric deformation 182 NOTE OF INTEREST .............................................. 184 REFERENCES ....................................................184 PROBLEMS ..................................................... 184 12 Formability ........................................... 186 12.1 Ductility 186 12.2 Metallurgy 186 12.3 Ductile fracture 190 12.4 Hydrostatic stress 191 12.5 Bulk formability tests 195 12.6 Formability in hot working 196 NOTE OF INTEREST .............................................. 197 REFERENCES ....................................................197 PROBLEMS ..................................................... 197
CONTENTS 年 13 Bending.··········· ...199 13.1 Sheet bending 199 13.2 Bending with super-imposed tension 202 13.3 Young's modulus on unloading 204 13.4 Reducing springback 204 13.5 Neutral axis shift 204 13.6 Bendability 206 13.7 Shape bending 207 13.8 Forming limits in bending 208 NOTE OF INTEREST.…· ..209 REFERENCES....................................................209 PR0 BLEMS...................209 14 Plastic Anisotropy············· ......212 14.1 Crystallographic basis 212 14.2 Measurement of R 214 14.3 Hill's anisotropic plasticity theory 214 14.4 Special cases of Hill's yield criterion 216 14.5 Nonquadratic yield criteria 217 14.6 Calculation of anisotropy from crystallographic considerations 220 N0TE0 F INTEREST........................221 REFERENCES...······· ...221 PROBLEMS.·… 221 15 Cupping,Redrawing,and Ironing...... 225 15.1 Cup drawing 225 15.2 Anisotropy effects in drawing 228 15.3 Effects of strain hardening in drawing 229 15.4 Analysis of assumptions 230 15.5 Effects of tooling on cup drawing 232 15.6 Earing 233 15.7 Redrawing 235 15.8 Ironing 236 15.9 Progressive forming 238 15.10 Tapered dies 238 15.11 Drawing boxes 240 15.12 Residual stresses 241 NOTES OF INTEREST...······ ...242 REFERENCES........... ....242 PROBLEMS.··· ....242 16 Forming Limit Diagrams.... ......245 16.1 Localized necking 245 16.2 Forming limit diagrams 249
CONTENTS ix 13 Bending ............................................. 199 13.1 Sheet bending 199 13.2 Bending with super-imposed tension 202 13.3 Young’s modulus on unloading 204 13.4 Reducing springback 204 13.5 Neutral axis shift 204 13.6 Bendability 206 13.7 Shape bending 207 13.8 Forming limits in bending 208 NOTE OF INTEREST .............................................. 209 REFERENCES ....................................................209 PROBLEMS ..................................................... 209 14 Plastic Anisotropy ...................................... 212 14.1 Crystallographic basis 212 14.2 Measurement of R 214 14.3 Hill’s anisotropic plasticity theory 214 14.4 Special cases of Hill’s yield criterion 216 14.5 Nonquadratic yield criteria 217 14.6 Calculation of anisotropy from crystallographic considerations 220 NOTE OF INTEREST .............................................. 221 REFERENCES ....................................................221 PROBLEMS ..................................................... 221 15 Cupping, Redrawing, and Ironing ........................... 225 15.1 Cup drawing 225 15.2 Anisotropy effects in drawing 228 15.3 Effects of strain hardening in drawing 229 15.4 Analysis of assumptions 230 15.5 Effects of tooling on cup drawing 232 15.6 Earing 233 15.7 Redrawing 235 15.8 Ironing 236 15.9 Progressive forming 238 15.10 Tapered dies 238 15.11 Drawing boxes 240 15.12 Residual stresses 241 NOTES OF INTEREST ..............................................242 REFERENCES ....................................................242 PROBLEMS ..................................................... 242 16 Forming Limit Diagrams..................................245 16.1 Localized necking 245 16.2 Forming limit diagrams 249
CONTENTS 16.3 Experimental determination of FLDs 250 16.4 Calculation of forming limit diagrams 252 16.5 Factors affecting forming limits 255 16.6 Changing strain paths 258 16.7 Stress-based forming limits 260 NOTE OF INTEREST ............ 260 REFERENCES.·········· .261 PROBLEMS ....261 17 Stamping········· ····.263 17.1 Stamping 263 17.2 Draw beads 263 17.3 Strain distribution 265 17.4 Loose metal and wrinkling 266 17.5 Springback 268 17.6 Strain signatures 269 17.7 Die design 270 17.8 Toughness and sheet tearing 272 17.9 General observations 274 N0TES0 F INTEREST.............275 REFERENCES.........................275 PROBLEMS.···· ....275 i8 Hydroforming,·············· ....277 18.1 General 277 18.2 Free expansion of tubes 277 18.3 Hydroforming into square cross sections 279 18.4 Bent sections 281 NOTE OF INTEREST ....................... 283 REFERENCES.··· .283 PROBLEMS.....................................................283 l9 Other Sheet Forming Operations.....··············· ......285 19.1 Roll forming 285 19.2 Spinning 286 19.3 Foldable shapes 288 19.4 Incremental sheet forming 288 19.5 Shearing 290 19.6 Flanging,hole expansion,and beading 291 19.7 Hemming 292 NOTE OF INTEREST .................... 293 REFERENCES………………·· .293 PROBLEMS.....................................................293
x CONTENTS 16.3 Experimental determination of FLDs 250 16.4 Calculation of forming limit diagrams 252 16.5 Factors affecting forming limits 255 16.6 Changing strain paths 258 16.7 Stress-based forming limits 260 NOTE OF INTEREST .............................................. 260 REFERENCES ....................................................261 PROBLEMS ..................................................... 261 17 Stamping ............................................ 263 17.1 Stamping 263 17.2 Draw beads 263 17.3 Strain distribution 265 17.4 Loose metal and wrinkling 266 17.5 Springback 268 17.6 Strain signatures 269 17.7 Die design 270 17.8 Toughness and sheet tearing 272 17.9 General observations 274 NOTES OF INTEREST ..............................................275 REFERENCES ....................................................275 PROBLEMS ..................................................... 275 18 Hydroforming ......................................... 277 18.1 General 277 18.2 Free expansion of tubes 277 18.3 Hydroforming into square cross sections 279 18.4 Bent sections 281 NOTE OF INTEREST .............................................. 283 REFERENCES ....................................................283 PROBLEMS ..................................................... 283 19 Other Sheet Forming Operations ............................ 285 19.1 Roll forming 285 19.2 Spinning 286 19.3 Foldable shapes 288 19.4 Incremental sheet forming 288 19.5 Shearing 290 19.6 Flanging, hole expansion, and beading 291 19.7 Hemming 292 NOTE OF INTEREST .............................................. 293 REFERENCES ................................................... .293 PROBLEMS ..................................................... 293
CONTENTS 20 Formability Tests..... ...294 20.1 Cupping tests 294 20.2 LDH test 294 20.3 Post-uniform elongation 297 20.4 OSU formability test 297 20.5 Hole expansion 298 20.6 Hydraulic bulge test 299 20.7 Duncan friction test 300 REFERENCES.·......·.··· ....301 PROBLEMS....······ ...301 21 Sheet Metal Properties.···· ,...304 21.1 Introduction 304 21.2 Surface appearance 305 21.3 Strain aging 305 21.4 Roller leveling and temper rolling 308 21.5 Properties of steels 309 21.6 Grades of low-carbon steel 309 21.7 Tailor-welded blanks 314 21.8 Special sheet steels 315 21.9 Surface treatment 315 21.10 Stainless steels 316 21.11 Aluminum alloys 317 21.12 Copper and brass 321 21.13 Hexagonal close-packed metals 322 21.14 Tooling 323 21.15 Product uniformity 323 21.16 Scrap 324 NOTES OF INTEREST.... ....324 REFERENCES.......... ...325 PROBLEMS.······· ..325 Index 327
CONTENTS xi 20 Formability Tests ....................................... 294 20.1 Cupping tests 294 20.2 LDH test 294 20.3 Post-uniform elongation 297 20.4 OSU formability test 297 20.5 Hole expansion 298 20.6 Hydraulic bulge test 299 20.7 Duncan friction test 300 REFERENCES ....................................................301 PROBLEMS ..................................................... 301 21 Sheet Metal Properties ................................... 304 21.1 Introduction 304 21.2 Surface appearance 305 21.3 Strain aging 305 21.4 Roller leveling and temper rolling 308 21.5 Properties of steels 309 21.6 Grades of low-carbon steel 309 21.7 Tailor-welded blanks 314 21.8 Special sheet steels 315 21.9 Surface treatment 315 21.10 Stainless steels 316 21.11 Aluminum alloys 317 21.12 Copper and brass 321 21.13 Hexagonal close-packed metals 322 21.14 Tooling 323 21.15 Product uniformity 323 21.16 Scrap 324 NOTES OF INTEREST ..............................................324 REFERENCES ....................................................325 PROBLEMS ..................................................... 325 Index 327
Preface to the Fourth Edition My coauthor,Robert Caddell,died in 1990,and I have greatly missed working with him. The most significant changes from the third edition are a new chapter on friction and lubrication and a major rearrangement of the last third of the book dealing with sheet forming.Most of the chapters in the last part of the book have been modified,with one whole chapter devoted to hydroforming.A new section is devoted to incremental forming.No attempt has been made to introduce numerical methods.Other books treat numerical methods.We feel that a thorough understanding of a process and the constitutive relations that are embedded in a computer program to analyze it are necessary.For example,the use of Hill's 1948 anisotropic yield criterion leads to significant errors. I wish to acknowledge my membership in the North American Deep Drawing Research Group from whom I have learned so much about sheet forming.Particular thanks are due to Alejandro Graf of ALCAN,Robert Wagoner of the Ohio State Uni- versity,John Duncan formerly with the University of Auckland,and Thomas Stoughton of General Motors. William F.Hosford xii
Preface to the Fourth Edition My coauthor, Robert Caddell, died in 1990, and I have greatly missed working with him. The most significant changes from the third edition are a new chapter on friction and lubrication and a major rearrangement of the last third of the book dealing with sheet forming. Most of the chapters in the last part of the book have been modified, with one whole chapter devoted to hydroforming. A new section is devoted to incremental forming. No attempt has been made to introduce numerical methods. Other books treat numerical methods. We feel that a thorough understanding of a process and the constitutive relations that are embedded in a computer program to analyze it are necessary. For example, the use of Hill’s 1948 anisotropic yield criterion leads to significant errors. I wish to acknowledge my membership in the North American Deep Drawing Research Group from whom I have learned so much about sheet forming. Particular thanks are due to Alejandro Graf of ALCAN, Robert Wagoner of the Ohio State University, John Duncan formerly with the University of Auckland, and Thomas Stoughton of General Motors. William F. Hosford xii
1 Stress and Strain An understanding of stress and strain is essential for the analysis of metal forming operations.Often the words stress and strain are used synonymously by the nonscientific public.In engineering usage,however,stress is the intensity of force and strain is a measure of the amount of deformation. 1.1 STRESS Stress o is defined as the intensity of force at a point. g=aF/8 A as A→0， (1.1) where F is the force acting on a plane of area,4. If the stress is the same everywhere in a body, G=F/A. (1.2) There are nine components of stress as shown in Figure 1.1.A normal stress component is one in which the force is acting normal to the plane.It may be tensile or compressive. A shear stress component is one in which the force acts parallel to the plane. Stress components are defined with two subscripts.The first denotes the normal to the plane on which the force acts and the second is the direction of the force.*For example,ox is a tensile stress in the x-direction.A shear stress acting on the x-plane in the y-direction is denoted by oy Repeated subscripts(e.g.)indicate normal stresses.They are tensile if both the plane and direction are positive or both are negative.If one is positive and the other is negative they are compressive.Mixed subscripts(e.g.,ox,oxy,oy=)denote shear stresses.A state of stress in tensor notation is expressed as Oxx Oyx Ozx Oij= Oxy Oyy Ozx (1.3) Oxz Oy 02z The use of the opposite convention should cause no problem because j=j
1 Stress and Strain An understanding of stress and strain is essential for the analysis of metal forming operations. Often the wordsstress and strain are used synonymously by the nonscientific public. In engineering usage, however, stress is the intensity of force and strain is a measure of the amount of deformation. 1.1 STRESS Stress σ is defined as the intensity of force at a point. σ = ∂F/∂ A as ∂ A → 0, (1.1) where F is the force acting on a plane of area, A. If the stress is the same everywhere in a body, σ = F/A. (1.2) There are nine components of stress as shown in Figure 1.1. A normal stress component is one in which the force is acting normal to the plane. It may be tensile or compressive. A shear stress component is one in which the force acts parallel to the plane. Stress components are defined with two subscripts. The first denotes the normal to the plane on which the force acts and the second is the direction of the force.∗ For example, σxx is a tensile stress in the x-direction. A shear stress acting on the x-plane in the y-direction is denoted by σx y . Repeated subscripts (e.g., σxx, σyy, σzz) indicate normal stresses. They are tensile if both the plane and direction are positive or both are negative. If one is positive and the other is negative they are compressive. Mixed subscripts (e.g., σzx , σx y , σyz) denote shear stresses. A state of stress in tensor notation is expressed as σi j = σx x σyx σzx σx y σyy σzx σx z σyz σzz , (1.3) ∗ The use of the opposite convention should cause no problem because σi j = σji . 1