INDEX Accuracy of simulation, 333 Air bending, 21, 141-147 Air rounding, 21 ALPID program, 136 Analysis in metal forming, 26-52 closed-die forging, 34, 35-36, 37 cold extrusion, 39-41 cold forging, 39-41 drawing of rod, wire, shapes, and tubes, 45-47 equations for, 73 flow stress of metals, 28-30 friction in metal forming, 30-33 hot extrusion of rods and shapes, 36-39 hot forming, 28-29, 34-35 impression, 35-36, 37 methods of. See Methods of analysis objectives of, 26 rolling of strip, plate, and shapes, 41-45 sheet-metal forming, 47-52 temperatures, 28, 29, 33-35 variables in, 26-28 Anisotropy, 189 plastic, 190-191 Area-weighted averaging, 126-127, 129, 255 Assemblage, element, 115-116 Automobile wheel center, 51-52 Axisymmetric forging of flange-hub shapes, 253-256, 257 in preform design, 309-315 of pulley blank, 256-259 See also Axisymmetric isothermal forging Axisymmetric isothermal forging, 151-172 cabbaging, 165-168 compression of cylinders, 153-159 371 compressor disk forging, 168-169 finite-element formulation, 151-153 flashless forging, 170, 171, 172 friction, evaluation of, 163-165, 166 heading of cylindrical bars, 153-159 ring compression, 159-163, 163-165 spike forging, 163, 165 Backward extrusion forging, 14 Backward tracing, 298-301,301-305 in hot forming, 316-317 in plane-strain rolling, 305-308 in shell nosing, 301-305 in symmetric forging, 311-312 Bar drawing, 178-183 multipass, 183-186 Bar extrusion, 176-178 multipass, 183-184 Beverage cans, 46 Block compression, 278-284 rectangular blocks, 278-281 wedge-shaped blocks, 281-284 Blocker operation, 35 Bore-expanding, 194-195, 196 Boundary conditions, 86-87, 117-121 with hemispheric punch, 201-203 Boundary lubrication, 31 Brake-bending, 21 Cabbaging, 165-168 CAD. See Computer-aided design Can ironing press, 46-47 Cauchy stress, 56, 57, 58, 208, 324 Closed-die forging analysis, 35-36, 37 load vs displacement curves in, 34 metal forming system using, 9 with flash, 14, 133-136 without flash, 12 Coining operation, 13 Cold extrusion, 39-41
372 Cold forging, 39-41 Compaction, 266-270 Compatibility conditions, 73 Compression block, 278-284 of cylinder, 75-78, 80-83, 153-159, 223-225, 229-234, 364-369 flat-tool, 314 of porous metals, 249-253 ring, 159-163, 163-165, 331-332, 333 square-ring, 284-287 See also Hot compression; Planestrain compression Compression test, 29 Compressor disk forging, 168-169 Computer-aided design (CAD), 35-36, 93-94 Computer-aided techniques, 1, 186. See also Extrusion dies, computer-aided design of Connectivity, element, 115 Constitutive equations, 73 Cup drawing, 48 square, 210-217 Cylinders compression of, 75-78, 80-83, 153-159, 223-225, 229-234, 364-369 steel, compression of, 229-234 Cylindrical bars, heading of, 153-159 Index square-cup, 210-217 of tubes, 45-47 of wire, 45-47 Dry conditions, 30 Effective strain, 66-68 interpolation of, 126-127 Effective strain-rate matrix, 107 Effective stress, 66-68 Efficiency of simulation, 333 Element assemblage, 115-116 Element connectivity, 115 Element strain-rate matrix, 101-107 Equilibrium, with tractions, 62 Equilibrium equations, 61-62, 73 Equipment, l0 Equivalent stress, 66 Euler equation, 86 Extremum principles, 68-70, 71-72, 74 Extrusion, 19 bar, 176-178 bar, multipass, 183-184 cold, 39-41 direct and indirect, 19 hot, 20, 36-39 of rods, 36-39 of shapes, 36-39 Extrusion dies, computer-aided design of, 186 Deep-drawing processes, 201-206 Deformation mechanics, 10. See also Plasticity Deformation speed, 26-28 Die bending, 21 Die rounding, 21 Direct iteration method, 121-122 Disk forging, 311,312 Distortion energy criterion, 59 Drawbead, 50, 51 Drawing, 20 bar, 178-183 bar, multipass, 183-186 cup, defects in, 48 deep, 23,201-206 flange, 194-195, 197 of rod, 45-47 of shapes, 45-47 FEM. See Finite-element method FEM code. See SPID Finite deformation, 56 Finite-element method (FEM), 83-87, 88, 90-110, 111-129 admissibility requirement for velocity field, 91-92 advantages of, 4 alternative approach, 84 assemblage, element, 115-I 16 axisymmetric isothermal forging, 151-153 axisymmetric out-of-plane deformation, 199-201 basic concept of, 3 basic equations, comments on, 85-86 basic equations, derivation of, 83
Index 373 basic principles and concepts in, 73-74 basis of, 4 boundary conditions, 86-87, 117-121 computer-aided, 93-94 construction of model, 3-4 direct iteration method, 121-122 discretization of problem, 90 element strain-rate matrix, 101-107 elemental stiffness equation, 108-1 l0 geometry updating, 123-125 history of, 90 in-plane deformation processes, 192-194 introduction to, 3-4 Lagrange multiplier method, 110 linear matrix solver, 115-117 linearization, 92 metal forming and, 5-6 nodal point velocities, 91 notation, 101 numerical integrations, l 11-114 penalty function method, 110 plane-strain problems, 131-133 porous metals, 246-249 procedures, 90-94 rectangular element family, 97-100, 105-106 rezoning, 126-129 rigid region, treatment of, 87 ring compression, 160-163 sheet-metal forming of general shapes, 209-210 stiffness equations, 92 three-dimensional brick element, 100-101, 106-107 three-dimensional problems, 276-278 time-increment, 123-125 triangular element family, 95-97, 104-105, 113 variational approach, 83-84, 86 Flange drawing, 194-195, 197 Flange-hub shapes, axisymmetric forging of, 253-256, 257 Flashless forging, 170, 171, 172 Flat-face die, 36, 38 Flat-tool compression, 314 Flat-tool forging, 293,294, 295 Flow formulation, 4-5 Flow rule, 63-66, 245-246 plasticity and, 63--66 porous metals, 245-246 Flow stress, 28-30, 66 Folding, 153 Forging. See also Axisymmetric forging; Closed-die forging classification of, 151 disk, 311,312 flashless, 170, 171, 172 flat-tool, 293, 294, 295 forward extrusion, 13 hot-die disk, 237, 238, 239 isothermal, 237, 238, 239. See also Axisymmetric isothermal forging open-die, 15 orbital, 16 process design objective, 35 process design steps, 35, 36 radial, 314 titanium alloy Ti6242, 234-237 Forward extrusion forging, 13 Friction, 10, 30-33. at tool-workpiece interface, 163-165, 166 Friction coefficient, 206 Friction hill, 138-139 Friction shear stress, 32 Frictional stress, 119-120 Gaussian elimination, 116-117 Gaussian quadrature formula, 111-113 Gear blank forging, 172 Generalized stress, 66 Geometry updating, 123-125 H cross sections, preforms for, 309 Hasek method, 50 Heading of cylindrical bars, 153-159 Heat transfer in porous metals, 259-262 in thermo-viscoplastic analysis, 225-227 Hexahedral element, 276 Hill's general method, 3, 78-83, 88 Hobbing, 14 Hooke's law, 63
374 Hot compression, of steel cylinder, 231-234 Hot-die disk forging, 237, 238, 239 Hot extrusion, 20 of rods and shapes, 36-39 Hot forming, 28-29, 34-35 preform design, 315-318 Hot nosing, 237-240 Hot pressing, under plane-strain compression, 262-266 Huber-Mises criterion, 59 Hydrodynamic conditions, 31 Impression, 35-36, 37 Infinitesimal deformation theory, 55-56 Infinitesimal plastic strain, 63--64 In-plane deformation processes, 192-195 Integration points, 111 Ironing, 21 Ironing press, 46-47 Isoparametric elements, 96, 98, 99 Isothermal forging, 237, 238, 239. See also Axisymmetric isothermal forging Jauman derivative, 324 Kirchhoff stress, definition of, 324 Lagrange multiplier method, 110 Lagrangian description, 55-57 Lagrangian family, 98 Lagrangian strain, 208 Levy-Mises equations, 66 Limiting drawing ratio, 205 Lubrication, basic types of, 30-31 Lubricity, 32-33 Maximum plastic work principle, 64-65 Maxwell-Heuber-Mises criterion, 59 Mechanics of deformation, 10. See also Plasticity Metal flow, 27 in non-steady-state upset forging, 27 in rolling, 42 in steady-state extrusion, 27 Index Metal forming and finite-element method, 5-6 Metal-forming processes, 8-24. See also specific processes backward extrusion forging, 14 brake-bending, 21 classification of, 11-12 closed-die forging, 9, 12, 13 closed-die forging, without flash, 12 coining operation, 13 deep drawing, 23 deformation mechanics, 10. See also Plasticity description of, 11-12 drawing, 20 extrusion process, 19, 20 forward extrusion forging, 13 friction, 10 hobbing, 14 ironing, 21 material variables, 9-10 nosing, 15 open-die forging, 15 orbital forging, 16 product properties, 10 radial forging of shaft, 16 ring rolling, 18 roll bending, 22 roll forming, 22 rolling, 17 rotary tube piercing, 18 rubber-diaphragm hydroforming, 24 rubber-pad forming, 24 shear forming from plate, 19 sheet-metal, 11, 12 spinning, 23 systems approach in, 8-10 tooling and equipment, I0 upsetting with flat-heading tool, 17 Metal powders. See Porous metals Methods of analysis, 73-88 equations for, 73 finite-element method, 83-87, 88. See also Finite-element method Hill's general method, 3, 78-83, 88 for non-steady-state processes, 175-176 for steady-state processes, 174-176 upper-bound method, 3, 74-78, 88 yon Mises criterion, 59
Index 375 Modeling, process, 1-3 Multipass bar drawing and extrusion, 183-186 Nakajima method, 50 Necking, 47 Newton-Raphson method, 92, 93, 121 Nominal stress, 324 Nonquadratic yield criterion, 217-220 Non-steady-state flow, 27 Nosing, 15 hot, 237-240 shell, 301-305 Open-die forging, 15 Orbital forging, 16 Parent element, 97 Penalty function method, 110 Piola-Kirchhoff stress, 208, 324 Piola-Kirchhoff stress tensor, 57 Plane plastic flow, 131. See also Plane-strain problems Plane-strain compression, hot pressing under, 262-266 Plane-strain problems, 131-149 closed-die forging with flash, 133-136 finite-element formulation, 131-133 plate bending, 141-147 sheet roiling, 137-141 side pressing, 148-149 Plane-strain rolling, 305-309 Plastic anisotropy, 190-191 Plastic strain-rate, 63-64 Plasticity, 54-70 effective strain, 66-68 effective stress, 66-68 equilibrium equations, 61-62 extremum principles, 68-70, 71-72 flow rule, 63-66 infinitesimal deformation theory, 55-56 Lagrangian description, 55-57 maximum plastic work principle, 64-65 plastic potential, 63--66 strain, 54-58 strain-hardening, 66-68 strain-rate, 54-58 stress, 54-58 virtual work-rate principle, 62-63 viscoplasticity, 70-72 yield criterion, 58--61 Plastometers, 29 Plate bending of, 141-147, 327-331 rolling of, 41-45 Porous metals, 244-270 axisymmetric forging of flange-hub shapes, 253-256, 257 axisymmetric forging of pulley blank, 256-259 compaction, 266-270 discretization, 247-248 finite-element modeling, 246-249 flow rules, 245-246 fully dense materials, 248 heat transfer in, 259-262 hot pressing under plane-strain condition, 262-266 numerical procedures, 246-249 simple compression, 249-253 updating relative density, 248 volume integration, 248-249 yield criterion, 245-246 Powder compaction, 266-270 Powder forming, 244. See also Porous metals Prandtl-Reuss equations, 65-66 Preform design, 298-320 axially symmetric forging, 309-315 backward tracing, 298-301, 301-305, 305-308, 311-312, 316-317, 318-320. See also Backward tracing definition of, 298 H cross sections, 309 hot forming, 315-318 method for design, 298-301 plane-strain rolling, 305-309 shell nosing, 301-305 Pressing, hot, 262-266 Process modeling, 1-3 Product properties, 10 Pseudoconcentrations, 334 Pulley blank, axisymmetric forging of, 256-259 Punch loads/displacements, 144-145, 146
376 Punch-stretching, 201-206 Radial forging, 314 of shaft, 16 Rectangular block compression, 278-281 Rectangular element family, 97-100 strain-rate matrix, 105-106 Reliability of simulation, 333-334 Rezoning, 126-129 Rigid-plastic formulation, 327-334 disadvantages of, 330 solid formulation vs., 327-334 Ring compression, 159-165, 331-332, 333 axisymmetric isothermal forging, 159-163 friction and, 163-165 solid formulation, 331-332, 333 Ring rolling, 18 Ring test, 33 Rods, 36-39, 45-47 drawing of, 45-47 hot extrusion of, 36-39 Roll bending, 22 Roll forming, 22 Rolling, 17 analysis of spread in, 289-292 metal flow in, 42 plane-strain, 305-309 plate, 17, 43-44 ring, 18 of shapes, 44-45 sheet, 17, 137-141 strip, 42-43, 45 Rotary tube piercing, 18 Rubber-diaphragm hydroforming, 24 Rubber-pad forming, 24 Shaft, radial forging of, 16 Shapes drawing of, 45-47 flange-hub, axisymmetric forging of, 253-256, 257 hot extrusion of, 36-39 rolling of, 44-45 sheet-metal forming of, 206-210 Shear forming from plate, 19 Shear strength of deforming material, 32 Index Shear stress criterion, 59 Sheet rolling, 137-141 Sheet-metal forming, 47-52, 189-220 axisymmetric out-of-plane deformation, 195-201 axisymmetric punch-stretching, 201-206 bore expanding, 194-195, 196 classification of processes, 11, 12 deep-drawing processes, 201-206 flange drawing, 194-195, 197 general shapes, 206-210 in-plane deformation processes, 192-195 nonquadratic yield criterion, 217-220 plastic anisotropy, 190-191 square-cup drawing, 210-217 Shell nosing, 301-305 Side pressing, 148-149 Simple compression, 249-253, 364-369 of cylinder, 364-369 of porous metals, 249-253 Simple plastic incremental deformation. See SPID Simpson's formula, 114 Simulation of forming processes, 332-334 Slab method, 1 Slip-line field method, 1-3, 138 Small-strain solid formulation, 321-323 Smooth entry dies, 38 Solid formulation, 4-5,321-335 large deformation: incremental form, 326-327 large deformation: rate form, 323-326 plate bending, 327-334 rigid-plastic (flow) solutions vs., 327-334 ring compression, 331-332, 333 small-strain, 321-323 three-dimensional analysis with, 334 SPID (simple plastic incremental deformation), 338-369 description of major variables, 342-343 example solution, 364-369
Index 377 input and output files, 340 input preparations, 340-342 program listings, 343-364 program structure, 339-340 simulation conditions, 364-369 Spike forging, 163, 165 Spinning, 23 Springback angle, 330 Square-cup drawing, 210-217 Square-ring compression, 284-287 Steady-state flow, 27 Steady-state processes, 174-187 applications to process design, 186-187 bar drawing, 178-183 bar drawing, multipass, 183-186 bar extrusion, 176-178 bar extrusion, multipass, 183-184 method of analysis, 174-176 Steel cylinder, compression of, 229-234 Stiffness equations, 120 assembled, 116-117 elemental, 108-110 Strain, 54-58 effective, 66-68 Strain-hardening, 66-68 Strain-rate, 54-58 in Cartesian coordinate system, 101 Strain-rate matrix, element, 101-107 Strain-rate vector, 103, 153 Stress, 54-58 effective, 66-68 Stress measures, 324 Strip, rolling of, 42-43, 45 Swift test, 205 Temperature in metal forming, 28, 29, 33-35. See also Thermoviscoplastic analysis Thermal conductivity. See Heat transfer Thermo-viscoplastic analysis, 222-242 applications of, 229-240 compression of cylinder, 223-225, 229-234 computational procedures for, 227-229 forging titanium alloy Ti6242, 234-237 heat transfer analysis, 225-227 hot nosing, 237-240 hot-die disk forging, 237, 238, 239 isothermal forging, 237, 238, 239 Three-dimensional brick element, 100-101 strain-rate matrix, 106-107 Three-dimensional problems, 275-296 block compressions, 278-284 brick element, 100-101, 106-107 finite-element formulation, 276-278 flat-tool forging, 293, 294, 295 hexahedral element, 276 rolling, analysis of spread in, 289-292 simplified elements, 287-289 solid formulation, 334 square-ring compression, 284-287 Time increment, 123-125 Titanium alloy Ti6242, 234-237 Tooling, l0 Tresca criterion (shear stress criterion), 59 Triangular elements, 95-97, 104-105, 113 strain-rate matrix of, 104-105 True stress, 56, 57, 58 Tubes, drawing of, 45-47 Upper-bound method, 3, 74-78, 88 Upsetting, with flat-heading tool, 17 Virtual work-rate principle, 62-63 Viscoplasticity, 70-72. See also Thermo-viscoplastic analysis definition of, 222 Visioplasticity method, 3 Volume strain-rate matrix, 107 von Mises criterion, 59 Wedge-shaped blocks, 281-284 Wheels, automobile, 51-52 Wire, drawing of, 45-47 Wrinkling, 47, 50 Yield criterion nonquadratic, 217-220 plasticity, 58-61 porous metals, 245-246
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