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أخوانى فى الله أحضرت لكم كتاب Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications D. W. A. Rees School of Engineering and Design, Brunel University, UK BASIC ENGINEERING PLASTICITY An Introduction with Engineering and Manufacturing Applications D. W. A. Rees School of Engineering and Design, Brunei University, UK
![كتاب Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications B_e_p_10](https://i.servimg.com/u/f61/15/36/74/14/b_e_p_10.jpg) و المحتوى كما يلي :
CONTENTS Preface xi Acknowledgements xii List of Symbols xiii C H A P T E R 1 STRESS ANALYSIS 1.1 Introduction 1 .,2 Cauchy Definition of Stress 4 1.3 Three Dimensional Stress Analysis 7 1.4 Principal Stresses and Invariants 15 1.5 Principal Stresses as Co-ordinates 21 1.6 Alternative Stress Definitions 27 Bibliography 31 Exercises 31 C H A P T E R 2 STRAIN ANALYSIS 2.1 Introduction 33 2.2 Infinitesimal Strain Tensor 33 2.3 Large Strain Definitions 40 2.4 Finite Strain Tensors 47 2.5 Polar Decomposition 58 2.6 Strain Definitions 62 References 62 Exercises 63vi CONTENTS CHAPTER 3 YIELD CRITERIA 3.1 Introduction 65 3.2 Yielding of Ductile Isotropie Materials 65 3.3 Experimental Verification 71 3.4 Anisotropic Yielding in Polyerystals 83 3.5 Choice of Yield Function 90 References 91 Exercises 93 C H A P T E R 4 NON-HARDENING PLASTICITY 4.1 Introduction 95 4.2 Classical Theories of Plasticity 95 4.3 Application of Classical Theory to Uniform Stress States 98 4.4 Application of Classical Theory to Non-Uniform Stress Slates 111 4.5 Hencky versus Prandtl-Reuss 123 References 124 Exercises 124 C H A P T E R 5 ELASTIC-PERFECT PLASTICITY 5.1 Introduction 127 5.2 Elastic-Plastic Bending of Beams 127 5.3 Elastic-Plastic Torsion 137 5.4 Thick-Walled, Pressurised Cylinder with Closed-Ends 144 5.5 Open-Ended Cylinder and Thin Disc Under Pressure 149 5.6 Rotating Disc 154 References 159 Exercises 159CONTENTS CHAPTER 6 SLIP LINK FIELDS 6.1 Introduction 161 6.2 Slip Line Field Theory 161 6.3 Frictionless Extrusion Through Parallel Dies 180 6.4 Frictionless Extrusion Through Inclined Dies 191 6.5 Extrusion With Friction Through Parallel Dies 195 6.6 Notched Bar in Tension 197 6.7 Die Indentation 199 6.8 Rough Die Indentation 204 6.9 Lubricated Die Indentation 207 References 210 Exercises 211 C H A P T E R 7 LIMIT ANALYSIS 7.1 Introduction 213 7.2 Collapse of Beams 213 7.3 Collapse of Structures 215 7.4 Die Indentation 221 7.5 Extrusion 225 7.6 Strip Rolling 230 7.7 Transverse Loading of Circular Plates 234 7.8 Concluding Remarks 238 References 239 Exercises 239vili CONTENTS CHAPTER 8 CRYSTAL PLASTICITY 8.1 Introduction 241 8.2 Resolved Shear Stress and Strain 242 8.3 Lattice Slip Systems 246 8.4 Hardening 248 8.5 Yield Surface 250 8.6 Flow Rule 255 8.7 Micro- to Macro-Plasticity 257 8.8 Subsequent Yield Surface 262 8.9 Summary 266 References 267 Exercises 268 C H A P T E R 9 THE FLOW CURVE 9.1 Introduction 269 9.2 Equivalence in Plasticity 269 9.3 Uniaxial Tests 274 9.4 Torsion Tests 280 9.5 Uniaxial and Torsional Equivalence 283 9.6 Modified Compression Tests 286 9.7 Bulge Test 290 9.8 Equations to the Flow Curve 294 9.9 Strain and Work Hardening Hypotheses 298 9.10 Concluding Remarks 304 References 304 Exercises 305 C H A P T E R 10 PLASTICITY WITH HARDENING 10.1 Introduction 309 10.2 Conditions Associated with the Yield Surface 309 10.3 Isotropic Hardening 313 10.4 Validation of Levy Mises and Drucker Flow Rules 318 10.5 Non-Associated Flow Rules 325 10.6 Prandtl-Reuss Flow Theory 326 10.7 Kinematic Hardening 331 10.8 Concluding Remarks 336 References 336 Exercises 337CONTENTS ta CHAPTER 11 ORTHOTROPIC PLASTICITY 11.1 Introduction 339 11.2 Ortnotropie Flow Potential 339 11.3 Qrtholropic How Curves 343 11.4 Planar Isotropy 348 11.5 Rolled Sheet Metals 351 11.6 Extruded Tubes 357 11.7 Non-Linear Strain Paths 362 11.8 Alternative Yield Criteria 365 11.9 Concluding Remarks 366 References 367 Exercises 368 C H A P T E R 12 PLASTIC INSTABILITY 12.1 Introduction 371 12.2 Inelastic Buckling of Struts 371 12.3 Buckling of Plates 378 12.4 Tensile Instability 388 12.5 Circular Bulge Instability 393 12.6 Ellipsoidal Bulging of Orthotropic Sheet 395 12.7 Plate Stretching 399 12.8 Concluding Remarks 408 References 409 Exercises 409 C H A P T E R 13 STRESS WAVES IN BARS 13.1 Introduction 411 13.2 The Wave Equation 411 13.3 Particle Velocity 412 13.4 Longitudinal Impact of Bars 415 13.5 Plastic Waves 421 13.6 Plastic Stress Levels 432 13.7 Concluding Remarks 436 References 436 Exercises 436CONTENTS CHAPTER 14 PRODUCTION PROCESSES 14.1 Introduction 439 14.2 Hot Forging 439 14.3 Cold Forging 442 14.4 Extrusion 444 14.5 Hot Rolling 448 14.6 Cold Rolling 454 14.7 Wire and Strip Drawing 457 14.8 Orthogonal Machining 461 14.9 Concluding Remarks 475 References 475 Exercises 475 C H A P T E R 15 APPLICATIONS OF FINITE ELEMENTS 15.1 Introduction 479 15.2 Elastic Stiffiiess Matrix 479 15.3 Energy Methods 482 15.4 Plane Triangular Element 484 15.5 Elastic-Plastic Stiffiiess Matrix 490 15.6 FE Simulations 496 15.7 Concluding Remarks 502 References 503 Exercises 503 Index 50 INDEX Acoustic impedance, 413 Admissible fields, 213 Almansi strain tensor, 52 Alternating tensor, 28 Anisotropy, 83, 89,342, 358,497,500 Angular distortion, 3 Angular twist, 144,281 Annealing, 289 Annular disc, 149 Anticlastic curvature, 377 Arc of contact, 451,456 Associated flow rule, 80, 96,256, 311 Autofrettage, 149 Axial symmetry, 144, 154, 177 Back extrapolation, 71 Bar in tension, 66,242, 274 Basal plane, 247 Bausehinger effect, 85,263,332 Beam collapse, 213,216,217 Bending moment, 127 Biaxial principal stress, 322, 357 Body centred cubic, 246, 249, 365 Boundary conditions, 145,156 Bridgeman correction, 277 Buckling; plates, 378, 383, 386 struts, 371,377 Built-up edge, 461,467 Bulge test, 290, 302,497 Bulk modulus, 95 Burger's vector, 257 Cantilever, 216 Cartesian: co-ordinates, 6 stress tensor, 13 Cauchy; strain tensor, 49, 52, S3 stress, 4,27 Cayley-Hamilton theorem, 17 Centroid, 136 Characteristic equation, 16 Chip formation, 461 Circular plates, 234 Classical plasticity theory, 95 Close-packed planes, 246 Coefficient of: elongation, 440 friction, 442,471 spread, 439,453 Cogging, 439,442 Cold forging, 442 Cold rolling, 454 Collapse of structures, 215 Column vectors, 37 Combined stress experiments, 98, 318,357 Compatibility condition, 415 Compressibility, 98, 111, 118, 123 Compression testing, 278,286,287 Conical dies, 457,446 Considere's construction, 276,391 Constant volume, 439,449,462 Constitutive relations, 68,487,492 Constrained paths, 99,102,107,118 Continuous beam, 217 Cooke and Larke test, 286,289, 302 Co-ordinate transformations, 14, 36,243 Co-ordinate systems: Cartesian, 6 curvilinear, 163 cylindrical, 149 spherical, 258 Coulomb: friction, 199,442 -Mohr criterion, SI yield criterion, 66 Cramar's rule, 16 Creep, 318, 360 Criteria of yielding, 65 Cross product, 47 Crystal: aggregate, 251 averaging, 252 lattice, 247 slip, 246 Curvature: beams, 133,482 bulge, 395 strut, 377 Curvilinear co-ordinates, 163506 BASIC ENGINEERING PLASTICITY Decomposition of strain, 35,245 Deformation gradient, 28,48 Degrees of freedom 485 Deviatoric: invariants, 69, 85,269 stress tensor, 69, 84,96,491 Die angle, 458 Die indentation: lower bound, 221 rough, 204 smooth, 199,207 upper bound, 224 Diffuse instability, 399 Direct stress and strain, 2 Direction cosines, 8, 10, 54 Discs: bending, 234 rotating, 154 Dislocations, 257 Displacement: gradient, 49 nodal 484 Distorted yield locus, 81 Distortions, 34,245 Dot product, 20 Double shear, 3 Draft, 450 Draw stress, 457 Drawing of wire and strip, 457 Drucker: compliance, 317 equivalence, 273 flow potential, 316 postulate, 311 yield function, 76,83,271 Dyadic product, 13 Earing, 85 Edge fixings, 378,379 Effective: length of strut, 371 strain increment, 273 stress, 270 Eigen values, 61 Elastic: limit, 71 matrix 487 moduli, 96 strain, 96 Elastic-plastic: bending, 127 disc under pressure, 149, 152 interface, 137,145,152,155 pressure cylinder, 144 rotating disc, 154 tension-torsion, 122 torsion, 137, 141 Ellipsoidal bulge, 395 Empirical strut theory, 377 Energy methods, 67, 235 Encastre fixing, 131 Engesser strut, 371 Engineering stress and strain, 2,275 Equilibrium equations, 146,165 Equivalence in flow, 269,289, 323 Equivalent: plastic strain, 71,273,281,491 stress, 270,281,491 width, 386 Erichsen test 499 Euler strut, 373 Eulerian strain tensors, 36,52 Extended Hencky equation, 467 Extension (stretch) ratio, 41,53 Extensometer, 284,352 Extruded: bar, 447 tube, 357 Extrusion: dead zone, 116 force, 185, 197 friction, 195,444 lower bound, 225 plastic work, 184,189,447 ratio, 186,195 upper bound, 227 Face centred cubic, 246,249 Finite elements: bending 482 codes 495,502 tension 480 torsion 481 triangle 484 Finite strain tensors, 42,47, 52 Flexural stiffness, 378 Flow curve: descriptions, 269,294 equivalence, 343,356 Flow rale, 255,310 Force resolution, 9 Forging, 439,442 Forming limit diagram, 404,408,498,501 Frame collapse, 218 Friction: angle, 458,464 coefficient, 204,458 Coulomb, 199,500 hill, 233,441,450,456INDEX 507 Frictioniess extrusion: inclined dies, 191 parallel dies, 180 Front and back tensions, 455 Fully plastic: beam, 129 pressured cylinder, 147 pressured disc, 157 rotating disc, 157 torsion bars, 138,142 Geiringer equations, 170 Grain boundary, 257 Greatest shear stress, 23,66,464 Green's strain, 42,46 Groove formation, 405 Guest, 66 Haigh, 74 Hardening: combined 336,339 isotropie, 262,313 kinematic, 265 plastic, 135,248,309 strain, 299, 324 work, 300,325 Heterogenous deformation, 258 Hencky: equations, 165 -Ilyushin deformation theory, 95 modified equations, 467 theorems, 164 yield criterion, 67 Hexagonal close packed, 247,249 Hill yield criteria, 83, 339,365,500 Hodograph, 171,185, 193, 199,206,228 Hollomon equation, 295 Homogoneous deformation, 447 Hookean material, 196 Hooke's law, 326,412 Hot rolling, 448 Huber, 67 Hydrostatic: pressure, 468 stress, 6i, 79,163 Hysterisis, 335 Impact: bars, 415 bar-target, 416,432 composite bars, 420 Inclination of principal axes, 17 Inclined dies, 191,458 Incompressibility, 83,96,102 Incremental: flow theory, 95, 310 plastic strain 310,316,491 Indicial notation, 1 Indentation, 199 Inelastic buckling: plates, 311 struts, 371 Inelastic deformation, 95,241,315 Infinitesimal strain, 31, 52,486 Instability: bulging, 333,335 pressure vessels, 391, 392 struts, 371,377 tension, 276,285 torsion, 281 Invariants: strain, 37,272 stress, 15, 80,272 Isotropic hardening, 90, 121,262,313 Isotropy, 269 Jacobean determinant, 28 Johnson's parabola, 374 Kinematic hardening, 77,265,331 Kinematic velocity field, 213,234 Kronecker delta, 69,315 Lagrangian: co-ordinates, 47,60,496 tensors, 51,62 Lame stresses, 144, 148,152,155 Large strains, 40 Latent hardening, 249 Lattice slip, 246 Left stretch tensor, 52,58 Levy-Mises: compliance, 315 equations, 161,288 flow rale, 311,494 Limit: forming strains, 396,404 load, 114 of proportionality, 72 Limit analysis, 213 Load factor, 129 Loading function, 309 Local buckling, 384 Local instability, 401 Lode's parameters, 320,321 Logarithmic (natural) strain, 40 Lower bound: beams, 213,214 die indentation, 221 extrusion, 225 yielding, 251508 BASIC ENGINEERING PLASTICITY Lubricated: dies, 207 indenters, 213 Ludwik law, 296 Machining: friction, 471 hodograph, 469 shear strain, 463,469 shear zone, 464,470 work rate, 472 Material co-ordinates, 48,54 Matrix notation, 2 Matrix of transformation: strain, 36 stress, 15,21 Maximum: shear stress, 23,464 work principle, 251,254 Maxwell, 67 Merchant's circle, 463 Miller indeces, 246,247 Minimum: shear principle, 253 surface, 263 weight, 219 work, 465,473 Modulus: elasticity, 128, 302 rigidity, 138,302 secant 381 tangent 371,493 Mohr"s circle: fracture, 82 local necking, 403 machining, 474 strain, 162 stress, 25, 162 Moment: curvature, 134 of area, 128,137 of resistance, 129 Nadai: construction, 281 parameters, 149 plasticity theory, 95, 120 Natural (logarithmic) strain, 40 Necking, 276 Neutral: axis, 136 loading, 105,310 point, 449 Node points 414 Nonassociated flow rule, 96,325 hardening, 95 uniform stress, 111 Normal anisotropy, 349 Normality rule, 263 Notched: beam, 215 tension bar, 197 Oblique plane stress, 9 Octahedral: plane, 24 shear strain, 273 shear stress, 24, 69, 273 Off-axis tension, 351 Offset strain, 71,88 Open section, 384 Optimum: die angle, 461 frame weight, 220 Orientation: flow stress, 356 grain, 250,254 r-values, 354 yield strength, 342 Orthogonal machining, 461 Orthotropic sheet: plastic buckling, 3S9, 395 sub-tangent, 391 Orthotropy: equivalent strain, 342, 355 equivalent stress, 340 flow potential, 339 flow rule, 341 incremental strains, 342 plasticity, 351 principal axes, 340 yield criteria, 365 Particle velocity: elastic, 412 plastic, 423 Petch equation, 250 Pinned-end strut, 371 Piola-Kirchoff tensors, 29,496 Planar isotropy, 348 Plane strain: analysis, 38,62 compression, 287, 302 element 487 extrusion, 448 forging, 440 indentation, 200 machining, 462 rolling, 230,448INDEX 509 Plane stress, 20, 26,487 Plastic: collapse, 130,215 flow curve, 269 hinge, 130,216 modulus, 428,493 penetration, 115 potential, 225 reduction factor, 382, 387 wave, 421 work, 190,311,491 Plastic strain paths: anisotropic, 353,360,363 isotropic, 318 Plate stretching, 399 Poisson's ratio, 96,337 Polar decomposition, 58 Polar moment of area, 138,142 Polycrystal, 250 Portal frames, 218 Post buckling of plates, 385 Prager: cycloid construction, 167 stress-strain law, 296 translation, 332 Prandtl-Reuss theory, 326,490,494 Pressure vessels, 391, 392 Principal: co-ordinates, 17,21 stress, 15,26,67 stress cubic, 16, 37,43 stretch tensors, 59 Proof stress, 71 Propped cantilever, 216 Pure shear, 387 rvalues, 90, 343,351,500 Radial: loading, 112,318, 322 stress path, 318 Radius of curvature, 128 Rake angle, 461,465 Ramberg-Osgood law, 295 Rankine-Gordon formula, 374 Rate of: hardening, 248, 297 strain, 446,452,454 work, 472 Reduction: factor, 387 ratio, 460 Residual: bending stress, 122,132 curvature, 133 pressure stress, 148,153 rotation stress, 158 strain distributions, 133,428 torsional stress, 138,143 Resolved shear: strain, 243 stress, 242,248 Restraint coefficient, 380 Reversed flow, 334 Right stretch tensor, 50,58 Rigid body motion, 173 Rigid boundary, 424 Roll: flattening, 454 force, 451,456 torque, 451,456 upper bound, 233 Rolled: sheet, 448 strip, 230,453 Rotation: matrix, 2,12,35 tensor, 35 Rotating disc, 154 Safety factor, 130 Sandhill analogy, 144 Scabbing, 411 Schmidt: law, 256 orientation factor, 244,248 Self-consistency, 259 Shaft loadings, 112,121 Shape: factor, 129, 138 function 486 Shear: angle, 3 displacement, 3 finite, 54 modulus, 71,327 plane, 462,464 strain, 3,37,43,404 strain energy, 68 stress, 3 yield stress, 76 zone, 464 Simple: compression, 278 supports, 386 tension, 274 Single: crystal, 248 shear, 3 Slendemess ratio, 371510 BASIC ENGINEERING PLASTICITY Slip; planes, 243 systems, 245,260 Slip line field: dry extrusion, 195,196 frictionless extrusion, 181, 184,188 inclined die extrusion, 191,193 indentation, 200,201 machining, 466 notched tensile bar, 198 rough indentation, 204,205 smooth indentation, 207, 208 theory, 161 Spacial: co-ordinates, 48, 54 deformation gradient, 48 displacement gradient, 49 Specific work, 457 Spread factor, 453 Stassi-D'Alia locus, 81, 86 Stationary potential energy 483,488 Stiffness matrix: elastic 479,488 elastic-plastic 490 overall 489 Stepped: impact bars, 433 loading, 121, 327, 330 Straight line formula, 374 Strain: definitions, 61 energy, 68 finite, 42 hardening hypothesis, 298,324 hardening rule, 299, 303 invariants, 37,43 matrix and tensor, 35 ratio, 342,358 signature 501 transformation, 36 Strength: proof, 71 ultimate, 82 yield, 71 Stress: definitions, 27 discontinuity, 168,226 intensity, 4 invariants, 16, 70,324 state, 6 -strain curves, 294,361 -strain relations, 96, 316 symmetry, 7 tensor, 7,243 transformation, 11 waves, 411 Stretch ratio, 53 Strip rolling, 230 Strut buckling theory: Engesser, 371,376 Euler, 373 parabola, 374,376 Rankine-Gordon, 374, 376 straight line, 374 Subsequent yield surface, 262 Sub-tangent, 391,397,401 Summation convention, 17 Symmettic sections, 385 Symmetry axis, 177 Swift law, 297, 500 Tangent modulus, 371 Taylor: hardening, 261 wave equation, 422 T-beam, 136 Temperature effects: cylindrical bodies, 85 extrusion, 446 Tensile instability, 388 Tension test, 66, 283 Tension-torsion, 318,327,358 Tensor: notation, 327 shear strain, 34, 37 Tensor transformations: strain, 36, 245 stress, 15,244 Texture, 254 Theory of bending, 128 Theory of torsion, 137 Thick-walled: cylinder, 144 disc, 148 Thin-walled: bulge, 290 cylinder, 98 pressure vessels, 391 Torque-twist diagram, 140, 281 Torsion: bar, 281 buckling, 318 solid bar, 281,283 tube, 280,285 Total strain, 96 Traction vector, 4 Transverse isotropy, 350 Transformation of: strain, 36 stress, 15INDEX 511 Translation rules, 332 Tresea: flow rule, 326 yield criterion, 66,149, 157,271,441 Triaxial stress, 67,144 True strain, 279 True stress, 275 Twinning, 247 Ultimate: load, 129,134 plastic moment, 130 tensile strength, 2 torque, 138 Uniaxial tests, 274, 302 Uniform stress state, 98,124 Uniformly distributed loading, 216 Unit normal, 9 Upper bound; circular plates, 235 die indentation, 224 extrusion, 227 strip rolling, 233 yielding, 251 Vector: co-ordinate direction, 11 displacement 481,485 force 481,487 normal, 8 strain 486,491 stress 487,490 unit, 8,12 Velocity: angular, 235 discontinuity, 172,196,227,229 particle, 412 Virtual work principal, 215,483,495 Voee law, 298 Volume averages, 257, 258 Volumetric strain, 43 von Karman, 422 von Mises: equivalent plastic strain, 273 equivalent stress, 270 flow potential, 270 yield criterion, 67, 70,145,157 Wave: equations, 411,422 interactions, 426,428 propagation velocity, 412,414 reflections, 413,417,424 unloading, 425 Weight function, 220 Wide strut, 377 Wire drawing, 457 Work: done, 245 frictional, 460,464 hardening, 298,300,303 hypothesis, 298, 325,343 principle, 251 specific, 457 violation, 190 virtual, 215 Workpiece, 461 Yield: criteria, 65 functions, 70,76, 84, 355 point definition, 71 stress, 66, 71 surface, 71,86,251 Young's modulus, 71,327 Ziegler translation, 332 Zorov, 471 LIST OF SYMBOLS The intention within the various theoretical developments given in this book has been to define each new symbol where it first appears in the text. In this regard each chapter should be treated as self-contained in its symbol content. There are, however, certain symbols that re-appear consistently throughout thetext, such as those representing force, stress and strain. These symbols are given in the following list along with others most commonly employed in plasticity theory. O.P aJ, ft? P>* 3 e,Y e a, T Of % °3 am * i&M p., v 4>,Ji 0 d,m fVPA I,H,F curvilinear co-ordinates (slip lines) kinematic hardening translations Schmidt's orientation factors friction and shear angles rolling draft normal and shear strains normal and shear rates of strain micro-plastic strain tensor equivalent plastic strain direct and shear stress principal stresses mean or hydrostatic stress micro-stress tensor transformed stress tensile and compressive strengths equivalent stress friction coefficient Lode's parameters scalar multipliers angular twist die angles hardening measure Poisson's ratio density extension (stretch) ratio hardening functions a, h, I, z lengths A section or surface area b, t breadth and thickness c propagation velocity C, T torque «?!, «j, e% principal engineering strains eti distortions ep subscripts denoting elastic-plastic E superscript denoting elasticMV E,G,K f F,G,H,L.. F,P Hfj, Cm Hm , HyUma I,J ?uh>h / „ J2, J% K l,m, n m, n MnPPQ Q,S rB,z r n r t RR UR2 u,v,w U v, a yW x,y,z x. X( x,z Y(=ao),k z LIST OF SYMBOLS elastic constants yield function (plastic potential) anisotropy parameters force orthotropic tensors orthotropic tensors continued second moments of area strain invariants stress invariants stress deviator invariants buckling coefficient direction cosines Pr. anisotropy parameters continued half-waves in buckling bending moment hardening exponent pressure superscript denoting plastic stress ratio shape and safety factors polar co-ordinates incremental strain ratios (r values) extrusion ratio radii of curvature back tensions displacements strain energy linear and angular velocities volume work done Cartesian co-ordinates spacial co-ordinates material co-ordinates equivalence coefficients tensile and shear yield stresses Considere's subtangent Q (= 6^) rotation tensor/matrix B, C, G, L deformation tensors E(=£g) infinitesimal strain tensor/matrix F, H deformation gradients m, n, u unit vectors M (= IQ) rotation matrix S nominal stress tensor T(=er9) stress tensor/matrix T (=00 deviatoric stress
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