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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
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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