Plasticity and creep : theory, examples, and problems

목차 (Table of Contents)

• CONTENTS
• Ⅰ BASIC DEFINITIONS = 1
• 1 STRESS AND STRAIN STATE = 3
• 1.1 Stress tensor = 3
• 1.1.1 Stress at a point; definitions and notation = 3
• 1.1.2 Equations of static equilibrium = 4
• 1.1.3 Transformation of stresses; principal stresses = 7
• 1.1.4 Stress axiator and stress deviator tensors = 8
• 1.1.5 Other stress invariants = 10
• 1.2 Strain tensor = 12
• 1.2.1 Strain at a point; small strain tensor = 12
• 1.2.2 Strain- displacements equations = 13
• 1.2.3 Compatibility of strain = 14
• 1.2.4 Principal strains; strain invariants = 15
• 1.2.5 Decomposition of the small strain tensor; other strain invariants = 16
• 2 FINITE DEFORMATIONS = 19
• 2.1 Finite strain tensors in material or spatial coordinates = 19
• 2.1.1 Displacement vector = 19
• 2.1.2 Finite strain tensors in Cartesian coordinates = 20
• 2.1.3 Finite strain tensors in curvilinear coordinates = 22
• 2.1.4 Example: Finite strains in cylindrical coordinates = 23
• 2.1.5 Generalized logarithmic strain measures = 26
• 2.1.6 Example: Application of logarithmic strains to changing principal directions = 27
• 2.1.7 Discussion; Comparison of various strain measures = 28
• 2.2 Strain rates tensors = 30
• 2.2.1 Material and spatial descriptions = 30
• 2.2.2 Rate-of-deformation tensor = 31
• 2.2.3 Example: Rate-of-deformation tensor in cylindrical and spherical coordinates = 32
• 2.2.4 Other strain rates measures = 34
• 2.3 Stress tensors in material or spatial descriptions = 34
• 2.3.1 Cauchy's and Piola-Kirchhoff's stress tensors = 34
• 2.3.2 Equations of equilibrium in material or spatial formulations = 35
• 2.3.3 Conjugate variables = 36
• Ⅱ FOUNDATIONS OF PLASTICITY = 37
• 3 BASIC EQUATIONS OF PERFECT PLASTICITY = 39
• 3.1 Uniaxial stress-strain behavior = 39
• 3.1.1 Tensile test = 39
• 3.1.2 True stress-strain curve = 42
• 3.1.3 Schematizations of the stress-strain curves = 43
• 3.1.4 Bauschinger's effect = 48
• 3.2 Criteria for yielding in perfect plasticity = 50
• 3.2.1 Conditions of perfect plasticity for isotropic materials = 50
• 3.2.2 Cylindrical yield surfaces = 51
• 3.2.3 Rotationally symmetric yield surfaces = 53
• 3.2.4 Limit state conditions in soil and rock mechanics = 55
• 3.2.5 Conditions of perfect plasticity for anisotropic materials = 57
• 3.3 Stress-strain relations for perfect plasticity = 60
• 3.3.1 Hencky-Ilyushin deformation theory of plasticity = 60
• 3.3.2 Levy-Mises (LM) and Prandtl-Reuss (PR) theories of plastic flow = 62
• 3.3.3 Discussion: Deformation theory versus flow rule; Ilyushin theorem on simple loading = 64
• 3.3.4 Example: A thin-walled tube or a circular bar subject to non-proportional tension with torsion = 67
• 3.3.5 Simplest theories for finite strains = 70
• 3.4 Methods of reduction of equations of perfect plasticity = 72
• 3.4.1 Stress functions = 72
• 3.4.2 Parametrization of the yield condition = 76
• 3.4.3 Displacement equations in plasticity = 78
• 3.5 Problems = 79
• 4 BASIC EQUATIONS OF PLASTIC HARDENING = 87
• 4.1 Drucker's postulate and the associated flow rule = 87
• 4.1.1 Material stability in Drucker's sense = 87
• 4.1.2 Associated flow rule (AFR) = 89
• 4.1.3 Discussion: Generality of Drucker's postulate = 91
• 4.2 Subsequent yield surfaces for hardening material = 93
• 4.2.1 Experimental investigations of subsequent yield surfaces = 93
• 4.2.2 Isotropic hardening rule = 96
• 4.2.3 Kinematic hardening rule = 98
• 4.2.4 Mixed hardening rule = 100
• 4.2.5 General anisotropic hardening hypotheses = 101
• 4.2.6 Example: Some plane cases of anisotropic hardening = 103
• 4.2.7 Mr$$\acute o$$z's multisurface hardening theory = 105
• 4.2.8 Summary: Review of anisotropic hardening functions = 107
• 4.3 Theories of plastic hardening = 109
• 4.3.1 N$$\grave a$$dai-Ilyushin (NI) deformation theory of plastic hardening = 109
• 4.3.2 Incremental theories of plastic hardening = 109
• 4.3.3 Mixed hardening material based on the HMH yield criterion = 111
• 4.3.4 Matrix formulation of incremental stress-strain relations = 113
• 4.3.5 Example: Application of the associated incremential law to combined tension with torsion of a thin-walled tube = 116
• 4.3.6 Simplest theories of plastic hardening for finite strains = 123
• 4.3.7 Example: Application of the deformation theory of plastic hardening for finite strains when the principal directions change / Thin-walled cylinder subject to bi-axial tension and torsion = 125
• 4.3.8 Cyclic plasticity; Chaboche-Rousselier's model of mixed hardening = 128
• 4.3.9 Example: Cyclic hardening of a solid axial specimen = 132
• 4.4 Problems = 133
• 5 METHODS OF THE THEORY OF PLASTICITY = 137
• 5.1 Analysis on the level of a cross-section = 137
• 5.1.1 Generalized variables = 137
• 5.1.2 Incremental generalized stress-strain relations; multi-point substitutive cross-sections = 141
• 5.1.3 Plastic interaction surfaces and the associated flow rule = 145
• 5.1.4 Examples: Plastic interaction surfaces for a cylindrical shell made of HMH material under radial loadings = 150
• 5.2 Interaction curves on levels of a cross-section or a body = 153
• 5.2.1 Limit carrying capacity (LCC) = 153
• 5.2.2 Examples: Maximum carrying capacity (MCC) of elastic-plastic structures = 154
• 5.2.3 Discontinuities- Decohesive carrying capacity (DCC) = 161
• 5.2.4 Examples: Interaction curves corresponding to decohesive carrying capacity = 166
• 5.2.5 Discussion: Mechanism of decohesion = 172
• 5.3 Extremum theorems of limit analysis: statically or kinematically admissible solutions = 175
• 5.3.1 Fundamentals of limit analysis = 175
• 5.3.2 Statically or kinematically admissible solutions = 177
• 5.3.3 Application of the upper-bound theorem to the limit analysis of bar structures = 180
• 5.3.4 Examples: Lower and upper bounds to the limit analysis of statically indeterminate bar structures = 185
• 5.4 Shakedown analysis = 192
• 5.4.1 Theorems of shakedown and inadaptation of structures = 192
• 5.4.2 Example: Application of shakedown theorems to frame structures = 194
• 5.4.3 Example: Shakedown domain, incremental collapse and alternating plasticity in a bar structure = 199
• 5.4.4 Example: Shakedown and non-shakedown analysis of a tubular cylindrical bar subject to torsion = 203
• 5.5 Integration along characteristics in plane strain problems = 210
• 5.5.1 Characteristics in plane strain = 210
• 5.5.2 Construction of the net of characteristics = 213
• 5.5.3 Examples: Application of slipline fields to upper bound analysis to the yield point load = 218
• 5.6 Problems = 226
• Ⅲ SOLUTIONS OF ELASTIC-PLASTIC PROBLEMS = 235
• 6 ELASTIC-PLASTIC TORSION AND BENDING = 237
• 6.1 Elastic-plastic torsion of prismatic bars = 237
• 6.1.1 General relations = 237
• 6.1.2 Elastic torsion; the membrane analogy = 239
• 6.1.3 Elastoplastic torsion; the sand hill analogy and the membraneroof analogy = 240
• 6.1.4 Examples: Application of the mathematical analogy to elasticplastic torsion problems = 245
• 6.1.5 Elastoplastic torsion of a rotationally-symmetric bar = 249
• 6.1.6 Example: Secondary yielding and residual twist of a solid strain-hardening bar = 253
• 6.1.7 Examples: Torsion of statically indeterminate bars and tubes = 255
• 6.2 Problems = 257
• 6.3 Elastic-plastic bending of prismatic beams and plane frames = 262
• 6.3.1 Perfectly elastic-plastic beams of doubly- symmetric cross-sections = 262
• 6.3.2 Perfectly elastic-plastic beams of mono-symmetric cross-sections = 264
• 6.3.3 Bending of prismatic beams of hardening material = 267
• 6.3.4 Examples: Elastic-plastic bending of statically determinate beams of rectangular cross-section = 270
• 6.3.5 Limit analysis of statically indeterminate beams and plane frames = 273
• 6.3.6 Plastic interaction curves for combined bending with normal force of doubly-symmetric cross-section = 280
• 6.4 Problems = 283
• 7 ELASTIC-PLASTIC ANALYSIS OF CYLINDERS, DISKS AND PLATES = 295
• 7.1 Thick-walled tubes, spherical shells and disks = 295
• 7.1.1 Elastic analysis and initial yielding of a thick-walled tube = 295
• 7.1.2 Elastic-plastic analysis of a thick-walled tube = 297
• 7.1.3 A thick-walled tube of hardening material = 301
• 7.1.4 Combined loadings of a thick-walled tube = 303
• 7.1.5 Examples: Particular cases of combined loadings of a thick-walled tube = 305
• 7.1.6 Elastic-plastic analysis of a thick-walled spherical shell = 308
• 7.1.7 A thick-walled spherical shell of hardening material = 310
• 7.1.8 Elastic-plastic analysis of a thin rotating circular disk according to the TG criterion = 311
• 7.1.9 Example: Limit state of a circular disk governed by the HMH yield criterion = 315
• 7.1.10 Example: Interaction curves of an annular disk under combined rotation and normal tractions = 317
• 7.2 Problems = 320
• 7.3 Limit analysis of plates = 325
• 7.3.1 Basic equations in Cartesian coordinates = 325
• 7.3.2 Discontinuities: Plastic hinge lines = 329
• 7.3.3 Rotationally symmetric plates = 330
• 7.3.4 Example: Circular plate of Tresca material = 332
• 7.3.5 Yield line method = 335
• 7.3.6 Examples: Upper bound estimation to the collapse load of a plate by the use of the yield line method = 341
• 7.4 Problems = 345
• Ⅳ FOUNDATIONS OF CREEP = 355
• 8 BASIC EQUATlONS OF UNIAXIAL CREEP MODELS = 357
• 8.1 Creep phenomenon = 357
• 8.1.1 Creep tests = 357
• 8.1.2 Mechanism of creep = 362
• 8.2 Schematizations of creep at constant uniaxial stress = 363
• 8.2.1 Functions of stress = 363
• 8.2.2 Functions of time = 364
• 8.2.3 Functions of temperature = 364
• 8.2.4 Convenient uniaxial creep relationships = 365
• 8.3 Modelling of creep at varying uniaxial stress = 366
• 8.3.1 Total strain (TS) model = 366
• 8.3.2 Time hardening (TH) model = 367
• 8.3.3 Strain hardening (SH) model = 367
• 8.3.4 Integral viscoelastic models; Rabotnov's hereditary equations = 370
• 8.3.5 Discussion: Comparison of creep models at varying uniaxial stress = 373
• 8.4 Linear uniaxial viscoelastic models = 375
• 8.4.1 Boltzmann's superposition principle = 375
• 8.4.2 Some particular models of the linear viscoelastic material = 377
• 8.4.3 Discussion: Equivalence of the integral and the differential representations = 383
• 8.5 Modelling of viscoplastic materials = 384
• 8.5.1 Dynamic uniaxial tests for metals = 384
• 8.5.2 Uniaxial relationships for viscoplastic response of metals = 387
• 8.5.3 Example: Simple elastic/viscoplastic models = 390
• 8.5.4 Examples: Four-parameter elastic-viscoplastic models = 394
• 8.6 Problems = 396
• 9 CREEP CONSTITUTIVE EQUATIONS UNDER MULTIAXIAL LOADING = 407
• 9.1 Classical multiaxial creep theories = 407
• 9.1.1 Introductory remarks = 407
• 9.1.2 Deformation or total strain (TS) theory = 408
• 9.1.3 Flow rule (FR) and creep potential = 409
• 9.1.4 Isotropic strain hardening (SH) theory = 412
• 9.2 Developed multiaxial creep theories = 413
• 9.2.1 Kinematic-hardening (KH) theory = 413
• 9.2.2 Chaboche's viscoplastic constitutive equation = 416
• 9.3 Linear multiaxial viscoelastic equations = 417
• 9.3.1 Differential representation under a multiaxial stress state; the Laplace transform = 417
• 9.3.2 Integral representation under a multiaxial stress state = 421
• 9.3.3 Elastic-viscoelastic correspondence principle = 421
• 9.3.4 Example: Thick-walled tube made of the standard material under hydrostatic pressure = 423
• Ⅴ SOLUTION OF CREEP PROBLEMS = 425
• 10 BENDING, BUCKLING AND TORSION OF BARS UNDER CREEP CONDITIONS = 427
• 10.1 Bending and buckling of a prismatic bar made of the linear viscoelastic material = 427
• 10.1.1 Direct solution of a bar subject to bending with tension = 427
• 10.1.2 Example: Solution of a bending with tension problem of a beam using a differential representation for standard material = 430
• 10.1.3 Example: Solution of a pure bending problem using the correspondence principle = 431
• 10.1.4 Buckling of a linear viscoelastic column under the axial force = 432
• 10.1.5 Examples: Creep buckling of columns made of simple linear viscoelastic materials = 434
• 10.2 Bending of a prismatic beam made of the piece-wise linear elastic/viscoplastic material = 440
• 10.2.1 Pure bending of beams of doubly-symmetric cross-sections made of the elastic-Bingham material = 440
• 10.2.2 Bending of beams of mono-symmetric cross-sections made of the elastic-Bingham model = 441
• 10.2.3 Example: Bending of a Bingham-type beam of the rectangular cross-section = 443
• 10.3 Bending of a prismatic beam made of the time hardening material = 445
• 10.3.1 Bending with tension of a beam of mono-symmetric cross-sections = 445
• 10.3.2 Example: Transient creep of a bent beam of the rectangular cross-section = 447
• 10.4 Torsion of a circular bar made of the elastic-Bingham material = 448
• 10.4.1 General relationship for torque versus unit angle of twist = 448
• 10.4.2 Example: Transient creep torsion of a circular bar subject to the constant torque (cf. Rzhanitsyn (1968)) = 449
• 10.4.3 Example: Stationary creep of a twisted bar of the Hooke-Norton material = 450
• 10.5 Problems = 451
• 11 ROTATIONALLY SYMMETRIC CREEP PROBLEMS = 457
• 11.1 Creep of a thick-walled tube = 457
• 11.1.1 Steady creep of a thick-walled tube made of the Norton-Odqvist material = 457
• 11.1.2 Example: Limit solutions for a perfectly elastic and a perfectly plastic tube = 460
• 11.1.3 Transient creep of a thick-walled tube made of the time-hardening material = 461
• 11.2 General formulae for the rotationally-symmetric transient creep problems = 464
• Ⅵ CREEP RUPTURE = 465
• 12 CONSTITUTIVE EQUATIONS OF CREEP RUPTURE = 467
• 12.1 Creep rupture phenomenon = 467
• 12.1.1 Mechanism of creep rupture = 467
• 12.1.2 Creep rupture data = 469
• 12.2 Classical creep rupture theories = 478
• 12.2.1 Ductile rupture theories = 478
• 12.2.2 Brittle rupture theories = 481
• 12.2.3 Example: Brittle rupture under plane stress = 484
• 12.2.4 Ductile-brittle rupture theory = 486
• 12.2.5 Example: Brittle rupture under simple relaxation conditions = 487
• 12.3 Problems = 488
• 13 ROTATIONALLY SYMMETRIC CREEP RUPTURE PROBLEMS = 495
• 13.1 Mechanisms of brittle rupture of tubes and disks = 495
• 13.1.1 Example: Creep rupture of pressurised thick-walled tube under plane strain conditions = 495
• 13.1.2 Example: Brittle rupture of an annular disk subject to combined loadings under plane stress conditions = 497
• 13.1.3 Example: Brittle rupture of a clamped annular disk subject to a temperature gradient = 500
• 13.2 Design of disks with respect to creep rupture = 501
• 13.2.1 Introductory remarks = 501
• 13.2.2 Example: Effect of an initial prestressing on the creep rupture of a rotating disk = 502
• 13.2.3 Example: Disk of variable thickness of the uniform creep strength = 504
• 13.2.4 Example: Effect of an initial prestressing on the failure mechanism of disks of the uniform creep strength = 506
• REFERENCES = 511
• AUTHOR INDEX = 527
• SUBJECT INDEX = 531