Nonlinear Continuum Mechanics and Large Inelastic Deformations (eBook)

Preface 6
Contents 10
Chapter 1:Introduction: Fundamental Axioms of Continuum Mechanics 26
Chapter 2:Kinematics of Continua 30
2.1 Material and Spatial Descriptions of Continuum Motion 30
2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law 30
2.1.2 Material and Spatial Descriptions 34
2.1.3 Local Bases in K and K 34
2.1.4 Tensors and Tensor Fields in Continuum Mechanics 36
2.1.5 Covariant Derivatives in K and K 38
2.1.6 The Deformation Gradient 39
2.1.7 Curvilinear Spatial Coordinates 41
2.2 Deformation Tensors and Measures 49
2.2.1 Deformation Tensors 49
2.2.2 Deformation Measures 50
2.2.3 Displacement Vector 50
2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector 51
2.2.5 Physical Meaning of Components of the Deformation Tensor 53
2.2.6 Transformation of an Oriented Surface Element 55
2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor 58
2.3 Polar Decomposition 61
2.3.1 Theorem on Polar Decomposition 61
2.3.2 Eigenvalues and Eigenbases 65
2.3.3 Representation of the Deformation Tensors in Eigenbases 67
2.3.4 Geometrical Meaning of Eigenvalues 69
2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum 70
2.4 Rate Characteristics of Continuum Motion 74
2.4.1 Velocity 74
2.4.2 Total Derivative of a Tensor with Respect to Time 75
2.4.3 Differential of a Tensor 78
2.4.4 Properties of Derivatives with Respect to Time 79
2.4.5 The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor 81
2.4.6 Eigenvalues of the Deformation Rate Tensor 83
2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor 84
2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point 85
2.4.9 Kinematic Meaning of the Vorticity Vector 87
2.4.10 Tensor of Angular Rate of Rotation (Spin) 88
2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients 90
2.4.12 Trajectory of a Material Point, Streamlineand Vortex Line 98
2.4.13 Stream Tubes and Vortex Tubes 100
2.5 Co-rotational Derivatives 102
2.5.1 Definition of Co-rotational Derivatives 102
2.5.2 The Oldroyd Derivative (hi = ri) 104
2.5.3 The Cotter--Rivlin Derivative (hi = ri) 105
2.5.4 Mixed Co-rotational Derivatives 106
2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor 106
2.5.6 The Derivative in the Eigenbasis (hi =pi) of the Left Stretch Tensor 107
2.5.7 The Jaumann Derivative (hi = qi) 108
2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis 108
2.5.9 Spin Derivative 109
2.5.10 Universal Form of the Co-rotational Derivatives 110
2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient 110
Chapter 3:Balance Laws 114
3.1 The Mass Conservation Law 114
3.1.1 Integral and Differential Forms 114
3.1.2 The Continuity Equation in Lagrangian Variables 115
3.1.3 Differentiation of Integral over a Moving Volume 116
3.1.4 The Continuity Equation in Eulerian Variables 117
3.1.5 Determination of the Total Derivatives with respect to Time 118
3.1.6 The Gauss--Ostrogradskii Formulae 119
3.2 The Momentum Balance Law and the Stress Tensor 120
3.2.1 The Momentum Balance Law 120
3.2.2 External and Internal Forces 122
3.2.3 Cauchy's Theorems on Properties of the Stress Vector 123
3.2.4 Generalized Cauchy's Theorem 126
3.2.5 The Cauchy and Piola--Kirchhoff Stress Tensors 127
3.2.6 Physical Meaning of Components of the Cauchy Stress Tensor 128
3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions 132
3.3 The Angular Momentum Balance Law 134
3.3.1 The Integral Form 134
3.3.2 Tensor of Moment Stresses 135
3.3.3 Differential Form of the Angular Momentum Balance Law 136
3.3.4 Nonpolar and Polar Continua 137
3.3.5 The Angular Momentum Balance Equation in the Material Description 138
3.4 The First Thermodynamic Law 139
3.4.1 The Integral Form of the Energy Balance Law 139
3.4.2 The Heat Flux Vector 141
3.4.3 The Energy Balance Equation 142
3.4.4 Kinetic Energy and Heat Influx Equation 143
3.4.5 The Energy Balance Equation in LagrangianDescription 144
3.4.6 The Energy Balance Law for Polar Continua 146
3.5 The Second Thermodynamic Law 149
3.5.1 The Integral Form 149
3.5.2 Differential Form of the Second Thermodynamic Law 151
3.5.3 The Second Thermodynamic Law in the Material Description 152
3.5.4 Heat Machines and Their Efficiency 153
3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles 157
3.5.6 Truesdell's Theorem 161
3.6 Deformation Compatibility Equations 166
3.6.1 Compatibility Conditions 166
3.6.2 Integrability Condition for Differential Form 167
3.6.3 The First Form of Deformation CompatibilityConditions 167
3.6.4 The Second Form of Compatibility Conditions 168
3.6.5 The Third Form of Compatibility Conditions 170
3.6.6 Properties of Components of the Riemann--Christoffel Tensor 171
3.6.7 Interchange of the Second Covariant Derivatives 173
3.6.8 The Static Compatibility Equation 173
3.7 Dynamic Compatibility Equations 174
3.7.1 Dynamic Compatibility Equations in Lagrangian Description 174
3.7.2 Dynamic Compatibility Equations in Spatial Description 176
3.8 Compatibility Equations for Deformation Rates 177
3.9 The Complete System of Continuum Mechanics Laws 180
3.9.1 The Complete System in Eulerian Description 180
3.9.2 The Complete System in Lagrangian Description 181
3.9.3 Integral Form of the System of Continuum Mechanics Laws 182
Chapter 4:Constitutive Equations 185
4.1 Basic Principles for Derivation of Constitutive Equations 185
4.2 Energetic and Quasienergetic Couples of Tensors 186
4.2.1 Energetic Couples of Tensors 186
4.2.2 The First Energetic Couple (TI,) 188
4.2.3 The Fifth Energetic Couple (TV , C) 189
4.2.4 The Fourth Energetic Couple (TIV, (U -E)) 190
4.2.5 The Second Energetic Couple (TII, (E-U-1)) 191
4.2.6 The Third Energetic Couple (TIII, B) 191
4.2.7 General Representations for Energetic Tensors of Stresses and Deformations 192
4.2.8 Energetic Deformation Measures 197
4.2.9 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors 199
4.2.10 Quasienergetic Couples of Stress and Deformation Tensors 200
4.2.11 The First Quasienergetic Couple (SI, A) 201
4.2.12 The Second Quasienergetic Couple (SII, (E -V-1)) 202
4.2.13 The Third Quasienergetic Couple (Y, TS) 202
4.2.14 The Fourth Quasienergetic Couple (SIV, (V-E)) 203
4.2.15 The Fifth Quasienergetic Couple (SV, J) 204
4.2.16 General Representation of Quasienergetic Tensors 204
4.2.17 Quasienergetic Deformation Measures 206
4.2.18 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors 207
4.2.19 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors 209
4.2.20 The Generalized Form of Representation of the Stress Power 210
4.2.21 Representation of Stress Power in Terms of Co-rotational Derivatives 211
4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient 212
4.3 The Principal Thermodynamic Identity 220
4.3.1 Different Forms of the Principal Thermodynamic Identity 220
4.3.2 The Clausius--Duhem Inequality 222
4.3.3 The Helmholtz Free Energy 222
4.3.4 The Gibbs Free Energy 223
4.3.5 Enthalpy 225
4.3.6 Universal Form of the Principal Thermodynamic Identity 226
4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives 227
4.4 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action 229
4.4.1 Active and Reactive Variables 229
4.4.2 The Principle of Thermodynamically Consistent Determinism 230
4.4.3 The Principle of Equipresence 232
4.4.4 The Principle of Local Action 232
4.5 Definition of Ideal Continua 233
4.5.1 Classification of Types of Continua 233
4.5.2 General Form of Constitutive Equations for Ideal Continua 234
4.6 The Principle of Material Symmetry 237
4.6.1 Different Reference Configurations 237
4.6.2 H-indifferent and H-invariant Tensors 239
4.6.3 Symmetry Groups of Continua 243
4.6.4 The Statement of the Principle of Material Symmetry 244
4.7 Definition of Fluids and Solids 245
4.7.1 Fluids and Solids 245
4.7.2 Isomeric Symmetry Groups 246
4.7.3 Definition of Anisotropic Solids 250
4.7.4 H-indifference and H-invariance of Tensors Describing the Motion of a Solid 252
4.7.5 H-invariance of Rate Characteristics of a Solid 255
4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua 260
4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids 260
4.8.2 Scalar Indifferent Functions of Tensor Argument 261
4.8.3 Producing Tensors of Groups 263
4.8.4 Scalar Invariants of a Second-Order Tensor 264
4.8.5 Representation of a Scalar Indifferent Function in Terms of Invariants 267
4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua 268
4.8.7 Quasilinear and Linear Models An of Elastic Continua 273
4.8.8 Constitutive Equations for Models Bn of Elastic Continua 277
4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua 278
4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua 286
4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases 289
4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua `in Rates' 295
4.8.13 Application of the Principle of Material Symmetry to Fluids 300
4.8.14 Functional Energetic Couples of Tensors 307
4.9 Incompressible Continua 311
4.9.1 Definition of Incompressible Continua 311
4.9.2 The Principal Thermodynamic Identity for Incompressible Continua 312
4.9.3 Constitutive Equations for Ideal Incompressible Continua 313
4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids 315
4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases 316
4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids 319
4.9.7 Linear Models of Ideal Incompressible Elastic Continua 320
4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis 322
4.10 The Principle of Material Indifference 324
4.10.1 Rigid Motion 324
4.10.2 R-indifferent and R-invariant Tensors 325
4.10.3 Density and Deformation Gradient in Rigid Motion 326
4.10.4 Deformation Tensors in Rigid Motion 327
4.10.5 Stress Tensors in Rigid Motion 328
4.10.6 The Velocity in Rigid Motion 330
4.10.7 The Deformation Rate Tensorand the Vorticity Tensor in Rigid Motion 330
4.10.8 Co-rotational Derivatives in Rigid Motion 331
4.10.9 The Statement of the Principle of Material Indifference 336
4.10.10 Material Indifference of the Continuity Equation 337
4.10.11 Material Indifference for the MomentumBalance Equation 337
4.10.12 Material Indifference of the Thermodynamic Laws 340
4.10.13 Material Indifference of the Compatibility Equations 342
4.10.14 Material Indifference of Models An and Bn of Ideal Continua 343
4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua 344
4.10.16 Material Indifference for Incompressible Continua 346
4.10.17 Material Indifference for Models of Solids `in Rates' 346
4.11 Relationships in a Moving System 348
4.11.1 A Moving Reference System 348
4.11.2 The Euler Formula 350
4.11.3 The Coriolis Formula 351
4.11.4 The Nabla-Operator in a Moving System 353
4.11.5 The Velocity Gradient in a Moving System 354
4.11.6 The Continuity Equation in a Moving System 354
4.11.7 The Momentum Balance Equation in a Moving System 355
4.11.8 The Thermodynamic Laws in a Moving System 355
4.11.9 The Equation of Deformation Compatibility in a Moving System 356
4.11.10 The Kinematic Equation in a Moving System 359
4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System 359
4.11.12 Constitutive Equations in a Moving System 359
4.11.13 General Remarks 362
4.12 The Onsager Principle 363
4.12.1 The Onsager Principle and the Fourier Law 363
4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law 365
4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law 367
4.12.4 The Fourier Law for Fluids 368
4.12.5 The Fourier Law for Solids 369
Chapter 5:Relations at Singular Surfaces 371
5.1 Relations at a Singular Surface in the Material Description 371
5.1.1 Singular Surfaces 371
5.1.2 The First Classification of Singular Surfaces 371
5.1.3 Axiom on the Class of Functions across a Singular Surface 374
5.1.4 The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface 376
5.1.5 Relations at a Coherent Singular Surface in K 379
5.1.6 Relation Between Velocities of a Singular Surface in K and K 381
5.2 Relations at a Singular Surface in the Spatial Description 382
5.2.1 Relations at a Coherent Singular Surface in the Spatial Description 382
5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface 384
5.3 Explicit Form of Relations at a Singular Surface 386
5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration 386
5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration 387
5.3.3 Mass Rate of Propagation of a Singular Surface 387
5.3.4 Relations at a Singular Surface Without Transition of Material Points 389
5.4 The Main Types of Singular Surfaces 390
5.4.1 Jump of Density 390
5.4.2 Jumps of Radius-Vector and Displacement Vector 391
5.4.3 Semicoherent and Completely Incoherent Singular Surfaces 393
5.4.4 Nondissipative and Homothermal Singular Surfaces 393
5.4.5 Surfaces with Ideal Contact 394
5.4.6 On Boundary Conditions 396
5.4.7 Equation of a Singular Surface in K 396
5.4.8 Equation of a Singular Surface in K 398
Chapter 6:Elastic Continua at Large Deformations 400
6.1 Closed Systems in the Spatial Description 400
6.1.1 RUVF-system of Thermoelasticity 400
6.1.2 RVF-, RUV-, and UV-Systems of Dynamic Equations of Thermoelasticity 404
6.1.3 TRUVF-system of Dynamic Equations of Thermoelasticity 407
6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description 408
6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations 411
6.2 Closed Systems in the Material Description 413
6.2.1 UVF-system of Dynamic Equations of Thermoelasticity in the Material Description 413
6.2.2 UV- and U-systems of Thermoelasticity in the Material Description 417
6.2.3 TUVF-system of Thermoelasticity in the Material Description 418
6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description 419
6.3 Statements of Problems for Elastic Continua at LargeDeformations 422
6.3.1 Boundary Conditions in the Spatial Description 422
6.3.2 Boundary Conditions in the Material Description 425
6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description 428
6.3.4 Statements of Thermoelasticity Problems in the Material Description 431
6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations 433
6.3.6 Conditions on External Forces in Quasistatic Problems 435
6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description 436
6.3.8 Variational Statement of Quasistatic Problem in the Material Description 440
6.3.9 Variational Statement for Incompressible Continua in the Material Description 440
6.4 The Problem on an Elastic Beam in Tension 444
6.4.1 Semi-Inverse Method 444
6.4.2 Deformation of a Beam in Tension 444
6.4.3 Stresses in a Beam 445
6.4.4 The Boundary Conditions 447
6.4.5 Resolving Relation 1 k1 447
6.4.6 Comparative Analysis of Different Models An 448
6.5 Tension of an Incompressible Beam 453
6.5.1 Deformation of an Incompressible Elastic Beam 453
6.5.2 Stresses in an Incompressible Beam for Models Bn 454
6.5.3 Resolving Relation 1(k1) 455
6.5.4 Comparative Analysis of Models Bn 455
6.5.5 Stresses in an Incompressible Beam for Models An 459
6.6 Simple Shear 461
6.6.1 Deformations in Simple Shear 461
6.6.2 Stresses in the Problem on Shear 462
6.6.3 Boundary Conditions in the Problem on Shear 464
6.6.4 Comparative Analysis of Different Models An for the Problem on Shear 465
6.6.5 Shear of an Incompressible Elastic Continuum 466
6.7 The Lamé Problem 469
6.7.1 The Motion Law for a Pipe in the Lamé Problem 469
6.7.2 The Deformation Gradient and Deformation Tensors in the Lamé Problem 471
6.7.3 Stresses in the Lamé Problem for Models An 472
6.7.4 Equation for the Function f 473
6.7.5 Boundary Conditions of the Weak Type 474
6.7.6 Boundary Conditions of the Rigid Type 476
6.8 The Lamé Problem for an Incompressible Continuum 477
6.8.1 Equation for the Function f 477
6.8.2 Stresses in the Lamé Problem for an Incompressible Continuum 477
6.8.3 Equation for Hydrostatic Pressure p 479
6.8.4 Analysis of the Problem Solution 479
Chapter 7:Continua of the Differential Type 484
7.1 Models An and Bn of Continua of the Differential Type 484
7.1.1 Constitutive Equations for Models An of Continua of the Differential Type 484
7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type 486
7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type 488
7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases 490
7.1.5 Models Bn of Solids of the Differential Type 493
7.1.6 Models Bn of Incompressible Continua of the Differential Type 494
7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type 495
7.2 Models An and Bn of Fluids of the Differential Type 496
7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type 496
7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type 497
7.2.3 Simultaneous Invariants for Fluids of the Differential Type 499
7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type 501
7.2.5 Viscous Coefficients in Model AV of a Fluid of the Differential Type 502
7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type 503
7.2.7 Constitutive Equations for Incompressible Viscous Fluids 504
7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type 504
7.3 Models Cn and Dn of Continua of the Differential Type 505
7.3.1 Models Cn of Continua of the Differential Type 505
7.3.2 Models Cnh of Solids with Co-rotational Derivatives 507
7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids 508
7.3.4 Viscosity Tensor in Models Cnh 511
7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids 512
7.3.6 Models Dnh of Isotropic Solids 514
7.4 The Problem on a Beam in Tension 515
7.4.1 Rate Characteristics of a Beam 515
7.4.2 Stresses in the Beam 515
7.4.3 Resolving Relation (k1, 1) 516
7.4.4 Comparative Analysis of Creep Curves for Different Models Bn 516
7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type 519
Chapter 8:Viscoelastic Continua at Large Deformations 520
8.1 Viscoelastic Continua of the Integral Type 520
8.1.1 Definition of Viscoelastic Continua 520
8.1.2 Tensor Functional Space 521
8.1.3 Continuous and Differentiable Functionals 522
8.1.4 Axiom of Fading Memory 526
8.1.5 Models An of Viscoelastic Continua 528
8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua 529
8.1.7 General Representation of Functional of Free Energy in Models An 530
8.1.8 Model An of Stable Viscoelastic Continua 533
8.1.9 Model An of a Viscoelastic Continuum with Difference Cores 534
8.1.10 Model An of a Thermoviscoelastic Continuum 536
8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum 537
8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua 539
8.2.1 Principal Models An of Viscoelastic Continua 539
8.2.2 Principal Model An of an Isotropic Thermoviscoelastic Continuum 541
8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum 542
8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum 543
8.2.5 Quadratic Models An of Thermoviscoelastic Continua 545
8.2.6 Linear Models An of Viscoelastic Continua 545
8.2.7 Representation of Linear Models An in the Boltzmann Form 548
8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua 551
8.2.9 Linear Models An for Isotropic Viscoelastic Continua 553
8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua 554
8.2.11 Linear Models An of Orthotropic Viscoelastic Continua 555
8.2.12 The Tensor of Relaxation Functions 556
8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua 558
8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations 562
8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua 565
8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids 572
8.3.1 Models An of Incompressible Viscoelastic Continua 572
8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua 572
8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua 574
8.3.4 Models Bn of Viscoelastic Continua 575
8.3.5 Models An and Bn of Viscoelastic Fluids 577
8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua 579
8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations 581
8.4.1 Statements of Dynamic Problems in the Spatial Description 581
8.4.2 Statements of Dynamic Problems in the Material Description 585
8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description 587
8.4.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description 589
8.5 The Problem on Uniaxial Deforming of a Viscoelastic Beam 591
8.5.1 Deformation of a Viscoelastic Beam in Uniaxial Tension 591
8.5.2 Viscous Stresses in Uniaxial Tension 592
8.5.3 Stresses in a Viscoelastic Beam in Tension 592
8.5.4 Resolving Relation 1(k1) for a Viscoelastic Beam 593
8.5.5 Method of Calculating the Constants B() and () 594
8.5.6 Method for Evaluating the Constants , l1, l2 and , m 597
8.5.7 Computations of Relaxation Curves 598
8.5.8 Cyclic Deforming of a Beam 600
8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming 602
8.6.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming 602
8.6.2 Fast and Slow Times in Multicycle Deforming 603
8.6.3 Differentiation and Integration of Quasiperiodic Functions 603
8.6.4 Heat Conduction Equation for a Thin Viscoelastic Beam 604
8.6.5 Dissipation Function for a Viscoelastic Beam 605
8.6.6 Asymptotic Expansion in Terms of a Small Parameter 606
8.6.7 Averaged Heat Conduction Equation 607
8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle 608
8.6.9 Regimes of Dissipative Heating Without Heat Removal 608
8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal 609
8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies 611
Chapter 9:Plastic Continua at Large Deformations 613
9.1 Models An of Plastic Continua at Large Deformations 613
9.1.1 Main Assumptions of the Models 613
9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua 616
9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua 619
9.1.4 Models An of Plastic Yield 620
9.1.5 Associated Model of Plasticity An 621
9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity 625
9.1.7 Associated Models of Plasticity An for Isotropic Continua 627
9.1.8 The Huber--Mises Model for Isotropic Plastic Continua 629
9.1.9 Associated Models of Plasticity An for Transversely Isotropic Continua 632
9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum 634
9.1.11 Associated Models of Plasticity An for Orthotropic Continua 636
9.1.12 The Orthotropic Unipotential Huber--Mises Model for Plastic Continua 638
9.1.13 The Principle of Material Indifference for Models An of Plastic Continua 640
9.2 Models Bn of Plastic Continua 645
9.2.1 Representation of Stress Power for Models Bn of Plastic Continua 645
9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua 649
9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua 651
9.2.4 Associated Models Bn of Plastic Continua 653
9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity 654
9.2.6 Associated Models of Plasticity Bn with Proper Strengthening 657
9.2.7 Associated Models of Plasticity Bn for Isotropic Continua 657
9.2.8 Associated Models of Plasticity Bn for Transversely Isotropic Continua 659
9.2.9 Associated Models of Plasticity Bn for Orthotropic Continua 660
9.2.10 The Principle of Material Indifference for Models Bn of Plastic Continua 661
9.3 Models Cn and Dn of Plastic Continua 662
9.3.1 General Representation of Constitutive Equations for Models Cn of Plastic Continua 662
9.3.2 Constitutive Equations for Models Cn of Isotropic Plastic Continua 665
9.3.3 General Representation of Constitutive Equations for Models Dn of Plastic Continua 668
9.3.4 Constitutive Equations for Models Dn of Isotropic Plastic Continua 672
9.3.5 The Principles of Material Symmetry and Material Indifference for Models Cn and Dn 673
9.4 Constitutive Equations of Plasticity Theory `in Rates' 674
9.4.1 Representation of Models An of Plastic Continua `in Rates' 674
9.5 Statements of Problems in Plasticity Theory 677
9.5.1 Statements of Dynamic Problems for Models An of Plasticity 677
9.5.2 Statements of Quasistatic Problems for Models An of Plasticity 679
9.6 The Problem on All-Round Tension--Compression of a Plastic Continuum 681
9.6.1 Deformation in All-Round Tension--Compression 681
9.6.2 Stresses in All-Round Tension--Compression 682
9.6.3 The Case of a Plastically Incompressible Continuum 683
9.6.4 The Case of a Plastically Compressible Continuum 684
9.6.5 Cyclic Loading of a Plastically Compressible Continuum 686
9.7 The Problem on Tension of a Plastic Beam 688
9.7.1 Deformation of a Beam in Uniaxial Tension 688
9.7.2 Stresses in a Plastic Beam 689
9.7.3 Plastic Deformations of a Beam 690
9.7.4 Change of the Density 692
9.7.5 Resolving Equation for the Problem 693
9.7.6 Numerical Method for the Resolving Equation 694
9.7.7 Method for Determination of Constants H0, n0, and s 697
9.7.8 Comparison with Experimental Data for Alloys 698
9.7.9 Comparison with Experimental Data for Grounds 699
9.8 Plane Waves in Plastic Continua 701
9.8.1 Formulation of the Problem 701
9.8.2 The Motion Law and Deformation of a Plate 702
9.8.3 Stresses in the Plate 703
9.8.4 The System of Dynamic Equations for the Plane Problem 704
9.8.5 The Statement of Problem on Plane Waves in Plastic Continua 706
9.8.6 Solving the Problem by the Characteristic Method 707
9.8.7 Comparative Analysis of the Solution for Different Models An 711
9.8.8 Plane Waves in Models AIV and AV 713
9.8.9 Shock Waves in Models AI and AII 715
9.8.10 Shock Adiabatic Curves for Models AI and AII 717
9.8.11 Shock Adiabatic Curves at a Given Rate of Impact 719
9.9 Models of Viscoplastic Continua 721
9.9.1 The Concept of a Viscoplastic Continuum 721
9.9.2 Model An of Viscoplastic Continua of the Differential Type 721
9.9.3 Model of Isotropic Viscoplastic Continua of the Differential Type 723
9.9.4 General Model An of Viscoplastic Continua 724
9.9.5 Model An of Isotropic Viscoplastic Continua 725
9.9.6 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type 725
References 729
Basic Notation 729
Index 734