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ОбложкаGurtin M.E. The mechanics and thermodynamics of continua / M.E.Gurtin, E.Fried, L.Anand. - New York: Cambridge University Press, 2010 (2011). - xxi, 694 p.: ill. - Ref.: p.671-681. - Ind.: p.683-694. - ISBN 978-0-521-40598-0
Шифр: (И/В25-G95) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
Preface ....................................................... xix

PART I. VECTOR AND TENSOR ALGEBRA ............................... 1
1  Vector Algebra ............................................... 3
   1.1  Inner Product. Cross Product ............................ 3
   1.2  Cartesian Coordinate Frames ............................. 6
   1.3  Summation Convention. Components of a Vector and
        a Point ................................................. 6
2  Tensor Algebra ............................................... 9
   2.1  What Is a Tensor? ....................................... 9
   2.2  Zero and Identity Tensors. Tensor Product of Two
        Vectors. Projection Tensor. Spherical Tensor ........... 10
   2.3  Components of a Tensor ................................. 11
   2.4  Transpose of & Tensor. Symmetric and Skew Tensors ...... 12
   2.5  Product of Tensors ..................................... 13
   2.6  Vector Cross. Axial Vector of a Skew Tensor ............ 15
   2.7  Trace of a Tensor. Deviatoric Tensors .................. 16
   2.8  Inner Product of Tensors. Magnitude of a Tensor ........ 17
   2.9  Invertible Tensors ..................................... 19
   2.10 Determinant of a Tensor ................................ 21
   2.11 Cofactor of a Tensor ................................... 22
   2.12 Qrthogonal Tensors ..................................... 25
   2.13 Matrix of a Tensor ..................................... 26
   2.14 Eigenvalues and Eigenvectors of a Tensor. Spectral
        Theorem ................................................ 28
   2.15 Square Root of a Symmetric, Positive-Definite Tensor.
        Polar Decomposition Theorem ............................ 31
   2.16 Principal Invariants of a Tensor. Cayley-Hamilton
        Equation ............................................... 35

PART II. VECTOR AND TENSOR ANALYSIS ............................ 39
3  Differentiation ............................................. 41
   3.1  Differentiation of Functions of a Scalar ............... 41
   3.2  Differentiation of Fields. Gradient .................... 43
   3.3  Divergence and Curl. Vector and Tensor Identities ...... 46
   3.4  Differentiation of a Scalar Function of a Tensor ....... 49
4  Integral Theorems ........................................... 52
   4.1  The Divergence Theorem ................................. 52
   4.2  Line Integrals. Stokes' Theorem ........................ 53

PART III. KINEMATICS ........................................... 59
5  Motion of a Body ............................................ 61
   5.1  Reference Body. Material Points ........................ 61
   5.2  Basic Quantities Associated with the Motion of a Body .. 61
   5.3  Convection of Sets with the Body ....................... 63
6  The Deformation Gradient .................................... 64
   6.1  Approximation of a Deformation by a Homogeneous
        Deformation ............................................ 64
        6.1.1  Homogeneous Deformations ........................ 64
        6.1.2  General Deformations ............................ 65
   6.2  Convection of Geometric Quantities ..................... 66
        6.2.1  Infinitesimal Fibers ............................ 66
        6.2.2  Curves .......................................... 67
        6.2.3  Tangent Vectors ................................. 67
        6.2.4  Bases ........................................... 68
7  Stretch, Strain, and Rotation ............................... 69
   7.1  Stretch and Rotation Tensors. Strain ................... 69
   7.2  Fibers. Properties of the Tensors U and С .............. 70
        7.2.1  Infinitesimal Fibers ............................ 70
        7.2.2  Finite Fibers ................................... 71
   7.3  Principal Stretches and Principal Directions ........... 73
8  Deformation of Volume and Area .............................. 75
   8.1  Deformation of Normals ................................. 75
   8.2  Deformation of Volume .................................. 76
   8.3  Deformation of Area .................................... 77
9  Material and Spatial Descriptions of Fields ................. 80
   9.1  Gradient, Divergence, and Curl ......................... 80
   9.2  Material and Spatial Time Derivatives .................. 81
   9.3  Velocity Gradient ...................................... 82
   9.4  'Commutator Identities ................................. 84
   9.5  Particle Paths ......................................... 85
   9.6  Stretching of Deformed Fibers .......................... 85
10 Special Motions ............................................. 86
   10.1 Rigid Motions .......................................... 86
   10.2 Motions Whose Velocity Gradient is Symmetric and
        Spatially Constant ..................................... 87
11 Stretching and Spin in an Arbitrary Motion .................. 89
   11.1 Stretching and Spin as Tensor Fields ................... 89
   11.2 Properties of D ........................................ 90
   11.3 Stretching and Spin Using the Current Configuration
        as Reference ........................................... 92
12 Material and Spatial Tensor Fields. Pullback and
   Pushforward Operations ...................................... 95
   12.1 Material and Spatial Tensor Fields ..................... 95
   12.2 Pullback and Pushforward Operations .................... 95
13 Modes of Evolution for Vector and Tensor Fields ............. 98
   13.1 Vector and Tensor Fields That Convect With the Body .... 98
        13.1.1 Vector Fields That Convect as Tangents .......... 98
        13.1.2 Vector Fields That Convect as Normals ........... 99
        13.1.3 Tangentially Convecting Basis and Its Dual
               Basis. Covariant and Contravariant Components
               of Spatial Fields ............................... 99
        13.1.4 Covariant and Contravariant Convection of
               Tensor Fields .................................. 102
   13.2 Corotational Vector and Tensor Fields ................. 105
14 Motions with Constant Velocity Gradient .................... 107
   14.1 Motions ............................................... 107
15 Material and Spatial Integration ........................... 109
   15.1 Line Integrals ........................................ 109
   15.2 Volume and Surface Integrals .......................... 109
        15.2.1 Volume Integrals ............................... 110
        15.2.2 Surface Integrals .............................. 111
   15.3 Localization of Integrals ............................. 111
   16 Reynolds' Transport Relation. Isochoric Motions ......... 113
17 More Kinematics ............................................ 115
   17.1 Vorticity ............................................. 115
   17.2 Transport Relations for Spin and Vorticity ............ 115
   17.3 Irrotational Motions .................................. 117
   17.4 Circulation ........................................... 118
   17.5 Vortex Lines .......................................... 120
   17.6 Steady Motions ........................................ 121
   17.7 A Class of Natural Reference Configurations for
        Fluids ................................................ 122
   17.8 The Motion Problem .................................... 122
        17.8.1 Kinematical Boundary Conditions ................ 122
        17.8.2 The Motion Problem in a Fixed Container ........ 123
        17.8.3 The Motion Problem in All of Space. Solution
               with Constant Velocity Gradient ................ 123

PART IV. BASIC MECHANICAL PRINCIPLES .......................... 125
18 Balance of Mass ............................................ 127
   18.1 Global Form of Balance of Mass ........................ 127
   18.2 Local Forms of Balance of Mass ........................ 128
   18.3 Simple Consequences of Mass Balance ................... 129
19 Forces and Moments. Balance Laws for Linear and Angular
   Momentum ................................................... 131
   19.1 Inertial Frames. Linear and Angular Momentum .......... 131
   19.2 Surface Tractions. Body Forces ........................ 132
   19.3 Balance Laws for Linear and Angular Momentum .......... 134
   19.4 Balance of Forces and Moments Based on the
        Generalized Body Force ................................ 136
   19.5 Cauchy's Theorem for the Existence of Stress .......... 137
   19.6 Local Forms of the Force and Moment Balances .......... 139
   19.7 Kinetic Energy. Conventional and Generalized External
        Power Expenditures .................................... 141
        19.7.1 Conventional Form of the External Power ........ 142
        19.7.2 Kinetic Energy and Inertial Power .............. 142
        19.7.3 Generalized Power Balance ...................... 143
        19.7.4 The Assumption of Negligible Inertial Forces ... 144
20 Frames of Reference ........................................ 146
   20.1 Changes of Frame ...................................... 146
   20.2 Frame-Indifferent Fields .............................. 147
   20.3 Transformation Rules for Kinematic Fields ............. 148
        20.3.1 Material Time-Derivatives of Frame-
               Indifferent Tensor Fields are Not Frame-
               Indifferent .................................... 151
        20.3.2 The Corotational, Covariant, and
               Contravariant Rates of a Tensor Field .......... 151
        20.3.3 Other Relations for the Corotational Rate ...... 152
        20.3.4 Other Relations for the Covariant Rate ......... 153
        20.3.5 Other Relations for the Contravariant Rate ..... 154
        20.3.6 General Tensorial Rate ......................... 155
21 Frame-Indifference Principle ............................... 157
   21.1 Transformation Rules for Stress and Body Force ........ 157
   21.2 Inertial Body Force in a Frame That Is Not Inertial ... 159
22 Alternative Formulations of the Force and Moment Balances .. 161
   22.1 Force and Moment Balances as a Consequence of
        Frame-Indifference of the Expended Power .............. 161
   22.2 Principle of Virtual Power ............................ 163
        22.2.1 Application to Boundary-Value Problems ......... 165
        22.2.2 Fundamental Lemma of the Calculus of
               Variations ..................................... 167
23 Mechanical Laws for a Spatial Control Volume ............... 168
   23.1 Mass Balance for a Control Volume .....................
   23.2 Momentum Balances for a Control Volume ................ 169
24 Referential Forms for the Mechanical Laws .................. 173
   24.1 Piola Stress. Force and Moment Balances ............... 173
   24.2 Expended Power ........................................ 175
25 Further Discussion of Stress ............................... 177
   25.1 Power-Conjugate Pairings. Second Piola Stress ......... 177
   25.2 Transformation Laws for the Piola Stresses ............ 178

PART V. BASIC THERMODYNAMICAL PRINCIPLES ...................... 181
26 The First Law: Balance of Energy ........................... 183
   26.1 Global and Local Forms of Energy Balance .............. 184
   26.2 Terminology for "Extensive" Quantities ................ 185
27 The Second Law: Nonnegative Production of Entropy .......... 186
   27.1 Global Form of the Entropy Imbalance .................. 187
   27.2 Temperature and the Entropy Imbalance ................. 187
   27.3 Free-Energy Imbalance. Dissipation .................... 188
28 General Theorems ........................................... 190
   28.1 Invariant Nature of the First Two Laws ................ 190
   28.2 Decay Inequalities for the Body Under Passive
        Boundary Conditions ................................... 191
        28.2.1 Isolated Body .................................. 191
        28.2.2 Boundary Essentially at Constant Pressure and
               Temperature .................................... 192
29 A Free-Energy Imbalance for Mechanical Theories ............ 194
   29.1 Free-Energy Imbalance. Dissipation .................... 194
   29.2 Digression: Role of the Free-Energy Imbalance within
        the General Thermodynamic Framework ................... 195
   29.3 Decay Inequalities .................................... 196
30 The First Two Laws for a Spatial Control Volume ............ 197
31 The First Two Laws Expressed Referentially ................. 199
   31.1 Global Forms of the First Two Laws .................... 200
   31.2 Local Forms of the First Two Laws ..................... 201
   31.3 Decay Inequalities for the Body Under Passive
        Boundary Conditions ................................... 202
   31.4 Mechanical Theory: Free-Energy Imbalance .............. 204

PART VI. MECHANICAL AND THERM OD YNAMICAL LAWS AT A SHOCK
WAVE .......................................................... 207
32 Shock Wave Kinematics ...................................... 209
   32.1 Notation. Terminology ................................. 209
   32.2 Hadamard's Compatibility Conditions ................... 210
   32.3 Relation Between the Scalar Normal Velocities VR and
        V ..................................................... 212
   32.4 Transport Relations in the Presence of a Shock Wave ... 212
   32.5 The Divergence Theorem in the Presence of a Shock
        Wave .................................................. 215
33 Basic Laws at a Shock Wave: Jump Conditions ................ 216
   33.1 Balance of Mass and Momentum .......................... 216
   33.2 Balance of Energy and the Entropy Imbalance ........... 218

PART VII. INTERLUDE: BASIC HYPOTHESES FOR DEVELOPING
PHYSICALLY MEANINGFUL CONSTITUTIVE THEORIES ................... 221
34 General Considerations ..................................... 223
35 Constitutive Response Functions ............................ 224
36 Frame-Indifference and Compatibility with Thermodynamics ... 225

PART VIII. RIGID HEAT CONDUCTORS .............................. 227
37 Basic Laws ................................................. 229
38 General Constitutive Equations ............................. 230
39 Thermodynamics and Constitutive Restrictions: The
   Coleman-Noll Procedure ..................................... 232
40 Consequences of the State Restrictions ..................... 234
41 Consequences of the Heat-Conduction Inequality ............. 236
42 Fourier's Law .............................................. 237

PART IX. THE MECHANICAL THEORY OF COMPRESSIBLE AND
INCOMPRESSIBLE FLUIDS ......................................... 239
43 Brief Review ............................................... 241
   43.1 Basic Kinematical Relations ........................... 241
   43.2 Basic Laws ............................................ 241
   43.3 Transformation Rules and Objective Rates .............. 242
44 Elastic Fluids ............................................. 244
   44.1 Constitutive Theory ................................... 244
   44.2 Consequences of Frame-Indifference .................... 244
   44.3 Consequences of Thermodynamics ........................ 245
   44.4 Evolution Equations ................................... 246
45 Compressible, Viscous Fluids ............................... 250
   45.1 General Constitutive Equations ........................ 250
   45.2 Consequences of Frame-Indifference .................... 251
   45.3 Consequences of Thermodynamics ........................ 253
   45.4 Compressible, Linearly Viscous Fluids ................. 255
   45.5 Compressible Navier-Stokes Equations .................. 256
   45.6 Vorticity Transport Equation .......................... 256
46 Incompressible Fluids ...................................... 259
   46.1 Free-Energy Imbalance for an Incompressible Body ...... 259
   46.2 Incompressible, Viscous Fluids ........................ 260
   46.3 Incompressible, Linearly Viscous Fluids ............... 261
   46.4 Incompressible Navier-Stokes Equations ................ 262
   46.5 Circulation. Vorticity-Transport Equation ............. 263
   46.6 Pressure Poisson Equation ............................. 265
   46.7 Transport Equations for the Velocity Gradient,
        Stretching, and Spin in a Linearly Viscous,
        Incompressible Fluid .................................. 265
   46.8 Impetus-Gauge Formulation of the Navier-Stokes
        Equations ............................................. 267
   46.9 Perfect Fluids ........................................ 268

PART X. MECHANICAL THEORY OF ELASTIC SOLIDS ................... 271
47 Brief Review ............................................... 273
   47.1 Kinematical Relations ................................. 273
   47.2 Basic Laws ............................................ 273
   47.3 Transformation Laws Under a Change in Frame ........... 274
48 Constitutive Theory ........................................ 276
   48.1 Consequences of Frame-Indifference .................... 276
   48.2 Thermodynamic Restrictions ............................ 278
        48.2.1 The Stress Relation ............................ 278
        48.2.2 Consequences of the Stress Relation ............ 280
        48.2.3 Natural Reference Configuration ................ 280
49 Summary of Basic Equations. Initial/Boundary-Value
   Problems ................................................... 282
   49.1 Basic Field Equations ................................. 282
   49.2 A Typical Initial/Boundary-Value Problem .............. 283
50 Material Symmetry .......................................... 284
   50.1 The Notion of a Group. Invariance Under a Group ....... 284
   50.2 The Symmetry Group Q .................................. 285
        50.2.1 Proof That Q Is a Group ........................ 288
   50.3 Isotropy .............................................. 288
        50.3.1 Free Energy Expressed in Terms of Invariants ... 290
        50.3.2 Free Energy Expressed in Terms of Principal
               stretches ...................................... 292
51 Simple Shear of a Homogeneous, Isotropic Elastic Body ...... 294
52 The Linear Theory of Elasticity ............................ 297
   52.1 Small Deformations .................................... 297
   52.2 The Stress-Strain Law for Small Deformations .......... 298
        52.2.1 The Elasticity Tensor .......................... 298
        52.2.2 The Compliance Tensor .......................... 300
        52.2.3 Estimates for the Stress and Free Energy ....... 300
   52.3 Basic Equations of the Linear Theory of Elasticity .... 302
   52.4 Special Forms for the Elasticity Tensor ............... 302
        52.4.1 Isotropic Material ............................. 303
        52.4.2 CubicCrystal ................................... 304
   52.5 Basic Equations of the Linear theory of Elasticity
        for an Isotropic Material ............................. 306
        52.5.1 Statical Equations ............................. 307
   52.6 Some Simple Statical Solutions ........................ 309
   52.7 Boundary-Value Problems ............................... 310
        52.7.1 Elastostatics .................................. 310
        52.7.2 Elastodynamics ................................. 313
   52.8 Sinusoidal Progressive Waves .......................... 313
53 Digression: Incompressibility .............................. 316
   53.1 Kinematics of Incompressibility ....................... 316
   53.2 Indeterminacy of the Pressure. Free-Energy Imbalance .. 317
   53.3 Changes in Frame ...................................... 318
54 Incompressible Elastic Materials ........................... 319
   54.1 Constitutive Theory ................................... 319
        54.1.1 Consequences of Frame-Indifference ............. 319
        54.1.2 Domain of Definition of the Response
               Functions ...................................... 320
        54.1.3 Thermodynamic Restrictions ..................... 321
   54.2 Incompressible Isotropic Elastic Bodies ............... 323
   54.3 Simple Shear of a Homogeneous, Isotropic,
        Incompressible Elastic Body ........................... 324
55 Approximately Incompressible Elastic Materials ............. 326

PART XI. THERMOELASTICITY ..................................... 331
56 Brief Review ............................................... 333
   56.1 Kinematical Relations ................................. 333
   56.2 Basic Laws ............................................ 333
57 Constitutive Theory ........................................ 335
   57.1 Consequences of Frame-Indifference .................... 335
   57.2 Thermodynamic Restrictions ............................ 336
   57.3 Consequences of the Thermodynamic Restrictions ........ 338
        57.3.1 Consequences of the State Relations ............ 338
        57.3.2 Consequences of the Heat-Conduction
               Inequality ..................................... 339
   57.4 Elasticity Tensor. Stress-Temperature Modulus. Heat
        Capacity .............................................. 341
   57.5 The Basic Thermoelastic Field Equations ............... 342
   57.6 Entropy as Independent Variable. Nonconductors ........ 343
   57.7 Nonconductors ......................................... 346
   57.8 Material Symmetry ..................................... 346
58 Natural Reference Configuration for a Given Temperature .... 348
   58.1 Asymptotic Stability and its Consequences. The
        Gibbs Function ........................................ 348
   58.2 Local Relations at a Reference Configuration that is
        Natural for a Temperature ν0 .......................... 349
59 Linear Thermoelasticity .................................... 354
   59.1 Approximate Constitutive Equations for the Stress
        and Entropy ........................................... 354
   59.2 Basic Field Equations of Linear Thermoelasticity ...... 356
   59.3 Isotropic linear Thermoelasticity ..................... 356

PART XII. SPECIES DIFFUSION COUPLED TO ELASTICITY ............. 361
60 Balance Laws for Forces, Moments, and the Conventional
   External Power ............................................. 363
61 Mass Balance for a Single Diffusing Species ................ 364
62 Free-Energy Imbalance Revisited. Chemical Potential ........ 366
63 Multiple Species ........................................... 369
   63.1 Species Mass Balances ................................. 369
   63.2 Free-Energy Imbalance ................................. 370
64 Digression: The Thermodynamic Laws in the Presence of
   Species Transport .......................................... 371
65 Referential Laws ........................................... 374
   65.1 Single Species ........................................ 374
   65.2 Multiple Species ...................................... 376
66 Constitutive Theory for a Single Species ................... 377
   66.1 Consequences of Frame-Indifference .................... 377
   66.2 Thermodynamic Restrictions ............................ 378
   66.3 Consequences of the Thermodynamic Restrictions ........ 380
   66.4 Fick's Law ............................................ 382
67 Material Symmetry .......................................... 385
68 Natural Reference Configuration ............................ 388
69 Summary of Basic Equations for a Single Species ............ 390
70 Constitutive Theory for Multiple Species ................... 391
   70.1 Consequences of Frame-Indifference and
        Thermodynamics ........................................ 391
   70.2 Fick's Law ............................................ 393
   70.3 Natural Reference Configuration ....................... 393
71 Summary of Basic Equations for N Independent Species ....... 396
72 Substitutional Alloys ...................................... 398
   72.1 Lattice Constraint .................................... 398
   72.2 Substitutional Flux Constraint ........................ 399
   72.3 Relative Chemical Potentials. Free-Energy Imbalance ... 399
   72.4 Elimination of the Lattice Constraint. Larche-Cahn
        Differentiation ....................................... 400
   72.5 General Constitutive Equations ........................ 403
   72.6 Thermodynamic Restrictions ............................ 404
   72.7 Verification of (t) ................................... 406
   72.8 Normalization Based on the Elimination of the
        Lattice Constraint .................................... 406
73 Linearization .............................................. 408
   73.1 Approximate Constitutive Equations for the Stress,
        Chemical Potentials, and Fluxes ....................... 408
   73.2 Basic Equations of the Linear Theory .................. 410
   73.3 Isotropic Linear Theory ............................... 411

PART XIII. THEORY OF ISOTROPIC PLASTIC SOLIDS UNDERGOING
SMALL DEFORMATIONS ............................................ 415
74 Some Phenomenological Aspects of the Elastic-Plastic
   Stress-Strain Response of Polycrystalline Metals ........... 417
   74.1 Isotropic and Kinematic Strain-Hardening .............. 419
   75 Formulation of the Conventional Theory. Preliminaries ... 422
   75.1 Basic Equations ....................................... 422
   75.2 Kinematical Assumptions that Define Plasticity
        Theory ................................................ 423
   75.3 Separability Hypothesis ............................... 424
   75.4 Constitutive Characterization of Elastic Response ..... 424
76 Formulation of the Mises Theory of Plastic Flow ............ 426
   76.1 General Constitutive Equations for Plastic Flow ....... 427
   76.2 Rate-Independence ..................................... 428
   76.3 Strict Dissipativity .................................. 430
   76.4 Formulation of the Mises Flow Equations ............... 431
   76.5 Initializing the Mises Flow Equations ................. 434
        76.5.1 Flow Equations With Y(S) not Identically
               Equal to S ..................................... 434
        76.5.2 Theory with Flow Resistance as Hardening
               Variable ....................................... 435
   76.6 Solving the Hardening Equation. Accumulated Plastic
        Strain is the Most General Hardening Variable ......... 435
   76.7 Flow Resistance as Hardening Variable, Revisited ...... 439
   76.8 Yield Surface. Yield Function. Consistency Condition .. 439
   76.9 Hardening and Softening ............................... 443
77 Inversion of the Mises Flow Rule: W in Terms of Ё and T .... 445
78 Rate-Dependent Plastic Materials ........................... 449
   78.1 Background ............................................ 449
   78.2 Materials with Simple Rate-Dependence ................. 449
   78.3 Power-Law Rate-Dependence ............................. 452
79 Maximum Dissipation ........................................ 454
   79.1 Basic Definitions ..................................... 454
   79.2 Warm-up: Derivation of the Mises Flow Equations
        Based on Maximum Dissipation .......................... 456
   79.3 More General Flow Rules. Drucker's Theorem ............ 458
        79.3.1 Yield-Set Hypotheses ........................... 458
        79.3.2 Digression: Some Definitions and Results
               Concerning Convex Surfaces ..................... 460
        79.3.3 Drucker's Theorem .............................. 461
   79.4 The Conventional Theory of Perfectly Plastic
        Materials v Fits within the Framework Presented Here .. 462
80 Hardening Characterized by a Defect Energy ................. 465
   80.1 Free-Energy Imbalance Revisited ....................... 465
   80.2 Constitutive Equations. Flow Rule ..................... 466
81 The Thermodynamics of Mises-Hill Plasticity ................ 469
   81.1 Background ............................................ 469
   81.2 Thermodynamics ........................................ 470
   81.3 Constitutive Equations ................................ 470
   81.4 Nature of the Defect Energy ........................... 472
   81.5 The Flow Rule and the Boundedness Inequality .......... 473
   81.6 Balance of Energy Revisited ........................... 473
   81.7 Thermally Simple Materials ............................ 475
   81.8 Determination of the Defect Energy by the Rosakis
        Brothers, Hodowany, and Ravichandran .................. 476
   81.9 Summary of the Basic Equations ........................ 477
82 Formulation of Initial/Boundary-Value Problems for the
   Mises Flow Equations as Variational Inequalities ........... 479
   82.1 Reformulation of the Mises Flow Equations in Terms
        of Dissipation ........................................ 479
   82.2 The Global Variational Inequality ..................... 482
   82.3 Alternative Formulation of the Global Variational
        Inequality When Hardening is Described by a Defect
        Energy ................................................ 483

PART XIV. SMALL DEFORMATION, ISOTROPIC PLASTICITY BASED ON
THE PRINCIPLE OF VIRTUAL POWER ................................ 485
83 Introduction ............................................... 487
84 Conventional Theory Based on the Principle of Virtual
   Power ...................................................... 489
   84.1 General Principle of Virtual Power .................... 489
   84.2 Principle of Virtual Power Based on the
        Codirectionality Constraint ........................... 493
        84.2.1 General Principle Based on Codirectionality .... 493
        84.2.2 Streamlined Principle Based on
               Codirectionality ............................... 495
   84.3 Virtual External Forces Associated with Dislocation
        Flow .................................................. 496
   84.4 Free-Energy Imbalance ................................. 497
   84.5 Discussion of the Virtual-Power Formulation ........... 498
85 Basic Constitutive Theory .................................. 499
   86 Material Stability and Its Relation to Maximum
      Dissipation ............................................. 501

PART XV. STRAIN GRADIENT PLASTICITY BASED ON THE PRINCIPLE
OF VIRTUAL POWER .............................................. 505
87 Introduction ............................................... 507
88 Kinematics ................................................. 509
   88.1 Characterization of the Burgers Vector ................ 509
   88.2 Irrotational Plastic Flow ............................. 511
89 The Gradient Theory of Aifantis ............................ 512
   89.1 The Virtual-Power Principle of Fleck and Hutchinson ... 512
   89.2 Free-Energy Imbalance ................................. 515
   89.3 Constitutive Equations ................................ 516
   89.4 Flow Rules ............................................ 517
   89.5 Microscopically Simple Boundary Conditions ............ 518
   89.6 Variational Formulation of the Flow Rule .............. 519
   89.7 Plastic Free-Energy Balance ........................... 520
   89.8 Spatial Oscillations. Shear Bands ..................... 521
        89.8.1 Oscillations ................................... 521
        89.8.2 Single Shear Bands and Periodic Arrays of
               Shear Bands .................................... 521
90 The Gradient Theory of Gurtin and Anand .................... 524
   90.1 Third-Order Tensors ................................... 524
   90.2 Virtual-Power Formulation: Macroscopic and
        Microscopic Force Balances ............................ 525
   90.3 Free-Energy Imbalance ................................. 528
   90.4 Energetic Constitutive Equations ...................... 528
   90.5 Dissipative Constitutive Equations .................... 530
   90.6 Flow Rule ............................................. 532
   90.7 Microscopically Simple Boundary Conditions ............ 533
   90.8 Variational Formulation of the Flow Rule .............. 534
   90.9 Plastic Free-Energy Balance. Flow-Induced
        Strengthening ......................................... 535
   90.10 Rate-Independent Theory .............................. 536

PART XVI. LARGE-DEFORMATION THEORY OF ISOTROPIC PLASTIC
SOLIDS ........................................................ 539
91 Kinematics ................................................. 541
   91.1 The Kröner Decomposition .............................. 541
   91.2 Digression: Single Crystals ........................... 543
   91.3 Elastic and Plastic Stretching and Spin. Plastic
        Incompressibility ..................................... 543
   91.4 Elastic and Plastic Polar Decompositions .............. 544
   91.5 Change in Frame Revisited in View of the Kröner
        Decomposition ......................................... 546
92 Virtual-Power Formulation of the Standard and Microscopic
   Force Balances ............................................. 548
   92.1 Internal and External Expenditures of Power ........... 548
   92.2 Principle of Virtual Power ............................ 549
        92.2.1 Consequences of Frame-Indifference ............. 550
        92.2.2 Macroscopic Force Balance ...................... 551
        92.2.3 Microscopic Force Balance ...................... 551
93 Free-Energy Imbalance ...................................... 553
   93.1 Free-Energy Imbalance Expressed in Terms of the
        Cauchy Stress ......................................... 553
94 Two New Stresses ........................................... 555
   94.1 The Second Piola Elastic-Stress Te .................... 555 
   94.2 The Mandel Stress Me .................................. 556
95 Constitutive Theory ........................................ 557
   95.1 General Separable Constitutive Theory ................. 557
   95.2 Structural Frame-Indifference and the
        Characterization of Polycrystalline Materials
        Without Texture ....................................... 559
   95.3 Interaction of Elasticity and Plastic Flow ............ 562
   95.4 Consequences of Rate-Independence ..................... 563
   95.5 Derivation of the Mises Flow Equations Based on
        Maximum-Dissipation ................................... 564
96 Summary of the Basic Equations. Remarks .................... 566
97 Plastic Irrotationality: The Condition Wp = 0 .............. 567
98 Yield Surface. Yield Function. Consistency Condition ....... 569
99 |Dp| in Terms of Ė and Me .................................. 571
   99.1 Some Important Identities ............................. 571
   99.2 Conditions that Describe Loading and Unloading ........ 571
   99.3 The Inverted Flow Rule ................................ 574
   99.4 Equivalent Formulation of the Constitutive Equations
        and Plastic Mises Flow Equations Based on the
        Inverted Flow Rule .................................... 574
100 Evolution Equation for the Second Piola Stress ............ 576
101 Rate-Dependent Plastic Materials .......................... 579
   101.1 Rate-Dependent Flow Rule ............................. 579
   101.2 Inversion of the Rate-Dependent Flow Rule ............ 579
   101.3 Summary of the Complete Constitutive Theory .......... 580

PART XVII. THEORY OF SINGLE CRYSTALS UNDERGOING SMALL
DEFORMATIONS .................................................. 583
   102.1 Introduction ......................................... 583
102 Basic Single-Crystal Kinematics ........................... 586
103 The Burgers Vector and the Flow of Screw and Edge
   Dislocations ............................................... 588
   103.1 Decomposition of the Burgers Tensor G into
         Distributions of Edge and Screw Dislocations ......... 588
   103.2 Dislocation Balances ................................. 590
   103.3 The Tangential Gradient fig.2α on the Slip Plane Πα ...... 590
104 Conventional Theory of Single-Crystals .................... 593
   104.1 Virtual-Power Formulation of the Standard and
         Microscopic Force Balances ........................... 593
   104.2 Free-Energy Imbalance ................................ 596
   104.3 General Separable Constitutive Theory ................ 596
   104.4 Linear Elastic Stress-Strain Law ..................... 597
   104.5 Constitutive Equations for Flow with Simple Rate-
         Dependence ........................................... 597
   104.6 Power-Law Rate Dependence ............................ 601
   104.7 Self-Hardening, Latent-Hardening ..................... 601
   104.8 Summary of the Constitutive Theory ................... 602
105 Single-Crystal Plasticity at Small Length-Scales: A
   Small-Deformation Gradient Theory .......................... 604
   105.1 Virtual-Power Formulation of the Standard and
         Microscopic Force Balances of the Gradient Theory .... 604
   105.2 Free-Energy Imbalance ................................ 607
   105.3 Energetic Constitutive Equations. Peach-Koehler
         Forces ............................................... 607
   105.4 Constitutive Equations that Account for Dissipation .. 609
   105.5 Viscoplastic Flow Rule ............................... 612
   105.6 Microscopically Simple Boundary Conditions ........... 615
   105.7 Variational Formulation of the Flow Rule ............. 616
   105.8 Plastic Free-Energy Balance .......................... 617
   105.9 Some Remarks ......................................... 618

PART XVIII. SINGLE CRYSTALS UNDERGOING LARGE DEFORMATIONS ..... 621
106 Basic Single-Crystal Kinematics ........................... 623
107 The Burgers Vector and the Flow of Screw and Edge
   Dislocations ............................................... 626
   107.1 Transformation of Vector Area Measures Between the
         Reference, Observed, and Lattice Spaces .............. 626
   107.2 Characterization of the Burgers Vector ............... 627
   107.3 The Plastically Convected Rate of G .................. 629
   107.4 Densities of Screw and Edge Dislocations ............. 631
   107.5 Comparison of Small- and Large-Deformation Results
         Concerning Dislocation Densities ..................... 633
108 Virtual-Power Formulation of the Standard and Microscopic
   Force Balances ............................................. 634
   108.1 Internal and External Expenditures of Power .......... 634
   108.2 Consequences of Frame-Indifference ................... 636
   108.3 Macroscopic and Microscopic Force Balances ........... 637
109 Free-Energy Imbalance ..................................... 639
110 Conventional Theory ....................................... 641
   110.1 Constitutive Relations ............................... 641
   110.2 Simplified Constitutive Theory ....................... 643
   110.3 Summary of Basic Equations ........................... 644
111 Taylor's Model of Polycrystal ............................. 646
   111.1 Kinematics of a Taylor Polycrystal ................... 646
   111.2 Principle of Virtual Power ........................... 648
   111.3 Free-Energy Imbalance ................................ 651
   111.4 Constitutive Relations ............................... 651
112 Single-Crystal Plasticity at Small Length Scales: A
   Large-Deformation Gradient Theory .......................... 653
   112.1 Energetic Constitutive Equations. Peach-Koehler
         Forces ............................................... 653
   112.2 Dissipative Constitutive Equations that Account for
         Slip-Rate Gradients .................................. 654
   l|2.3 Viscoplastic Flow Rule ............................... 656
   112.4 Microscopically Simple Boundary Conditions ........... 657
   112.5 Variational Formulation .............................. 658
   112.6 Plastic Free-Energy Balance .......................... 659
   112.7 Some Remarks ......................................... 660
113 Isotropic Functions ....................................... 665
   113.1 Isotropic Scalar Functions ........................... 666
   113.2 Isotropic Tensor Functions ........................... 666
   113.3 Isotropic Linear Tensor Functions .................... 668
114 The Exponential of a Tensor ............................... 669

   References ................................................. 671
   Index ...................................................... 683

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