The Mechanics and Thermodynamics of Continua

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
The Central Thrust of This Book......Page 21
For Whom Is This Book Meant?......Page 22
Our Debt......Page 23
Part I: Vector and tensor algebra......Page 25
1.1 Inner Product. Cross Product......Page 27
1.3 Summation Convention. Components of a Vector and a Point......Page 30
2.1 What Is a Tensor?......Page 33
2.2 Zero and Identity Tensors. Tensor Product of Two Vectors. Projection Tensor. Spherical Tensor......Page 34
2.3 Components of a Tensor......Page 35
2.4 Transpose of a Tensor. Symmetric and Skew Tensors......Page 36
2.5 Product of Tensors......Page 37
2.6 Vector Cross. Axial Vector of a Skew Tensor......Page 39
2.7 Trace of a Tensor. Deviatoric Tensors......Page 40
2.8 Inner Product of Tensors. Magnitude of a Tensor......Page 41
2.9 Invertible Tensors......Page 43
2.9.1 Proof of (‡)......Page 44
2.10 Determinant of a Tensor......Page 45
2.11 Cofactor of a Tensor......Page 46
2.12 Orthogonal Tensors......Page 49
2.13 Matrix of a Tensor......Page 50
2.14 Eigenvalues and Eigenvectors of a Tensor. Spectral Theorem......Page 52
2.15 Square Root of a Symmetric, Positive-Definite Tensor. Polar Decomposition Theorem......Page 55
2.16 Principal Invariants of a Tensor. Cayley–Hamilton Equation......Page 59
Part II: Vector and tensor analysis......Page 63
3.1 Differentiation of Functions of a Scalar......Page 65
3.2 Differentiation of Fields. Gradient......Page 67
3.3 Divergence and Curl. Vector and Tensor Identities......Page 70
3.4 Differentiation of a Scalar Function of a Tensor......Page 73
4.1 The Divergence Theorem......Page 76
4.2 Line Integrals. Stokes\' Theorem......Page 77
Part III: Kinematics......Page 83
5.2 Basic Quantities Associated with the Motion of a Body......Page 85
5.3 Convection of Sets with the Body......Page 87
6.1.1 Homogeneous Deformations......Page 88
6.1.2 General Deformations......Page 89
6.2.1 Infinitesimal Fibers......Page 90
6.2.3 Tangent Vectors......Page 91
6.2.4 Bases......Page 92
7.1 Stretch and Rotation Tensors. Strain......Page 93
7.2.1 Infinitesimal Fibers......Page 94
7.2.2 Finite Fibers......Page 95
7.3 Principal Stretches and Principal Directions......Page 97
8.1 Deformation of Normals......Page 99
8.2 Deformation of Volume......Page 100
8.3 Deformation of Area......Page 101
9.1 Gradient, Divergence, and Curl......Page 104
9.2 Material and Spatial Time Derivatives......Page 105
9.3 Velocity Gradient......Page 106
9.4 Commutator Identities......Page 108
9.6 Stretching of Deformed Fibers......Page 109
10.1 Rigid Motions......Page 110
10.2 Motions Whose Velocity Gradient is Symmetric and Spatially Constant......Page 111
11.1 Stretching and Spin as Tensor Fields......Page 113
11.2 Properties of D......Page 114
11.3 Stretching and Spin Using the Current Configuration as Reference......Page 116
12.2 Pullback and Pushforward Operations......Page 119
13.1.1 Vector Fields That Convect as Tangents......Page 122
13.1.3 Tangentially Convecting Basis and Its Dual Basis. Covariant and Contravariant Components of Spatial Fields......Page 123
13.1.4 Covariant and Contravariant Convection of Tensor Fields......Page 126
13.2 Corotational Vector and Tensor Fields......Page 129
14.1 Motions......Page 131
15.2 Volume and Surface Integrals......Page 133
15.2.1 Volume Integrals......Page 134
15.3 Localization of Integrals......Page 135
15.3.1 Verification of (15.9 Blackcolor push  Blackcolor pop) Blackcolor push  Blackcolor pop Blackcolor push  Blackcolor pop, (15.10 Blackcolor push  Blackcolor pop) Blackcolor push  Blackcolor pop Blackcolor push  Blackcolor pop, and (†)......Page 136
16 Reynolds\' Transport Relation. Isochoric Motions......Page 137
17.2 Transport Relations for Spin and Vorticity......Page 139
17.3 Irrotational Motions......Page 141
17.4 Circulation......Page 142
17.5 Vortex Lines......Page 144
17.6 Steady Motions......Page 145
17.8.1 Kinematical Boundary Conditions......Page 146
17.8.3 The Motion Problem in All of Space. Solution with Constant Velocity Gradient......Page 147
Part IV: Basic mechanical principles......Page 149
18.1 Global Form of Balance of Mass......Page 151
18.2 Local Forms of Balance of Mass......Page 152
18.3 Simple Consequences of Mass Balance......Page 153
19.1 Inertial Frames. Linear and Angular Momentum......Page 155
19.2 Surface Tractions. Body Forces......Page 156
19.3 Balance Laws for Linear and Angular Momentum......Page 158
19.4 Balance of Forces and Moments Based on the Generalized Body Force......Page 160
19.5 Cauchy\'s Theorem for the Existence of Stress......Page 161
19.6 Local Forms of the Force and Moment Balances......Page 163
19.7 Kinetic Energy. Conventional and Generalized External Power Expenditures......Page 165
19.7.2 Kinetic Energy and Inertial Power......Page 166
19.7.3 Generalized Power Balance......Page 167
19.7.4 The Assumption of Negligible Inertial Forces......Page 168
20.1 Changes of Frame......Page 170
20.2 Frame-Indifferent Fields......Page 171
20.3 Transformation Rules for Kinematic Fields......Page 172
20.3.2 The Corotational, Covariant, and Contravariant Rates of a Tensor Field......Page 175
20.3.3 Other Relations for the Corotational Rate......Page 176
20.3.4 Other Relations for the Covariant Rate......Page 177
20.3.5 Other Relations for the Contravariant Rate......Page 178
20.3.6 General Tensorial Rate......Page 179
21.1 Transformation Rules for Stress and Body Force......Page 181
21.2 Inertial Body Force in a Frame That Is Not Inertial......Page 183
22.1 Force and Moment Balances as a Consequence of Frame-Indifference of the Expended Power......Page 185
22.2 Principle of Virtual Power......Page 187
22.2.1 Application to Boundary-Value Problems......Page 189
22.2.2 Fundamental Lemma of the Calculus of Variations......Page 191
23 Mechanical Laws for a Spatial Control Volume......Page 192
23.2 Momentum Balances for a Control Volume......Page 193
24.1 Piola Stress. Force and Moment Balances......Page 197
24.2 Expended Power......Page 199
25.1 Power-Conjugate Pairings. Second Piola Stress......Page 201
25.2 Transformation Laws for the Piola Stresses......Page 202
Part V: Basic thermodynamical principles......Page 205
26 The First Law: Balance of Energy......Page 207
26.1 Global and Local Forms of Energy Balance......Page 208
26.2 Terminology for \"Extensive\'\' Quantities......Page 209
27 The Second Law: Nonnegative Production of Entropy......Page 210
27.2 Temperature and the Entropy Imbalance......Page 211
27.3 Free-Energy Imbalance. Dissipation......Page 212
28.1 Invariant Nature of the First Two Laws......Page 214
28.2.1 Isolated Body......Page 215
28.2.2 Boundary Essentially at Constant Pressure and Temperature......Page 216
29.1 Free-Energy Imbalance. Dissipation......Page 218
29.2 Digression: Role of the Free-Energy Imbalance within the General Thermodynamic Framework......Page 219
29.3 Decay Inequalities......Page 220
30 The First Two Laws for a Spatial Control Volume......Page 221
31 The First Two Laws Expressed Referentially......Page 223
31.1 Global Forms of the First Two Laws......Page 224
31.2 Local Forms of the First Two Laws......Page 225
31.3 Decay Inequalities for the Body Under Passive Boundary Conditions......Page 226
31.4 Mechanical Theory: Free-Energy Imbalance......Page 228
Part VI: Mechanical and thermodynamical laws at ashock wave......Page 231
32.1 Notation. Terminology......Page 233
32.2 Hadamard\'s Compatibility Conditions......Page 234
32.4 Transport Relations in the Presence of a Shock Wave......Page 236
32.5 The Divergence Theorem in the Presence of a Shock Wave......Page 239
33.1 Balance of Mass and Momentum......Page 240
33.2 Balance of Energy and the Entropy Imbalance......Page 242
Part VII: Interlude: Basic hypotheses for developingphysically meaningful constitutive theories......Page 245
34 General Considerations......Page 247
35 Constitutive Response Functions......Page 248
36 Frame-Indifference and Compatibility with Thermodynamics......Page 249
Part VIII: Rigid heat conductors......Page 251
37 Basic Laws......Page 253
38 General Constitutive Equations......Page 254
39 Thermodynamics and Constitutive Restrictions: The Coleman–Noll Procedure......Page 256
40 Consequences of the State Restrictions......Page 258
41 Consequences of the Heat-Conduction Inequality......Page 260
42 Fourier\'s Law......Page 261
Part IX: The mechanical theory of compressible and incompressible fluids......Page 263
43.2 Basic Laws......Page 265
43.3 Transformation Rules and Objective Rates......Page 266
44.2 Consequences of Frame-Indifference......Page 268
44.3 Consequences of Thermodynamics......Page 269
44.4 Evolution Equations......Page 270
45.1 General Constitutive Equations......Page 274
45.2 Consequences of Frame-Indifference......Page 275
45.3 Consequences of Thermodynamics......Page 277
45.4 Compressible, Linearly Viscous Fluids......Page 279
45.6 Vorticity Transport Equation......Page 280
46.1 Free-Energy Imbalance for an Incompressible Body......Page 283
46.2 Incompressible, Viscous Fluids......Page 284
46.3 Incompressible, Linearly Viscous Fluids......Page 285
46.4 Incompressible Navier--Stokes Equations......Page 286
46.5 Circulation. Vorticity-Transport Equation......Page 287
46.7 Transport Equations for the Velocity Gradient, Stretching, and Spin in a Linearly Viscous, Incompressible Fluid......Page 289
46.8 Impetus-Gauge Formulation of the Navier–Stokes Equations......Page 291
46.9 Perfect Fluids......Page 292
Part X: Mechanical theory of elastic solids......Page 295
47.2 Basic Laws......Page 297
47.3 Transformation Laws Under a Change in Frame......Page 298
48.1 Consequences of Frame-Indifference......Page 300
48.2.1 The Stress Relation......Page 302
48.2.3 Natural Reference Configuration......Page 304
48.2.4 Verification of (†)......Page 305
49.1 Basic Field Equations......Page 306
49.2 A Typical Initial/Boundary-Value Problem......Page 307
50.1 The Notion of a Group. Invariance Under a Group......Page 308
50.2 The Symmetry Group G......Page 309
50.3 Isotropy......Page 312
50.3.1 Free Energy Expressed in Terms of Invariants......Page 314
50.3.2 Free Energy Expressed in Terms of Principal stretches......Page 316
50.3.3 Verification of (50.43)......Page 317
51 Simple Shear of a Homogeneous, Isotropic Elastic Body......Page 318
52.1 Small Deformations......Page 321
52.2.1 The Elasticity Tensor......Page 322
52.2.3 Estimates for the Stress and Free Energy......Page 324
52.4 Special Forms for the Elasticity Tensor......Page 326
52.4.1 Isotropic Material......Page 327
52.4.2 Cubic Crystal......Page 328
52.5 Basic Equations of the Linear theory of Elasticity for an Isotropic Material......Page 330
52.5.1 Statical Equations......Page 331
52.6 Some Simple Statical Solutions......Page 333
52.7.1 Elastostatics......Page 334
52.8 Sinusoidal Progressive Waves......Page 337
53.1 Kinematics of Incompressibility......Page 340
53.2 Indeterminacy of the Pressure. Free-Energy Imbalance......Page 341
53.3 Changes in Frame......Page 342
54.1.1 Consequences of Frame-Indifference......Page 343
54.1.2 Domain of Definition of the Response Functions......Page 344
54.1.3 Thermodynamic Restrictions......Page 345
54.1.4 Verification of (†)......Page 346
54.2 Incompressible Isotropic Elastic Bodies......Page 347
54.3 Simple Shear of a Homogeneous, Isotropic, Incompressible Elastic Body......Page 348
55 Approximately Incompressible Elastic Materials......Page 350
Part XI: Thermoelasticity......Page 355
56.2 Basic Laws......Page 357
57.1 Consequences of Frame-Indifference......Page 359
57.2 Thermodynamic Restrictions......Page 360
57.3.1 Consequences of the State Relations......Page 362
57.3.2 Consequences of the Heat-Conduction Inequality......Page 363
57.4 Elasticity Tensor. Stress-Temperature Modulus. Heat Capacity......Page 365
57.5 The Basic Thermoelastic Field Equations......Page 366
57.6 Entropy as Independent Variable. Nonconductors......Page 367
57.8 Material Symmetry......Page 370
58.1 Asymptotic Stability and its Consequences. The Gibbs Function......Page 372
58.2 Local Relations at a Reference Configuration that is Natural for a Temperature theta0......Page 373
59.1 Approximate Constitutive Equations for the Stress and Entropy......Page 378
59.3 Isotropic Linear Thermoelasticity......Page 380
Part XII: Species diffusion coupled to elasticity......Page 385
60 Balance Laws for Forces, Moments, and the Conventional External Power......Page 387
61 Mass Balance for a Single Diffusing Species......Page 388
62 Free-Energy Imbalance Revisited. Chemical Potential......Page 390
63.1 Species Mass Balances......Page 393
63.2 Free-Energy Imbalance......Page 394
64 Digression: The Thermodynamic Laws in the Presence of Species Transport......Page 395
65.1 Single Species......Page 398
65.2 Multiple Species......Page 400
66.1 Consequences of Frame-Indifference......Page 401
66.2 Thermodynamic Restrictions......Page 402
66.3 Consequences of the Thermodynamic Restrictions......Page 404
66.4 Fick\'s Law......Page 406
67 Material Symmetry......Page 409
67.1 Verification of (‡)......Page 410
68 Natural Reference Configuration......Page 412
69 Summary of Basic Equations for a Single Species......Page 414
70.1 Consequences of Frame-Indifference and Thermodynamics......Page 415
70.3 Natural Reference Configuration......Page 417
71 Summary of Basic Equations for N Independent Species......Page 420
72.1 Lattice Constraint......Page 422
72.3 Relative Chemical Potentials. Free-Energy Imbalance......Page 423
72.4 Elimination of the Lattice Constraint. Larché--Cahn Differentiation......Page 424
72.5 General Constitutive Equations......Page 427
72.6 Thermodynamic Restrictions......Page 428
72.8 Normalization Based on the Elimination of the Lattice Constraint......Page 430
73.1 Approximate Constitutive Equations for the Stress, Chemical Potentials, and Fluxes......Page 432
73.2 Basic Equations of the Linear Theory......Page 434
73.3 Isotropic Linear Theory......Page 435
Part XIII: Theory of isotropic plastic solids undergoing small deformations......Page 439
74 Some Phenomenological Aspects of the Elastic-Plastic Stress-Strain Response of Polycrystalline Metals......Page 441
74.1 Isotropic and Kinematic Strain-Hardening......Page 443
75.1 Basic Equations......Page 446
75.2 Kinematical Assumptions that Define Plasticity Theory......Page 447
75.4 Constitutive Characterization of Elastic Response......Page 448
76 Formulation of the Mises Theory of Plastic Flow......Page 450
76.1 General Constitutive Equations for Plastic Flow......Page 451
76.2 Rate-Independence......Page 452
76.3 Strict Dissipativity......Page 454
76.4 Formulation of the Mises Flow Equations......Page 455
76.5.1 Flow Equations With Y(S) not Identically Equal to S......Page 458
76.6 Solving the Hardening Equation. Accumulated Plastic Strain is the Most General Hardening Variable......Page 459
76.8 Yield Surface. Yield Function. Consistency Condition......Page 463
76.9 Hardening and Softening......Page 467
77 Inversion of the Mises Flow Rule: Ep in Terms of E and T......Page 469
78.2 Materials with Simple Rate-Dependence......Page 473
78.3 Power-Law Rate-Dependence......Page 476
79.1 Basic Definitions......Page 478
79.2 Warm-up: Derivation of the Mises Flow Equations Based on Maximum Dissipation......Page 480
79.3.1 Yield-Set Hypotheses......Page 482
79.3.2 Digression: Some Definitions and Results Concerning Convex Surfaces......Page 484
79.3.3 Drucker\'s Theorem......Page 485
79.4 The Conventional Theory of Perfectly Plastic Materials Fits within the Framework Presented Here......Page 486
80.1 Free-Energy Imbalance Revisited......Page 489
80.2 Constitutive Equations. Flow Rule......Page 490
81.1 Background......Page 493
81.3 Constitutive Equations......Page 494
81.4 Nature of the Defect Energy......Page 496
81.6 Balance of Energy Revisited......Page 497
81.7 Thermally Simple Materials......Page 499
81.8 Determination of the Defect Energy by the Rosakis Brothers, Hodowany, and Ravichandran......Page 500
81.9 Summary of the Basic Equations......Page 501
82.1 Reformulation of the Mises Flow Equations in Terms of Dissipation......Page 503
82.2 The Global Variational Inequality......Page 506
82.3 Alternative Formulation of the Global Variational Inequality When Hardening is Described by a Defect Energy......Page 507
Part XIV: Small deformation, isotropic plasticity based on the principle of virtual power......Page 509
83 Introduction......Page 511
84.1 General Principle of Virtual Power......Page 513
84.2.1 General Principle Based on Codirectionality......Page 517
84.2.2 Streamlined Principle Based on Codirectionality......Page 519
84.3 Virtual External Forces Associated with Dislocation Flow......Page 520
84.4 Free-Energy Imbalance......Page 521
84.5 Discussion of the Virtual-Power Formulation......Page 522
85 Basic Constitutive Theory......Page 523
86 Material Stability and Its Relation to Maximum Dissipation......Page 525
Part XV: Strain gradient plasticity based on the principle of virtual power......Page 529
87 Introduction......Page 531
88.1 Characterization of the Burgers Vector......Page 533
88.2 Irrotational Plastic Flow......Page 535
89.1 The Virtual-Power Principle of Fleck and Hutchinson......Page 536
89.2 Free-Energy Imbalance......Page 539
89.3 Constitutive Equations......Page 540
89.4 Flow Rules......Page 541
89.5 Microscopically Simple Boundary Conditions......Page 542
89.6 Variational Formulation of the Flow Rule......Page 543
89.7 Plastic Free-Energy Balance......Page 544
89.8.2 Single Shear Bands and Periodic Arrays of Shear Bands......Page 545
90.1 Third-Order Tensors......Page 548
90.2 Virtual-Power Formulation: Macroscopic and Microscopic Force Balances......Page 549
90.4 Energetic Constitutive Equations......Page 552
90.5 Dissipative Constitutive Equations......Page 554
90.6 Flow Rule......Page 556
90.7 Microscopically Simple Boundary Conditions......Page 557
90.8 Variational Formulation of the Flow Rule......Page 558
90.9 Plastic Free-Energy Balance. Flow-Induced Strengthening......Page 559
90.10 Rate-Independent Theory......Page 560
Part XVI: Large-deformation theory of isotropic plastic solids......Page 563
91.1 The Kröner Decomposition......Page 565
91.3 Elastic and Plastic Stretching and Spin. Plastic Incompressibility......Page 567
91.4 Elastic and Plastic Polar Decompositions......Page 568
91.5 Change in Frame Revisited in View of the Kröner Decomposition......Page 570
92.1 Internal and External Expenditures of Power......Page 572
92.2 Principle of Virtual Power......Page 573
92.2.1 Consequences of Frame-Indifference......Page 574
92.2.3 Microscopic Force Balance......Page 575
93.1 Free-Energy Imbalance Expressed in Terms of the Cauchy Stress......Page 577
94.1 The Second Piola Elastic-Stress Te......Page 579
94.2 The Mandel Stress Me......Page 580
95.1 General Separable Constitutive Theory......Page 581
95.2 Structural Frame-Indifference and the Characterization of Polycrystalline Materials Without Texture......Page 583
95.3 Interaction of Elasticity and Plastic Flow......Page 586
95.4 Consequences of Rate-Independence......Page 587
95.5 Derivation of the Mises Flow Equations Based on Maximum-Dissipation......Page 588
96 Summary of the Basic Equations. Remarks......Page 590
97 Plastic Irrotationality: The Condition…......Page 591
98 Yield Surface. Yield Function. Consistency Condition......Page 593
99.2 Conditions that Describe Loading and Unloading......Page 595
99.4 Equivalent Formulation of the Constitutive Equations and Plastic Mises Flow Equations Based on the Inverted Flow Rule......Page 598
100 Evolution Equation for the Second Piola Stress......Page 600
101.2 Inversion of the Rate-Dependent Flow Rule......Page 603
101.3 Summary of the Complete Constitutive Theory......Page 604
102.1 Introduction......Page 607
102 Basic Single-Crystal Kinematics......Page 610
103.1 Decomposition of the Burgers Tensor G into Distributions of Edge and Screw Dislocations......Page 612
103.3 The Tangential Gradient  on the Slip Plane......Page 614
104.1 Virtual-Power Formulation of the Standard and Microscopic Force Balances......Page 617
104.3 General Separable Constitutive Theory......Page 620
104.5 Constitutive Equations for Flow with Simple Rate-Dependence......Page 621
104.7 Self-Hardening, Latent-Hardening......Page 625
104.8 Summary of the Constitutive Theory......Page 626
105.1 Virtual-Power Formulation of the Standard and Microscopic Force Balances of the Gradient Theory......Page 628
105.3 Energetic Constitutive Equations. Peach–Koehler Forces......Page 631
105.4 Constitutive Equations that Account for Dissipation......Page 633
105.5 Viscoplastic Flow Rule......Page 636
105.6 Microscopically Simple Boundary Conditions......Page 639
105.7 Variational Formulation of the Flow Rule......Page 640
105.8 Plastic Free-Energy Balance......Page 641
105.9 Some Remarks......Page 642
Part XVIII: Single Crystals Undergoing Large Deformations......Page 645
106 Basic Single-Crystal Kinematics......Page 647
107.1 Transformation of Vector Area Measures Between the Reference, Observed, and Lattice Spaces......Page 650
107.2 Characterization of the Burgers Vector......Page 651
107.3 The Plastically Convected Rate of G......Page 653
107.4 Densities of Screw and Edge Dislocations......Page 655
107.5 Comparison of Small- and Large-Deformation Results Concerning Dislocation Densities......Page 657
108.1 Internal and External Expenditures of Power......Page 658
108.2 Consequences of Frame-Indifference......Page 660
108.3 Macroscopic and Microscopic Force Balances......Page 661
109 Free-Energy Imbalance......Page 663
110.1 Constitutive Relations......Page 665
110.2 Simplified Constitutive Theory......Page 667
110.3 Summary of Basic Equations......Page 668
111.1 Kinematics of a Taylor Polycrystal......Page 670
111.2 Principle of Virtual Power......Page 672
111.4 Constitutive Relations......Page 675
112.1 Energetic Constitutive Equations. Peach–Koehler Forces......Page 677
112.2 Dissipative Constitutive Equations that Account for Slip-Rate Gradients......Page 678
112.3 Viscoplastic Flow Rule......Page 680
112.4 Microscopically Simple Boundary Conditions......Page 681
112.5 Variational Formulation......Page 682
112.6 Plastic Free-Energy Balance......Page 683
112.7 Some Remarks......Page 684
113 Isotropic Functions......Page 689
113.2 Isotropic Tensor Functions......Page 690
113.3 Isotropic Linear Tensor Functions......Page 692
114 The Exponential of a Tensor......Page 693
References......Page 695
Index......Page 707