Mechanics of solids and materials

Cover\r......Page 1
Foreword\r......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 20
1.1 Vector Algebra......Page 22
1.2 Coordinate Transformation: Rotation of Axes......Page 25
1.4 Symmetric and Antisymmetric Tensors......Page 26
1.5 Prelude to Invariants of Tensors......Page 27
1.7 Additional Proofs......Page 28
1.8 Additional Lemmas for Vectors......Page 29
1.9 Coordinate Transformation of Tensors......Page 30
1.11 Tensor Product......Page 31
1.12 Orthonormal Basis......Page 32
1.13 Eigenvectors and Eigenvalues......Page 33
1.15 Positive Definiteness of a Tensor......Page 35
1.16.1 Eigenvectors of W......Page 36
1.17 Orthogonal Tensors......Page 38
1.18 Polar Decomposition Theorem......Page 40
1.19 Polar Decomposition: Physical Approach......Page 41
1.19.2 Principal Stretches......Page 42
1.20 The Cayley–Hamilton Theorem......Page 43
1.22 Identities and Relations Involving Operator......Page 44
1.23 Suggested Reading......Page 46
2.1 Gauss and Stokes\'s Theorems......Page 47
2.2 Vector and Tensor Fields: Physical Approach......Page 48
2.3 Surface Integrals: Gauss Law......Page 49
2.4 Evaluating Surface Integrals......Page 50
2.5 The Divergence......Page 52
2.6 Divergence Theorem: Relation of Surface to Volume Integrals......Page 54
2.7 More on Divergence Theorem......Page 55
2.8 Suggested Reading......Page 56
3.1 Fourier Series......Page 57
3.2 Double Fourier Series......Page 58
3.2.1 Double Trigonometric Series......Page 59
3.3 Integral Transforms......Page 60
3.4 Dirichlet\'s Conditions......Page 63
3.5 Integral Theorems......Page 67
3.6 Convolution Integrals......Page 69
3.7 Fourier Transforms of Derivatives of f (x)......Page 70
3.8 Fourier Integrals as Limiting Cases of Fourier Series......Page 71
3.9 Dirac Delta Function......Page 72
3.10 Suggested Reading......Page 73
4.1 Preliminaries......Page 76
4.2 Uniaxial Strain......Page 77
4.3 Deformation Gradient......Page 78
4.4 Strain Tensor......Page 79
4.6 Angle Change and Shear Strains......Page 81
4.7 Infinitesimal Strains......Page 82
4.8 Principal Stretches......Page 83
4.10 Volume Changes......Page 84
4.11 Area Changes......Page 85
4.12 Area Changes: Alternative Approach......Page 86
4.13 Simple Shear of a Thick Plate with a Central Hole......Page 87
4.14 Finite vs. Small Deformations......Page 89
4.15 Reference vs. Current Configuration......Page 90
4.17 Velocity Gradient......Page 92
4.18 Deformation Rate and Spin......Page 95
4.19 Rate of Stretching and Shearing......Page 96
4.20 Material Derivatives of Strain Tensors: vs. D......Page 97
4.21 Rate of F in Terms of Principal Stretches......Page 99
4.21.1 Spins of Lagrangian and Eulerian Triads......Page 102
4.22 Additional Connections Between Current and Reference State Representations......Page 103
4.23 Transport Formulae......Page 104
4.24 Material Derivatives of Volume, Area, and Surface Integrals: Transport Formulae Revisited......Page 105
4.25 Analysis of Simple Shearing......Page 106
4.26 Examples of Particle and Plane Motion......Page 108
4.27 Rigid Body Motions......Page 109
4.28 Behavior under Superposed Rotation......Page 110
4.29 Suggested Reading......Page 111
5.1 Traction Vector and Stress Tensor......Page 113
5.2 Equations of Equilibrium......Page 115
5.3 Balance of Angular Momentum: Symmetry of.........Page 116
5.4 Principal Values of Cauchy Stress......Page 117
5.5 Maximum Shear Stresses......Page 118
5.6 Nominal Stress......Page 119
5.7 Equilibrium in the Reference State......Page 120
5.8 Work Conjugate Connections......Page 121
5.10 Frame Indifference......Page 123
5.11 Continuity Equation and Equations of Motion......Page 128
5.12 Stress Power......Page 129
5.13 The Principle of Virtual Work......Page 130
5.15 Suggested Reading......Page 132
6.1 First Law of Thermodynamics: Energy Equation......Page 134
6.2 Second Law of Thermodynamics: Clausius–Duhem Inequality......Page 135
6.3.1 Thermodynamic Potentials......Page 137
6.3.2 Specific and Latent Heats......Page 139
6.3.3 Coupled Heat Equation......Page 140
6.4 Thermodynamic Relationships with p, V, T, and s......Page 141
6.4.1 Specific and Latent Heats......Page 142
6.4.2 Coefficients of Thermal Expansion and Compressibility......Page 143
6.5 Theoretical Calculations of Heat Capacity......Page 144
6.6 Third Law of Thermodynamics......Page 146
6.7 Irreversible Thermodynamics......Page 148
6.8 Gibbs Conditions of Thermodynamic Equilibrium......Page 150
6.9 Linear Thermoelasticity......Page 151
6.10.1 Internal Energy......Page 153
6.10.2 Helmholtz Free Energy......Page 154
6.10.3 Gibbs Energy......Page 155
6.10.4 Enthalpy Function......Page 156
6.11 Uniaxial Loading and Thermoelastic Effect......Page 157
6.12 Thermodynamics of Open Systems: Chemical Potentials......Page 160
6.13 Gibbs–Duhem Equation......Page 162
6.14 Chemical Potentials for Binary Systems......Page 163
6.15 Configurational Entropy......Page 164
6.16 Ideal Solutions......Page 165
6.17 Regular Solutions for Binary Alloys......Page 166
6.18 Suggested Reading......Page 168
7.1 Green Elasticity......Page 169
7.2 Isotropic Green Elasticity......Page 171
7.3 Constitutive Equations in Terms of B......Page 172
7.4 Constitutive Equations in Terms of Principal Stretches......Page 173
7.6 Elastic Moduli Tensors......Page 174
7.8 Elastic Pseudomoduli......Page 176
7.9 Elastic Moduli of Isotropic Elasticity......Page 177
7.10 Elastic Moduli in Terms of Principal Stretches......Page 178
7.11 Suggested Reading......Page 179
8.1 Elementary Theory of Isotropic Linear Elasticity......Page 182
8.2 Elastic Energy in Linear Elasticity......Page 184
8.3.1 Material Symmetry......Page 185
8.3.2 Restrictions on the Elastic Constants......Page 189
8.4 Compatibility Relations......Page 190
8.5 Compatibility Conditions: Cesaro Integrals......Page 191
8.7 Navier Equations of Motion......Page 193
8.8.2 Uniqueness of the Solution......Page 195
8.9 Potential Energy and Variational Principle......Page 196
8.10 Betti\'s Theorem of Linear Elasticity......Page 198
8.11 Plane Strain......Page 199
8.11.1 Plane Stress......Page 200
8.13 Thermal Distortion of a Simple Beam......Page 201
8.14 Suggested Reading......Page 203
9.1 A Simple 2D Beam Problem......Page 205
9.2 Polynomial Solutions to.........Page 206
9.3 A Simple Beam Problem Continued......Page 207
9.3.1 Strains and Displacements for 2D Beams......Page 208
9.4 Beam Problems with Body Force Potentials......Page 209
9.5 Beam under Fourier Loading......Page 211
9.6 Complete Boundary Value Problems for Beams......Page 214
9.6.1 Displacement Calculations......Page 217
9.7 Suggested Reading......Page 219
10.1 Polar Components of Stress and Strain......Page 220
10.2.1 Far Field Shear......Page 222
10.2.2 Far Field Tension......Page 224
10.3 Degenerate Cases of Solution in Polar Coordinates......Page 225
10.4 Curved Beams: Plane Stress......Page 227
10.4.1 Pressurized Cylinder......Page 230
10.4.2 Bending of a Curved Beam......Page 231
10.5 Axisymmetric Deformations......Page 232
10.6 Suggested Reading......Page 234
11.1 Torsion of Prismatic Rods......Page 235
11.2 Elastic Energy of Torsion......Page 237
11.3 Torsion of a Rod with Rectangular Cross Section......Page 238
11.4 Torsion of a Rod with Elliptical Cross Section......Page 242
11.5 Torsion of a Rod with Multiply Connected Cross Sections......Page 243
11.5.1 Hollow Elliptical Cross Section......Page 245
11.6 Bending of a Cantilever......Page 246
11.7 Elliptical Cross Section......Page 248
11.8 Suggested Reading......Page 249
12.1 Fourier Transform of Biharmonic Equation......Page 250
12.2 Loading on a Half-Plane......Page 251
12.3 Half-Plane Loading: Special Case......Page 253
12.4 Symmetric Half-Plane Loading......Page 255
12.5 Half-Plane Loading: Alternative Approach......Page 256
12.6 Additional Half-Plane Solutions......Page 258
12.6.1 Displacement Fields in Half-Spaces......Page 259
12.6.2 Boundary Value Problem......Page 260
12.6.3 Specific Example......Page 261
12.7 Infinite Strip......Page 263
12.7.1 Uniform Loading:.........Page 264
12.7.2 Symmetrical Point Loads......Page 265
12.8 Suggested Reading......Page 266
13.1 Displacement-Based Equations of Equilibrium......Page 267
13.2 Boussinesq–Papkovitch Solutions......Page 268
13.3 Spherically Symmetrical Geometries......Page 269
13.3.1 Internally Pressurized Sphere......Page 270
13.4 Pressurized Sphere: Stress-Based Solution......Page 272
13.4.1 Pressurized Rigid Inclusion......Page 273
13.4.2 Disk with Circumferential Shear......Page 274
13.5 Spherical Indentation......Page 275
13.5.1 Displacement-Based Equilibrium......Page 276
13.5.2 Strain Potentials......Page 277
13.5.3 Point Force on a Half-Plane......Page 278
13.5.4 Hemispherical Load Distribution......Page 279
13.5.5 Indentation by a Spherical Ball......Page 280
13.6 Point Forces on Elastic Half-Space......Page 282
13.7 Suggested Reading......Page 284
14.2 Green\'s Function......Page 285
14.3 Isotropic Green\'s Function......Page 289
14.4 Suggested Reading......Page 291
15.1 Wedge Problem......Page 292
15.2 Distributed Contact Forces......Page 295
15.2.1 Uniform Contact Pressure......Page 296
15.3 Displacement-Based Contact: Rigid Flat Punch......Page 298
15.4 Suggested Reading......Page 300
16.1 Stresses and Strains of Bent Plates......Page 301
16.2 Energy of Bent Plates......Page 302
16.3 Equilibrium Equations for a Plate......Page 303
16.4 Shear Forces and Bending and Twisting Moments......Page 306
16.5.1 Clamped Circular Plate......Page 308
16.5.4 Peeled Surface Layer......Page 309
16.6 Rectangular Plates......Page 310
16.6.1 Uniformly Loaded Rectangular Plate......Page 311
16.7 Suggested Reading......Page 312
17.1 Dislocations......Page 314
17.1.1 Derivation of the Displacement Field......Page 315
17.2 Tensile Cracks......Page 316
17.3 Suggested Reading......Page 319
18.1 Dislocation Character and Geometry......Page 320
18.2.1 Infinitely Long Screw Dislocations......Page 323
18.2.3 Infinitely Long Mixed Segments......Page 324
18.3 Planar Geometric Theorem......Page 326
18.4 Applications of the Planar Geometric Theorem......Page 329
18.4.1 Angular Dislocations......Page 332
18.5 A 3D Geometrical Theorem......Page 333
18.6 Suggested Reading......Page 335
19.1 Dislocation Mechanics: Reviewed......Page 336
19.2 Freely Slipping Crack......Page 337
19.3 Crack Extension Force......Page 340
19.4 Crack Faces Loaded by Tractions......Page 341
19.5 Stress Intensity Factors and Crack Extension Force......Page 343
19.5.1 Computation of the Crack Extension Force......Page 344
19.7 Dislocation Energy Factor Matrix......Page 346
19.8 Inversion of a Singular Integral Equation......Page 349
19.9 2D Anisotropic Elasticity – Stroh Formalism......Page 350
19.9.1 Barnett–Lothe Tensors......Page 353
19.10 Suggested Reading......Page 355
20.1 The Problem......Page 356
20.2 Eshelby\'s Solution Setup......Page 357
20.3 Calculation of the Constrained Fields: uc, ec, and.........Page 359
20.4 Components of the Eshelby Tensor for Ellipsoidal Inclusion......Page 362
20.6 Inhomogeneous Inclusion: Uniform Transformation Strain......Page 364
20.7 Nonuniform Transformation Strain Inclusion Problem......Page 366
20.7.1 The Cases M = 0, 1......Page 370
20.8.1 Constrained Elastic Field......Page 371
20.8.2 Field in the Matrix......Page 372
20.8.3 Field at the Interface......Page 373
20.8.4 Isotropic Spherical Inclusion......Page 374
20.9 Suggested Reading......Page 375
21.1 Free Energy and Mechanical Potential Energy......Page 376
21.2 Forces of Translation......Page 378
21.2.1 Force on an Interface......Page 380
21.2.2 Finite Deformation Energy Momentum Tensor......Page 381
21.3 Interaction Between Defects and Loading Mechanisms......Page 383
21.3.1 Interaction Between Dislocations and Inclusions......Page 385
21.3.2 Force on a Dislocation Segment......Page 386
21.4 Elastic Energy of a Dislocation......Page 387
21.5 In-Plane Stresses of Straight Dislocation Lines......Page 388
21.6 Chemical Potential......Page 390
21.6.1 Force on a Defect due to a Free Surface......Page 392
21.7.2 Application of the Interface Force to Precipitation......Page 393
21.8 Suggested Reading......Page 395
22.1 Introduction......Page 396
22.2 Basic Equations of Couple-Stress Elasticity......Page 397
22.3 Displacement Equations of Equilibrium......Page 398
22.4 Correspondence Theorem of Couple-Stress Elasticity......Page 399
22.5 Plane Strain Problems of Couple-Stress Elasticity......Page 400
22.5.1 Mindlin\'s Stress Functions......Page 401
22.6 Edge Dislocation in Couple-Stress Elasticity......Page 402
22.6.1 Strain Energy......Page 403
22.7 Edge Dislocation in a Hollow Cylinder......Page 405
22.8 Governing Equations for Antiplane Strain......Page 407
22.8.1 Expressions in Polar Coordinates......Page 409
22.8.2 Correspondence Theorem for Antiplane Strain......Page 410
22.9 Antiplane Shear of Circular Annulus......Page 411
22.10 Screw Dislocation in Couple-Stress Elasticity......Page 412
22.11 Configurational Forces in Couple-Stress Elasticity......Page 413
22.11.1 Reciprocal Properties......Page 414
22.11.3 Energy due to Internal and External Sources of Stress......Page 415
22.11.4 The Force on an Elastic Singularity......Page 416
22.12 Energy-Momentum Tensor of a Couple-Stress Field......Page 417
22.13 Basic Equations of Micropolar Elasticity......Page 419
22.14 Noether\'s Theorem of Micropolar Elasticity......Page 421
22.15 Conservation Integrals in Micropolar Elasticity......Page 424
22.17 M Integral of Micropolar Elasticity......Page 425
22.18 Suggested Reading......Page 427
23.2 Screw Dislocation Near a Bimaterial Interface......Page 428
23.2.3 Screw Dislocation Near a Free Surface......Page 430
23.3 Edge Dislocation (bx) Near a Bimaterial Interface......Page 431
23.3.1 Interface Edge Dislocation......Page 436
23.3.3 Edge Dislocation Near a Free Surface......Page 438
23.3.4 Edge Dislocation Near a Rigid Boundary......Page 439
23.4 Edge Dislocation (by) Near a Bimaterial Interface......Page 440
23.4.1 Interface Edge Dislocation......Page 441
23.4.3 Edge Dislocation Near a Free Surface......Page 443
23.5 Strain Energy of a Dislocation Near a Bimaterial Interface......Page 444
23.5.1 Strain Energy of a Dislocation Near a Free Surface......Page 447
23.6 Suggested Reading......Page 448
24.1 Dislocation Array Beneath the Free Surface......Page 449
24.2 Energy of a Dislocation Array......Page 451
24.3 Strained-Layer Epitaxy......Page 452
24.4 Conditions for Dislocation Array Formation......Page 453
24.5 Frank and van der Merwe Energy Criterion......Page 455
24.6 Gradual Strain Relaxation......Page 457
24.8 Stronger Stability Criteria......Page 460
24.9.1 Lower Bound......Page 462
24.9.2 Upper Bound......Page 464
24.10 Suggested Reading......Page 467
25.1 Stressed Surface Problem......Page 468
25.2 Chemical Potential......Page 470
25.3 Surface Diffusion and Interface Stability......Page 471
25.4 Volume Diffusion and Interface Stability......Page 472
25.5 2D Surface Profiles and Surface Stability......Page 476
25.6 Asymptotic Stresses for 1D Surface Profiles......Page 478
25.7 Suggested Reading......Page 480
26 Phenomenological Plasticity......Page 482
26.1 Yield Criteria for Multiaxial Stress States......Page 483
26.2 Von Mises Yield Criterion......Page 484
26.3 Tresca Yield Criterion......Page 486
26.4 Mohr–Coulomb Yield Criterion......Page 488
26.4.1 Drucker–Prager Yield Criterion......Page 489
26.6 Anisotropic Yield Criteria......Page 491
26.7 Elastic-Plastic Constitutive Equations......Page 492
26.8 Isotropic Hardening......Page 494
26.8.1 J2 Flow Theory of Plasticity......Page 495
26.9 Kinematic Hardening......Page 496
26.9.1 Linear and Nonlinear Kinematic Hardening......Page 498
26.10 Constitutive Equations for Pressure-Dependent Plasticity......Page 499
26.12 Plastic Potential for Geomaterials......Page 501
26.13 Rate-Dependent Plasticity......Page 503
26.14 Deformation Theory of Plasticity......Page 505
26.14.1 Rate-Type Formulation of Deformation Theory......Page 506
26.14.2 Application beyond Proportional Loading......Page 507
26.15 J2 Corner Theory......Page 508
26.16.1 Multiplicative Decomposition.........Page 510
26.17 Elastic and Plastic Constitutive Contributions......Page 512
26.17.1 Rate-Dependent J2 Flow Theory......Page 513
26.18 A Rate Tangent Integration......Page 514
26.19 Plastic Void Growth......Page 516
26.19.1 Ideally Plastic Material......Page 518
26.19.2 Incompressible Linearly Hardening Material......Page 519
26.20 Suggested Reading......Page 522
27.1 Early Observations......Page 523
27.2 Dislocations......Page 529
27.2.1 Some Basic Properties of Dislocations in Crystals......Page 532
27.2.2 Strain Hardening, Dislocation Interactions, and Dislocation Multiplication......Page 535
27.3 Other Strengthening Mechanisms......Page 538
27.4 Measurements of Latent Hardening......Page 540
27.5 Observations of Slip in Single Crystals and Polycrystals at Modest Strains......Page 544
27.6 Deformation Mechanisms in Nanocrystalline Grains......Page 546
27.6.1 Background: AKK Model......Page 551
27.6.3 Dislocation and Partial Dislocation Slip Systems......Page 556
27.7 Suggested Reading......Page 558
28.1 Basic Kinematics......Page 559
28.2 Stress and Stress Rates......Page 562
28.2.1 Resolved Shear Stress......Page 563
28.2.2 Rate-Independent Strain Hardening......Page 565
28.3 Convected Elasticity......Page 566
28.4 Rate-Dependent Slip......Page 568
28.4.1 A Rate Tangent Modulus......Page 569
28.5 Crystalline Component Forms......Page 571
28.5.1 Additional Crystalline Forms......Page 574
28.6 Suggested Reading......Page 576
29.1 Perspectives on Nonuniform and Localized Plastic Flow......Page 578
29.1.1 Coarse Slip Bands and Macroscopic Shear Bands in Simple Crystals......Page 579
29.1.2 Coarse Slip Bands and Macroscopic Shear Bands in Ordered Crystals......Page 580
29.2.2 Plastic Shearing with Non-Schmid Effects......Page 581
29.2.3 Conditions for Localization......Page 584
29.2.4 Expansion to the Order of.........Page 586
29.2.5 Perturbations about the Slip and Kink Plane Orientations......Page 588
29.2.6 Isotropic Elastic Moduli......Page 591
29.2.7 Particular Cases for Localization......Page 592
29.3.1 Double Slip Model......Page 597
29.3.2 Constitutive Law for the Double Slip Crystal......Page 598
29.4.1 Additional Experimental Observations......Page 601
29.4.2 Numerical Observations......Page 603
29.5 Suggested Reading......Page 605
30.1 Perspectives on Polycrystalline Modeling and Texture Development......Page 607
30.2 Polycrystal Model......Page 609
30.3 Extended Taylor Model......Page 611
30.4 Model Calculational Procedure......Page 613
30.4.1 Texture Determinations......Page 614
30.4.2 Yield Surface Determination......Page 615
30.5 Deformation Theories and Path-Dependent Response......Page 617
30.5.1 Specific Model Forms......Page 618
30.5.3 Nonproportional Loading......Page 619
30.6 Suggested Reading......Page 621
31.1 Laminate Model......Page 622
31.3 Final Constitutive Forms......Page 625
31.3.1 Rigid-Plastic Laminate in Single Slip......Page 626
31.4 Suggested Reading......Page 628
32.1 Introduction......Page 630
32.2.1 Material Form of Continuity Equation......Page 631
32.3 Reynolds Transport Theorem......Page 633
32.4 Momentum Principles......Page 635
32.5 Energy Equation......Page 636
32.5.1 Material Form of Energy Equation......Page 637
32.6 Entropy Equation......Page 638
32.6.1 Material Form of Entropy Equation......Page 639
32.7 General Constitutive Framework......Page 640
32.7.1 Thermodynamic Potentials per Unit Initial Mass......Page 641
32.7.2 Equivalence of the Constitutive Structures......Page 642
32.8 Multiplicative Decomposition of Deformation Gradient......Page 643
32.8.1 Strain and Strain-Rate Measures......Page 644
32.9 Density Expressions......Page 645
32.10 Elastic Stress Response......Page 646
32.11 Partition of the Rate of Deformation......Page 647
32.12 Elastic Moduli Tensor......Page 648
32.13 Elastic Strain Energy Representation......Page 650
32.14 Evolution Equation for Stretch Ratio......Page 651
32.15 Suggested Reading......Page 652
33.1 Biological Membranes......Page 654
33.2 Membrane Kinematics......Page 655
33.3 Constitutive Laws for Membranes......Page 658
33.4 Limited Area Compressibility......Page 659
33.5 Simple Triangular Networks......Page 660
33.6 Suggested Reading......Page 661
CHAPTER 1......Page 662
CHAPTER 2......Page 670
CHAPTER 3......Page 673
CHAPTER 4......Page 676
CHAPTER 5......Page 694
CHAPTER 6......Page 707
CHAPTER 7......Page 715
CHAPTER 8......Page 731
CHAPTER 9......Page 736
CHAPTER 10......Page 738
CHAPTER 11......Page 746
CHAPTER 12......Page 753
CHAPTER 13......Page 757
CHAPTER 14......Page 764
CHAPTER 15......Page 770
CHAPTER 16......Page 774
CHAPTER 17......Page 777
CHAPTER 18......Page 781
CHAPTER 19......Page 783
CHAPTER 20......Page 785
CHAPTER 21......Page 791
CHAPTER 22......Page 802
CHAPTER 23......Page 803
CHAPTER 24......Page 806
CHAPTER 25......Page 810
CHAPTER 26......Page 811
CHAPTER 27......Page 832
CHAPTER 28......Page 835
CHAPTER 29......Page 838
CHAPTER 30......Page 841
CHAPTER 31......Page 846
CHAPTER 32......Page 848
CHAPTER 33......Page 851
Bibliography......Page 854
Index......Page 874