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ОбложкаPhillips R. Crystals, defects and microstructures: modeling across scales. - Cambridge; New York: Cambridge University Press, 2001. - xxvi, 780 p.: ill. - Ref.: p.757-770. - Ind.: p.771-780. - ISBN 978-0-521-79357-5
Шифр: (И/В37-Р56) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
Preface ........................................................ xv
Acknowledgements .............................................. xxi
Notes on Units, Scales and Conventions ....................... xxiv

Part one: Thinking About the Material World ..................... 1
1    Idealizing Material Response ............................... 3
1.1  A Material World ........................................... 3
     1.1.1  Materials: A Databook Perspective ................... 3
     1.1.2  The Structure-Properties Paradigm ................... 8
     1.1.3  Controlling Structure: The World of Heat and Beat .. 12
1.2  Modeling of Materials ..................................... 14
     1.2.1  The Case for Modeling .............................. 14
     1.2.2  Modeling Defined: Contrasting Perspectives ......... 15
     1.2.3  Case Studies in Modeling ........................... 18
     1.2.4  Modeling and the Computer: Numerical Analysis vs
            Simulation ......................................... 25
1.3  Further Reading ........................................... 26

2    Continuum Mechanics Revisited ............................. 29
2.1  Continuum Mechanics as an Effective Theory ................ 29
2.2  Kinematics: The Geometry of Deformation ................... 31
     2.2.1  Deformation Mappings and Strain .................... 32
     2.2.2  Geometry of Rigid Deformation ...................... 35
     2.2.3  Geometry of Slip and Twinning ...................... 36
     2.2.4  Geometry of Structural Transformations ............. 37
2.3  Forces and Balance Laws ................................... 39
     2.3.1  Forces Within Continua: Stress Tensors ............. 39
     2.3.2  Equations of Continuum Dynamics .................... 41
     2.3.3  Configurational Forces and the Dynamics of
            Defects ............................................ 44
2.4  Continuum Descriptions of Deformation and Failure ......... 51
     2.4.1    Constitutive Modeling ............................ 51
     2.4.2  Linear Elastic Response of Materials ............... 51
     2.4.3  Plastic Response of Crystals and Polycrystals ...... 54
     2.4.4  Continuum Picture of Fracture ...................... 60
2.5  Boundary Value Problems and Modeling ...................... 64
     2.5.1  Principle of Minimum Potential Energy and
            Reciprocal Theorem ................................. 64
     2.5.2  Elastic Green Function ............................. 66
     2.5.3  Method of Eigenstrains ............................. 69
     2.5.4  Numerical Solutions: Finite Element Method ......... 72
2.6  Difficulties with the Continuum Approach .................. 75
2.7  Further Reading ........................................... 76
2.8  Problems .................................................. 78

3    Quantum and Statistical Mechanics Revisited ............... 81
3.1  Background ................................................ 81
3.2  Quantum Mechanics ......................................... 82
     3.2.1  Background and Formalism ........................... 82
     3.2.2  Catalog of Important Solutions ..................... 87
     3.2.3  Finite Elements and Schrödinger .................... 94
     3.2.4  Quantum Corrals: A Finite Element Analysis ........ 101
     3.2.5  Metals and the Electron Gas ....................... 103
     3.2.6  Quantum Mechanics of Bonding ...................... 109
3.3  Statistical Mechanics .................................... 115
     3.3.1  Background ........................................ 115
     3.3.2  Entropy of Mixing ................................. 119
     3.3.3  The Canonical Distribution ........................ 122
     3.3.4  Information Theoretic Approach to Statistical
            Mechanics ......................................... 126
     3.3.5  Statistical Mechanics Models for Materials ........ 129
     3.3.6  Bounds and Inequalities: The Bogoliubov
            Inequality ........................................ 135
     3.3.7  Correlation Functions: The Kinematics of Order .... 137
     3.3.8  Computational Statistical Mechanics ............... 139
3.4  Further Reading .......................................... 142
3.5  Problems ................................................. 144

Part two: Energetics of Crystalline Solids .................... 147
4    Energetic Description of Cohesion in Solids .............. 149
4.1  The Role of the Total Energy in Modeling Materials ....... 149
4.2  Conceptual Backdrop for Characterizing the Total
     Energy ................................................... 152
     4.2.1  Atomistic and Continuum Descriptions Contrasted ... 152
     4.2.2  The Many-Particle Hamiltonian and Degree of
            Freedom Reduction ................................. 154
4.3  Pair Potentials .......................................... 156
     4.3.1  Generic Pair Potentials ........................... 156
     4.3.2  Free Electron Pair Potentials ..................... 158
4.4  Potentials with Environmental and Angular Dependence ..... 164
     4.4.1  Diagnostics for Evaluating Potentials ............. 164
     4.4.2  Pair Functionals .................................. 165
     4.4.3  Angular Forces: A First Look ...................... 172
4.5  Tight-Binding Calculations of the Total Energy ........... 176
     4.5.1  The Tight-Binding Method .......................... 176
     4.5.2  An Aside on Periodic Solids: k-space Methods ...... 184
     4.5.3  Real Space Tight-Binding Methods .................. 189
4.6  First-Principles Calculations of the Total Energy ........ 197
     4.6.1  Managing the Many-Particle Hamiltonian ............ 198
     4.6.2  Total Energies in the Local Density
            Approximation ..................................... 200
4.7  Choosing a Description of the Total Energy: Challenges
     and Conundrums ........................................... 203
4.8  Further Reading .......................................... 204
4.9  Problems ................................................. 206

5    Thermal and Elastic Properties of Crystals ............... 210
5.1  Thermal and Elastic Material Response .................... 210
5.2  Mechanics of the Harmonic Solid .......................... 213
     5.2.1  Total Energy of the Thermally Fluctuating Solid ... 214
     5.2.2  Atomic Motion and Normal Modes .................... 216
     5.2.3  Phonons ........................................... 228
     5.2.4  Buckminsterfullerene and Nanotubes: A Case Study
            in Vibration ...................................... 229
5.3  Thermodynamics of Solids ................................. 231
     5.3.1  Harmonic Approximation ............................ 231
     5.3.2  Beyond the Harmonic Approximation ................. 239
5.4  Modeling the Elastic Properties of Materials ............. 244
     5.4.1  Linear Elastic Moduli ............................. 244
     5.4.2  Nonlinear Elastic Material Response: Cauchy-Born
            Elasticity ........................................ 248
5.5  Further Reading .......................................... 250
5.6  Problems ................................................. 251

6    Structural Energies and Phase Diagrams ................... 253
6.1  Structures in Solids ..................................... 253
6.2  Atomic-Level Geometry in Materials ....................... 254
6.3  Structural energies of solids ............................ 260
     6.3.1  Pair Potentials and Structural Stability .......... 261
     6.3.2  Structural Stability in Transition Metals ......... 264
     6.3.3  Structural Stability Reconsidered: The Case of
            Elemental Si ...................................... 265
6.4  Elemental Phase Diagrams ................................. 268
     6.4.1  Free Energy of the Crystalline Solid .............. 268
     6.4.2  Free Energy of the Liquid ......................... 275
     6.4.3  Putting It All Together ........................... 277
     6.4.4  An Einstein Model for Structural Change ........... 278
     6.4.5  A Case Study in Elemental Mg ...................... 280
6.5  Alloy Phase Diagrams ..................................... 282
     6.5.1  Constructing the Effective Energy: Cluster
            Expansions ........................................ 283
     6.5.2  Statistical Mechanics for the Effective
            Hamiltonian ....................................... 291
     6.5.3  The Effective Hamiltonian Revisited: Relaxations
            and Vibrations .................................... 297
     6.5.4  The Alloy Free Energy ............................. 299
6.5.5  Case Study: Oxygen Ordering in High TC
            Superconductors ................................... 300
6.6  Summary .................................................. 304
6.7  Further Reading .......................................... 304
6.8  Problems ................................................. 305

Part three: Geometric Structures in Solids: Defects and
Microstructures ............................................... 309
7    Point Defects in Solids .................................. 311
7.1  Point Defects and Material Response ...................... 311
     7.1.1  Material Properties Related to Point Disorder ..... 312
7.2  Diffusion ................................................ 318
     7.2.1  Effective Theories of Diffusion ................... 318
7.3  Geometries and Energies of Point Defects ................. 326
     7.3.1  Crystallographic Preliminaries .................... 327
     7.3.2  A Continuum Perspective on Point Defects .......... 328
     7.3.3  Microscopic Theories of Point Defects ............. 332
     7.3.4  Point Defects in Si: A Case Study ................. 341
7.4  Point Defect Motions ..................................... 344
     7.4.1  Material Parameters for Mass Transport ............ 345
     7.4.2  Diffusion via Transition State Theory ............. 346
     7.4.3  Diffusion via Molecular Dynamics .................. 351
     7.4.4  A Case Study in Diffusion: Interstitials in Si .... 353
7.5  Defect Clustering ........................................ 356
7.6  Further Reading .......................................... 356
7.7  Problems ................................................. 359

8    Line Defects in Solids ................................... 362
8.1  Permanent Deformation of Materials ....................... 362
     8.1.1  Yield and Hardening ............................... 363
     8.1.2  Structural Consequences of Plastic Deformation .... 365
     8.1.3  Single Crystal Slip and the Schmid Law ............ 367
8.2  The Ideal Strength Concept and the Need for
     Dislocations ............................................. 369
8.3  Geometry of Slip ......................................... 371
     8.3.1  Topological Signature of Dislocations ............. 372
     8.3.2  Crystallography of Slip ........................... 375
8.4  Elastic Models of Single Dislocations .................... 382
     8.4.1  The Screw Dislocation ............................. 382
     8.4.2  The Volterra Formula .............................. 388
     8.4.3  The Edge Dislocation .............................. 391
     8.4.4  Mixed Dislocations ................................ 392
8.5  Interaction Energies and Forces .......................... 393
     8.5.1  The Peach-Koehler Formula ......................... 395
     8.5.2  Interactions and Images: Peach-Koehler Applied .... 398
     8.5.3  The Line Tension Approximation .................... 402
8.6  Modeling the Dislocation Core: Beyond Linearity .......... 404
     8.6.1  Dislocation Dissociation .......................... 404
     8.6.2  The Peierls-Nabarro Model ......................... 406
     8.6.3  Structural Details of the Dislocation Core ........ 412
8.7  Three-Dimensional Dislocation Configurations ............. 415
     8.7.1  Dislocation Bow-Out ............................... 416
     8.7.2  Kinks and Jogs .................................... 418
     8.7.3  Cross Slip ........................................ 423
     8.7.4  Dislocation Sources ............................... 426
     8.7.5  Dislocation Junctions ............................. 430
8.8  Further Reading .......................................... 435
8.9  Problems ................................................. 437

9    Wall Defects in Solids ................................... 441
9.1  Interfaces in Materials .................................. 441
     9.1.1  Interfacial Confinement ........................... 442
9.2  Free Surfaces ............................................ 446
     9.2.1  Crystallography and Energetics of Ideal Surfaces .. 447
     9.2.2  Reconstruction at Surfaces ........................ 452
     9.2.3  Steps on Surfaces ................................. 474
9.3  Stacking Faults and Twins ................................ 476
     9.3.1  Structure and Energetics of Stacking Faults ....... 477
     9.3.2  Planar Faults and Phase Diagrams .................. 484
9.4  Grain Boundaries ......................................... 487
     9.4.1  Bicrystal Geometry ................................ 489
     9.4.2  Grain Boundaries in Polycrystals .................. 492
     9.4.3  Energetic Description of Grain Boundaries ......... 494
     9.4.4  Triple Junctions of Grain Boundaries .............. 500
9.5  Diffuse Interfaces ....................................... 501
9.6  Modeling Interfaces: A Retrospective ..................... 502
9.7  Further Reading .......................................... 503
9.8  Problems ................................................. 505

10   Microstructure and its Evolution ......................... 507
10.1 Microstructures in Materials ............................. 508
     10.1.1 Microstructural Taxonomy .......................... 508
     10.1.2 Microstructural Change ............................ 516
     10.1.3 Models of Microstructure and its Evolution ........ 519
10.2 Inclusions as Microstructure ............................. 520
     10.2.1 Eshelby and the Elastic Inclusion ................. 520
     10.2.2 The Question of Equilibrium Shapes ................ 527
     10.2.3 Precipitate Morphologies and Interfacial Energy ... 528
     10.2.4 Equilibrium Shapes: Elastic and Interfacial
            Energy ............................................ 529
     10.2.5 A Case Study in Inclusions: Precipitate
            Nucleation ........................................ 537
     10.2.6 Temporal Evolution of Two-Phase Microstructures ... 540
10.3 Microstructure in Martensites ............................ 546
     10.3.1 The Experimental Situation ........................ 547
     10.3.2 Geometrical and Energetic Preliminaries ........... 551
     10.3.3 Twinning and Compatibility ........................ 554
     10.3.4 Fine-Phase Microstructures and Attainment ......... 560
     10.3.5 The Austenite-Martensite Free Energy
            Reconsidered ...................................... 565
10.4 Microstructural Evolution in Polycrystals ................ 566
     10.4.1 Phenomenology of Grain Growth ..................... 567
     10.4.2 Modeling Grain Growth ............................. 568
10.5 Microstructure and Materials ............................. 580
10.6 Further Reading .......................................... 580
10.7 Problems ................................................. 582

Part four: Facing the Multiscale Challenge of Real Material
Behavior ...................................................... 585
11   Points, Lines and Walls: Defect Interactions and
     Material Response ........................................ 587
11.1 Defect Interactions and the Complexity of Real Material
     Behavior ................................................. 587
11.2 Diffusion at Extended Defects ............................ 588
     11.2.1 Background on Short-Circuit Diffusion ............. 588
     11.2.2 Diffusion at Surfaces ............................. 589
11.3 Mass Transport Assisted Deformation ...................... 592
     11.3.1 Phenomenology of Creep ............................ 593
     11.3.2 Nabarro-Herring and Coble Creep ................... 595
11.4 Dislocations and Interfaces .............................. 599
     11.4.1 Dislocation Models of Grain Boundaries ............ 600
     11.4.2 Dislocation Pile-Ups and Slip Transmission ........ 604
11.5 Cracks and Dislocations .................................. 609
     11.5.1 Variation on a Theme of Irwin ..................... 610
     11.5.2 Dislocation Screening at a Crack Tip .............. 611
     11.5.3 Dislocation Nucleation at a Crack Tip ............. 615
11.6 Dislocations and Obstacles: Strengthening ................ 620
     11.6.1 Conceptual Overview of the Motion of Dislocations
            Through a Field of Obstacles ...................... 622
     11.6.2 The Force Between Dislocations and Glide
            Obstacles ......................................... 625
     11.6.3 The Question of Statistical Superposition ......... 628
     11.6.4 Solution Hardening ................................ 633
     11.6.5 Precipitate Hardening ............................. 636
     11.6.6 Dislocation-Dislocation Interactions and Work
            Hardening ......................................... 642
11.7 Further Reading .......................................... 644
     11.8 Problems ............................................ 647

12   Bridging Scales: Effective Theory Construction ........... 649
12.1 Problems Involving Multiple Length and Time Scales ....... 651
     12.1.1 Problems with Multiple Temporal Scales: The
            Example of Diffusion .............................. 652
     12.1.2 Problems with Multiple Spatial Scales: The
            Example of Plasticity ............................. 653
     12.1.3 Generalities on Modeling Problems Involving
            Multiple Scales ................................... 655
12.2 Historic Examples of Multiscale Modeling ................. 658
12.3 Effective Theory Construction ............................ 668
     12.3.1 Degree of Freedom Selection: State Variables,
            Order Parameters and Configurational Coordinates .. 669
     12.3.2 Dynamical Evolution of Relevant Variables:
            Gradient Flow Dynamics and Variational
            Principles ........................................ 674
     12.3.3 Inhomogeneous Systems and the Role of Locality .... 685
     12.3.4 Models with Internal Structure .................... 688
     12.3.5 Effective Hamiltonians ............................ 697
12.4 Bridging Scales in Microstructural Evolution ............. 701
     12.4.1 Hierarchical Treatment of Diffusive Processes ..... 701
     12.4.2 From Surface Diffusion to Film Growth ............. 709
     12.4.3 Solidification Microstructures .................... 711
     12.4.4 Two-Phase Microstructures Revisited ............... 715
     12.4.5 A Retrospective on Modeling Microstructural
            Evolution ......................................... 718
12.5 Bridging Scales in Plasticity ............................ 719
     12.5.1 Mesoscopic Dislocation Dynamics ................... 720
     12.5.2 A Case Study in Dislocations and Plasticity:
            Nanoindentation ................................... 728
     12.5.3 A Retrospective on Modeling Plasticity Using
            Dislocation Dynamics .............................. 731
12.6 Bridging Scales in Fracture .............................. 732
     12.6.1 Atomic-Level Bond Breaking ........................ 732
     12.6.2 Cohesive Surface Models ........................... 734
     12.6.3 Cohesive Surface Description of Crack Tip
            Dislocation Nucleation ............................ 735
12.7 Further Reading .......................................... 736
12.8 Problems ................................................. 738

13   Universality and Specificity in Materials ................ 742
13.1 Materials Observed ....................................... 743
     13.1.1 What is a Material: Another Look .................. 743
     13.1.2 Structural Observations ........................... 744
     13.1.3 Concluding Observations on the Observations ....... 746
13.2 How Far Have We Come? .................................... 748
     13.2.1 Universality in Materials ......................... 749
     13.2.2 Specificity in Materials .......................... 750
     13.2.3 The Program Criticized ............................ 751
13.3 Intriguing Open Questions ................................ 752
13.4 In Which the Author Takes His Leave ...................... 754

References .................................................... 757
Index ......................................................... 111

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