Random heterogeneous materials: microstructure and macroscopic properties
Saved in:
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2002
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| Series: | Interdisciplinary applied mathematics
16 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 701 S. graph. Darst. |
| ISBN: | 9780387951676 0387951679 |
Staff View
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| 100 | 1 | |a Torquato, Salvatore |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Random heterogeneous materials |b microstructure and macroscopic properties |c Salvatore Torquato |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2002 | |
| 300 | |a XXI, 701 S. |b graph. Darst. | ||
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| 490 | 1 | |a Interdisciplinary applied mathematics |v 16 | |
| 490 | 0 | |a Mechanics and materials | |
| 650 | 4 | |a Inhomogenes Medium - Mikrostruktur - Physikalische Eigenschaft | |
| 650 | 4 | |a Inhomogeneous materials | |
| 650 | 4 | |a Microstructure | |
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| adam_text | Contents
Preface vii
1 Motivation and Overview 1
1.1 What Is a Heterogeneous Material? 1
1.2 Effective Properties and Applications 3
1.2.1 Conductivity and Analogous Properties 6
1.2.2 Elastic Moduli 7
1.2.3 Survival Time or Trapping Constant 8
1.2.4 Fluid Permeability 8
1.2.5 Diffusion and Viscous Relaxation Times 9
1.2.6 Definitions of Effective Properties 9
1.3 Importance of Microstructure 10
1.4 Development of a Systematic Theory 12
1.4.1 Microstructural Details 12
1.4.2 Multidisciplinary Research Area 14
1.5 Overview of the Book 17
1.5.1 Parti 17
1.5.2 Part II 18
1.5.3 Scope 19
1 Microstructure Characterization 21
2 Microstructural Descriptors 23
2.1 Preliminaries 24
xii Contents
2.2 rz Point Probability Functions 25
2.2.1 Definitions 25
2.2.2 Symmetries and Ergodicity 28
2.2.3 Geometrical Probability Interpretation 32
2.2.4 Asymptotic Properties and Bounds 33
2.2.5 Two Point Probability Function 34
2.3 Surface Correlation Functions 43
2.4 Lineal Path Function 44
2.5 Chord Length Density Function 45
2.6 Pore Size Functions 48
2.7 Percolation and Cluster Functions 50
2.8 Nearest Neighbor Functions 50
2.9 Point/g Particle Correlation Functions 57
2.10 Surface/Particle Correlation Function 58
3 Statistical Mechanics of Many Particle Systems 59
3.1 Many Particle Statistics 60
3.1.1 n Particle Probability Densities 60
3.1.2 Pair Potentials 65
3.2 Ornstein Zernike Formalism 72
3.3 Equilibrium Hard Sphere Systems 75
3.3.1 Low Density Expansions 79
3.3.2 Arbitrary Fluid Densities 81
3.4 Random Sequential Addition Processes 83
3.4.1 One Dimensional Identical Hard Rods 85
3.4.2 Identical Hard Spheres in Higher Dimensions 87
3.4.3 General Hard Particle Systems 88
3.5 Maximally Random Jammed State 88
3.5.1 Random Close Packing Is Ill Defined 89
3.5.2 Definition of Maximally Random Jammed State 90
3.5.3 Order Metrics 92
3.5.4 Molecular Dynamics Simulations 93
3.5.5 Concluding Remarks 95
4 Unified Approach to Characterize Microstructure 96
4.1 Volume Fraction and Specific Surface 97
4.1.1 Bounding Properties 100
4.1.2 Example Calculations 102
4.2 Canonical Correlation Function Hn 104
4.2.1 Definitions 105
4.2.2 Asymptotic Properties 109
4.3 Series Representations of Hn 109
4.3.1 Mayer Representation 110
Contents xiii
4.3.2 Kirkwood Salsburg Representation Ill
4.3.3 Bounding Properties 112
4.4 Special Cases of Hn 114
4.5 Polydispersivity 116
4.6 Other Model Microstructures 118
5 Monodisperse Spheres 119
5.1 Fully Penetrable Spheres 120
5.1.1 K Point Probability Functions 122
5.1.2 Surface Correlation Functions 124
5.1.3 Lineal Path Function 125
5.1.4 Chord Length Density Function 127
5.1.5 Nearest Neighbor Functions 128
5.1.6 Pore Size Functions 128
5.1.7 Point/g Particle Correlation Functions 129
5.2 Totally Impenetrable Spheres 129
5.2.1 n Point Probability Functions 130
5.2.2 Surface Correlation Functions 134
5.2.3 Lineal Path Function 136
5.2.4 Chord Length Density Function 137
5.2.5 Nearest Neighbor Functions 139
5.2.6 Pore Size Functions 151
5.2.7 Point/g Particle Correlation Functions 152
5.3 Interpenetrable Spheres 153
5.3.1 Nearest Neighbor Functions 154
5.3.2 Volume Fraction 155
5.3.3 Specific Surface 155
5.3.4 Pore Size Functions 157
5.3.5 Other Statistical Descriptors 157
5.4 Statistically Inhomogeneous Systems 158
6 Polydisperse Spheres 160
6.1 Fully Penetrable Spheres 161
6.1.1 M Point Probability Functions 163
6.1.2 Surface Correlation Functions 164
6.1.3 Lineal Path Function 165
6.1.4 Chord Length Density Function 166
6.1.5 Nearest Surface Functions 166
6.1.6 Pore Size Functions 167
6.1.7 Point/^ Particle Correlation Functions 167
6.2 Totally Impenetrable Spheres 167
6.2.1 H Point Probability Functions 169
6.2.2 Surface Correlation Functions 170
xiv Contents
6.2.3 Lineal Path Function 171
6.2.4 Chord Length Density Function 171
6.2.5 Nearest Surface Functions 172
6.2.6 Pore Size Functions 176
6.2.7 Point/g Particle Correlation Functions 176
7 Anisotropic Media 177
7.1 General Considerations 177
7.2 Fully Penetrable Oriented Inclusions 179
7.3 Impenetrable Oriented Inclusions 181
7.4 Hierarchical Laminates 183
8 Cell and Random Field Models 188
8.1 Cell Models . 188
8.1.1 Voronoi and Delaunay Tessellations 189
8.1.2 Cell Statistics 192
8.1.3 Symmetric Cell Materials 194
8.1.4 Random Checkerboard 199
8.1.5 Ising Model 201
8.2 Random Field Models 203
8.2.1 General Considerations 203
8.2.2 Gaussian Convolved Intensities 207
9 Percolation and Clustering 210
9.1 Lattice Percolation 211
9.1.1 Bond and Site Percolation 211
9.1.2 Percolation Properties 215
9.1.3 Scaling and Critical Exponents 217
9.1.4 Infinite Cluster and Fractality 222
9.1.5 Finite Size Scaling 223
9.2 Continuum Percolation 224
9.2.1 Percolation Properties 227
9.2.2 Two Point Cluster Function 230
9.2.3 Critical Exponents 231
10 Some Continuum Percolation Results 234
10.1 Exact Results for Overlapping Spheres 234
10.1.1 One Dimension 235
10.1.2 Higher Dimensions 240
10.1.3 Low Density Expansions of Cluster Statistics 242
10.2 Ornstein Zernike Formalism 243
10.3 Percus Yevick Approximations 245
10.3.1 Permeable Sphere Model 246
Contents xv
10.3.2 Cherry Pit Model 248
10.3.3 Sticky Hard Sphere Model 249
10.4 Beyond Percus Yevick Approximations 250
10.5 Two Point Cluster Function 250
10.6 Percolation Threshold Estimates 251
10.6.1 Overlapping Disks and Spheres 252
10.6.2 Nonspherical Overlapping Particles 254
10.6.3 Interacting Particle Systems 255
11 Local Volume Fraction Fluctuations 257
11.1 Definitions 258
11.2 Coarseness 260
11.2.1 General Formula 260
11.2.2 Asymptotic Formula 261
11.2.3 Calculations 262
11.3 Moments of Local Volume Fraction 264
11.4 Evaluations of Full Distribution 265
12 Computer Simulations, Image Analyses, and Reconstructions 269
12.1 Monte Carlo Simulations 270
12.1.1 Introduction 270
12.1.2 Importance Sampling 271
12.2 Metropolis Method for Gibbs Ensembles 273
12.2.1 Markov Chain 273
12.2.2 Algorithm 275
12.2.3 Practical Implementation 275
12.2.4 Hard Spheres 277
12.2.5 Other Particle Systems 278
12.2.6 Cell Models 279
12.3 Methods for Generating Nonequilibrium Ensembles 279
12.4 Sampling in Particle Systems 281
12.4.1 Radial Distribution Function 281
12.4.2 n point Probability Functions 283
12.4.3 Surface Con elation Functions 285
12.4.4 Cluster Type Functions 285
12.4.5 Other Correlation Functions 286
12.5 Sampling Images and Digitized Media 287
12.5.1 Two Point Probability Function 289
12.5.2 Lineal Path Function 291
12.5.3 Chord Length Density Function 292
12.5.4 Pore Size Functions 292
12.5.5 Two Point Cluster Function 293
xvi Contents
12.6 Reconstructing Heterogeneous Materials 294
12.6.1 Reconstruction Procedure 295
12.6.2 Illustrative Examples 297
II Microstructure/Property Connection 303
13 Local and Homogenized Equations 305
13.1 Preliminaries 306
13.2 Conduction Problem 308
13.2.1 Local Relations 308
13.2.2 Conduction Symmetry 311
13.2.3 Model One Dimensional Problem 313
13.2.4 Homogenization of Periodic Problem in did 315
13.2.5 Homogenization of Random Problem in Wrf 318
13.2.6 Frequency Dependent Conductivity 321
13.3 Elastic Problem 321
13.3.1 Local Relations 321
13.3.2 Elastic Symmetry 324
13.3.3 Homogenization of Random Problem in ffirf 332
13.3.4 Heterogeneous Materials 334
13.3.5 Relationship Between Elasticity and Viscous Fluid Theory . . . 337
13.3.6 Viscosity of a Suspension 338
13.3.7 Viscoelasticity 339
13.4 Steady State Trapping Problem 339
13.4.1 Local Relations 341
13.4.2 Homogenization of Random Problem in !)irf 341
13.5 Steady State Fluid Permeability Problem 344
13.5.1 Local Relations 345
13.5.2 Homogenization of Random Problem in Md 346
13.5.3 Relationship to Sedimentation Rate 348
13.6 Classification of Steady State Problems 349
13.7 Time Dependent Trapping Problem 350
13.7.1 Basic Equations 350
13.7.2 Relationship Between Survival and Relaxation Times 353
13.8 Time Dependent Flow Problem 354
13.8.1 Basic Equations 354
13.8.2 Relationship Between Permeability and Relaxation Times . . . 356
14 Variational Principles 357
14.1 Conductivity 359
14.1.1 Field Fluctuations 359
14.1.2 Energy Representation 361
Contents xvii
14.1.3 Minimum Energy Principles 363
14.1.4 Hashin Shtrikman Principle 367
14.2 Elastic Moduli 368
14.2.1 Field Fluctuations 369
14.2.2 Energy Representation 370
14.2.3 Minimum Energy Principles 373
14.2.4 Hashin Shtrikman Principle 377
14.3 Trapping Constant 379
14.3.1 Energy Representation 379
14.3.2 Minimum Energy Principles 380
14.4 Fluid Permeability 383
14.4.1 Energy Representation 383
14.4.2 Minimum Energy Principles 385
15 Phase Interchange Relations 390
15.1 Conductivity 390
15.1.1 Duality for Two Dimensional Media 390
15.1.2 Three Dimensional Media 397
15.2 Elastic Moduli 398
15.2.1 Two Dimensional Media 398
15.2.2 Three Dimensional Media 401
15.3 Trapping Constant and Fluid Permeability 402
16 Exact Results 403
16.1 Conductivity 404
16.1.1 Coated Spheres Model 404
16.1.2 Simple Laminates 407
16.1.3 Higher Order Laminates and Attaimibility 410
16.1.4 Fiber Reinforced Materials 413
16.1.5 Periodic Arrays of Inclusions 413
16.1.6 Low Density Cellular Solids 415
16.1.7 Field Fluctuations 416
16.2 Elastic Moduli 417
16.2.1 Coated Spheres Model 417
16.2.2 Simple Laminates 419
16.2.3 Higher Order Laminates and Attainability 424
16.2.4 Periodic Arrays of Inclusions 426
16.2.5 Low Density Cellular Solids 428
16.2.6 Equal Phase Shear Moduli 429
16.2.7 Sheets with Holes 429
16.2.8 Dispersions of Particles in a Liquid 429
16.2.9 Cavities (Bubbles) in an Incompressible Matrix (Liquid) . . . 429
16.2.10 Field Fluctuations 430
xviii Contents
16.2.11 Link to Two Dimensional Conductivity 430
16.2.12 Link to Thermoelastic Constants 431
16.3 Trapping Constant 432
16.3.1 Diffusion Inside Hyperspheres 432
16.3.2 Periodic Arrays of Traps 433
16.4 Fluid Permeability 434
16.4.1 Flow Between Plates and Inside Tubes 434
16.4.2 Periodic Arrays of Obstacles 436
17 Single Inclusion Solutions 437
17.1 Conduction Problem 437
17.1.1 Spherical Inclusion 437
17.1.2 Polarization Within an Ellipsoid 441
17.2 Elasticity Problem 442
17.2.1 Spherical Inclusion 442
17.2.2 Polarization Within an Ellipsoid 448
17.3 Trapping Problem 451
17.3.1 Spherical Trap 451
17.3.2 Spheroidal Trap 453
17.4 Flow Problem 455
17.4.1 Spherical Obstacle 455
17.4.2 Spheroidal Obstacle 457
18 Effective Medium Approximations 459
18.1 Conductivity 459
18.1.1 Maxwell Approximations 460
18.1.2 Self Consistent Approximations 462
18.1.3 Differential Effective Medium Approximations 467
18.2 Elastic Moduli 470
18.2.1 Maxwell Approximations 470
18.2.2 Self Consistent Approximations 474
18.2.3 Differential Effective Medium Approximations 477
18.3 Trapping Constant 479
18.4 Fluid Permeability 481
19 Cluster Expansions 485
19.1 Conductivity 486
19.1.1 Dilute Dispersions of Spheres 488
19.1.2 Dilute Dispersions of Ellipsoids 490
19.1.3 Nondilute Concentrations 491
19.2 Elastic Moduli 496
19.2.1 Dilute Dispersions of Spheres 497
19.2.2 Dilute Dispersions of Ellipsoids 500
Contents xix
19.2.3 Nondilute Concentrations 501
19.3 Trapping Constant 502
19.3.1 Dilute Dispersions of Spherical Traps 502
19.3.2 Dilute Dispersions of Spheroidal Traps 503
19.3.3 Nondilute Concentrations 504
19.4 Fluid Permeability 505
19.4.1 Dilute Beds of Spheres 505
19.4.2 Dilute Beds of Spheroids 506
19.4.3 Nondilute Concentrations 507
20 Exact Contrast Expansions 509
20.1 Conductivity Tensor 510
20.1.1 Integral Equation for Cavity Electric Field 511
20.1.2 Strong Contrast Expansions 514
20.1.3 Some Tensor Properties 519
20.1.4 Weak Contrast Expansions 520
20.1.5 Expansion of Local Electric Field 521
20.1.6 Isotropic Media 521
20.2 Stiffness Tensor 530
20.2.1 Integral Equation for the Cavity Strain Field 530
20.2.2 Strong Contrast Expansions 534
20.2.3 Weak Contrast Expansions 539
20.2.4 Expansion of Local Strain Field 540
20.2.5 Isotropic Media 541
21 Rigorous Bounds 552
21.1 Conductivity 554
21.1.1 General Considerations 554
21.1.2 Contrast Bounds 555
21.1.3 Cluster Bounds 563
21.1.4 Security Spheres Bounds 564
21.2 Elastic Moduli 566
21.2.1 General Considerations 566
21.2.2 Contrast Bounds 568
21.2.3 Cluster Bounds 576
21.2.4 Security Spheres Bounds 577
21.3 Trapping Constant 578
21.3.1 Interfacial Surface Lower Bound 579
21.3.2 Void Lower Bound 580
21.3.3 Cluster Lower Bounds 581
21.3.4 Security Spheres Upper Bound 582
21.3.5 Pore Size Upper Bound 584
xx Contents
21.4 Fluid Permeability 585
21.4.1 Interfacial Surface Upper Bound 585
21.4.2 Void Upper Bound 586
21.4.3 Cluster Upper Bounds 587
21.4.4 Security Spheres Lower Bound 589
21.5 Structural Optimization 590
21.6 Utility of Bounds 592
22 Evaluation of Bounds 593
22.1 Conductivity 594
22.1.1 Contrast Bounds 594
22.1.2 Cluster Bounds 609
22.1.3 Security Spheres Bounds 610
22.2 Elastic Moduli 611
22.2.1 Contrast Bounds 611
22.2.2 Cluster Bounds 620
22.2.3 Security Spheres Bounds 620
22.3 Trapping Constant 621
22.3.1 Interfacial Surface Lower Bound 621
22.3.2 Void Lower Bound 623
22.3.3 Cluster Lower Bounds 624
22.3.4 Security Spheres Upper Bound 625
22.3.5 Pore Size Upper Bound 625
22.4 Fluid Permeability 627
22.4.1 Interfacial Surface Upper Bound 627
22.4.2 Void Upper Bound 629
22.4.3 Cluster Upper Bounds 630
22 A A Security Spheres Lower Bound 631
23 Cross Property Relations 632
23.1 Conductivity and Elastic Moduli 633
23.1.1 Elementary Bounds 633
23.1.2 Translation Bounds fore? = 2 636
23.1.3 Translation Bounds for d = 3 642
23.2 Flow and Diffusion Parameters 647
23.2.1 Permeability and Survival Time 647
23.2.2 Permeability, Formation Factor,
and Viscous Relaxation Times 650
23.2.3 Viscous and Diffusion Relaxation Times 654
A Equilibrium Hard Disk Program 656
B Interrelations Among Two and Three Dimensional Moduli 661
Contents xxi
References 663
Index 693
|
| any_adam_object | 1 |
| author | Torquato, Salvatore |
| author_facet | Torquato, Salvatore |
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| discipline | Physik Werkstoffwissenschaften |
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| id | DE-604.BV013972420 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:55:22Z |
| institution | BVB |
| isbn | 9780387951676 0387951679 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009563271 |
| oclc_num | 248518949 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-526 DE-91 DE-BY-TUM DE-83 |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-29T DE-19 DE-BY-UBM DE-526 DE-91 DE-BY-TUM DE-83 |
| physical | XXI, 701 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series | Interdisciplinary applied mathematics |
| series2 | Interdisciplinary applied mathematics Mechanics and materials |
| spelling | Torquato, Salvatore Verfasser aut Random heterogeneous materials microstructure and macroscopic properties Salvatore Torquato New York [u.a.] Springer 2002 XXI, 701 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Interdisciplinary applied mathematics 16 Mechanics and materials Inhomogenes Medium - Mikrostruktur - Physikalische Eigenschaft Inhomogeneous materials Microstructure Heterogenes System (DE-588)4159737-0 gnd rswk-swf Mikrostruktur (DE-588)4131028-7 gnd rswk-swf Physikalische Eigenschaft (DE-588)4134738-9 gnd rswk-swf Inhomogenes Medium (DE-588)4228459-4 gnd rswk-swf Medium Physik (DE-588)4224991-0 gnd rswk-swf Stochastisches System (DE-588)4057635-8 gnd rswk-swf Inhomogenes Medium (DE-588)4228459-4 s Mikrostruktur (DE-588)4131028-7 s Physikalische Eigenschaft (DE-588)4134738-9 s DE-604 Medium Physik (DE-588)4224991-0 s Heterogenes System (DE-588)4159737-0 s Stochastisches System (DE-588)4057635-8 s 1\p DE-604 Interdisciplinary applied mathematics 16 (DE-604)BV004216726 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009563271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Torquato, Salvatore Random heterogeneous materials microstructure and macroscopic properties Interdisciplinary applied mathematics Inhomogenes Medium - Mikrostruktur - Physikalische Eigenschaft Inhomogeneous materials Microstructure Heterogenes System (DE-588)4159737-0 gnd Mikrostruktur (DE-588)4131028-7 gnd Physikalische Eigenschaft (DE-588)4134738-9 gnd Inhomogenes Medium (DE-588)4228459-4 gnd Medium Physik (DE-588)4224991-0 gnd Stochastisches System (DE-588)4057635-8 gnd |
| subject_GND | (DE-588)4159737-0 (DE-588)4131028-7 (DE-588)4134738-9 (DE-588)4228459-4 (DE-588)4224991-0 (DE-588)4057635-8 |
| title | Random heterogeneous materials microstructure and macroscopic properties |
| title_auth | Random heterogeneous materials microstructure and macroscopic properties |
| title_exact_search | Random heterogeneous materials microstructure and macroscopic properties |
| title_full | Random heterogeneous materials microstructure and macroscopic properties Salvatore Torquato |
| title_fullStr | Random heterogeneous materials microstructure and macroscopic properties Salvatore Torquato |
| title_full_unstemmed | Random heterogeneous materials microstructure and macroscopic properties Salvatore Torquato |
| title_short | Random heterogeneous materials |
| title_sort | random heterogeneous materials microstructure and macroscopic properties |
| title_sub | microstructure and macroscopic properties |
| topic | Inhomogenes Medium - Mikrostruktur - Physikalische Eigenschaft Inhomogeneous materials Microstructure Heterogenes System (DE-588)4159737-0 gnd Mikrostruktur (DE-588)4131028-7 gnd Physikalische Eigenschaft (DE-588)4134738-9 gnd Inhomogenes Medium (DE-588)4228459-4 gnd Medium Physik (DE-588)4224991-0 gnd Stochastisches System (DE-588)4057635-8 gnd |
| topic_facet | Inhomogenes Medium - Mikrostruktur - Physikalische Eigenschaft Inhomogeneous materials Microstructure Heterogenes System Mikrostruktur Physikalische Eigenschaft Inhomogenes Medium Medium Physik Stochastisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009563271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004216726 |
| work_keys_str_mv | AT torquatosalvatore randomheterogeneousmaterialsmicrostructureandmacroscopicproperties |