Stochastic modeling of microstructures/
Kazimierz Sobczyk, David J. Kirkner.
- Boston : Birkhäuser, c2001.
- viii, 270 p. : ill. ; 25 cm.
1. Probability and Random Variables: A Short Resume 5 1.1 Basic Concepts .................. ... ........................... .......... ....5 1.2 Some Probability Distributions ............. .......1......................... 11 1.3 Convergence of Sequences of Random Variables ................................ 14 1.4 Stochastic Processes .............................................................................17 1.4.1 Basic Concepts ....................................... 17 1.4.2 Mean and Covariance Function .............................................21 1.4.3 Gaussian Processes .................................................................23 1.4.4 Stationary Processes ..............................................................25 1.4.5 Markov Processes ...................................................................29 2. Continuous Random Fields 33 2.1 Basic Concepts ......................... ..........................................................33 2.2 Homogeneous Random Fields ..............................................................37 2.3 Isotropic Random Fields ......................................................................41 2.3.1 Definition and Spectral Analysis ............................................41 2.3.2 Special Cases .........................................................................45 2.3.3 Variance of Isotropic Fields .............................................. 49 2.4 Locally Homogeneous and Isotropic Random Fields................ ..........51 2.4.1 Locally Homogeneous Fields and the Structure Function .....51 2.4.2 Locally Isotropic Fields ..........................................................53 2.5 Space-Time Random Fields ..................................................................56 2.6 Vector-Valued Random Fields ..............................................................59
2.6.1 Basic Concepts ............................ .................................. 59 2.6.2 Isotropic Vector Random Fields ........................................ 60 2.7 Tensor-Valued Random Fields ............................................................. 65 2.8 Markov Random Fields ........................................................................ 71 2.8.1 Basic Concepts................................................ .. ..... 71 2.8.2 Brownian Fields .....................................................................73 2.9 Fields Governed by Stochastic Equations . ........................... 74 3. Random Point Fields 79 3.1 Basic Properties........................... .................................. 79 3.2 Poisson Random Fields ....................................................................... 85 3.2.1 Homogeneous Poisson Random Field .................................. 85 3.2.2 Inhomogeneous Poisson Field ...................................... 88 3.2.3 Doubly Stochastic Poisson Field ....................................... . 90 3.2.4 Poisson Cluster Field ........................................................... 92 3.2.5 Poisson Hard-Core Field................................................... 93 3.3 Boolean Random Fields ............................................................... 94 3.4 Markov and Gibbs Fields ..................................................................... 98 3.4.1 Markov Fields ........................................................................ 98 3.4.2 Gibbs Fields ......................................................................... 102 3.5 Random Configurations of Objects ........................ ................. 104 3.5.1 Packing Problems ................................................................ 104 3.5.2 General Probability Distributions ........................................10 3.6 Random Set Patterns ........................................................................... 115 3.6.1 Random Sets .........................................................................115 3.6.2 Random Lines in a Plane ......................................................118 3.6.3 Random Tessellations ...........................................................119 4. Statistical Inference 123 4.1 Introductory Remarks......................................................................... 123 4.2 Estimation of Mean and Covariance .................................................. 124 4.3 Estimation of Spectral Density........................................................... 127
4.4 Prediction Problems: Kriging ............................................................. 130 4.5 Spatial Sampling Design.................................................................. 134 4.6 Inference for Point Fields ................................................................... 137 4.7 Stereology ....................................... 139 4.8 Simulation and Remarks ..................................................................... 146 5. Material Media Microstructure: Modeling Issues 149 5.1 Basic Characteristics of Microstructure . .................................. 149 5.1.1 Introductory Remarks ........................................................... 149 5.1.2 Correlation Structure .......... ................................. 150 5.1.3 Scales of Spatial Variation.................................................... 156 5.1.4 Self-Similarity and Fractals .................................................. 160 5.2 Averaging Procedures ........................................................................ 164 5.2.1 Representative Elementary Volume (REV) .......................... 164 5.2.2 Volume Averaging ................................................................ 166 5.2.3 Probabilistic Averaging ....................................................... 168 5.3 Homogenization and Smoothing ........................................................ 172 5.3.1 Homogenization: Effective Parameters ................................ 172 5.3.2 Equations for the Mean Field ............................................... 176 5.3.3 Perturbation Smoothing ........................................................ 180 5.4 Random Porous Media........................................................................ 182 5.4.1 Basic Properties .................................................................... 182 5.4.2 Darcy's Law and Its Justification ........................................ 184 5.4.3 Stochastic Models of Porous Media ..................................... 188 5.4.4 Other Problems and Remarks ............................................... 193 5.5 Spatial Randomness in Solid Materials .............................................. 194 5.5.1 Modeling Approaches .......................................................... 194 5.5.2 Random Elastic Solid Media ................................................ 196 6. Physical Phenomena in Random Microstructures: Selected Applications 203 6.1 Introductory Remarks ........................................ .... ....203 6.2 Wave Propagation .......................................205
6.2.1 Sound-Wave Effective Parameters...................................... 205 6.2.2 Scattering by Randomly Distributed Inclusions .................. 209 6.3 Pollution Transport in Groundwater.......... .............................. 218 6.3.1 Stochastic Flow and Transport Models................................ 218 6.3.2 Non-Local Mean Concentration Field .................................221 6.4 Deformation of Random Elastic Materials . ................. 227 6.5 Random Microstructure and Fracture . . ............................... 234 6.5.1 General Remarks ................. ....234 6.5.2 Crack Growth in Random Microstructure ...........................235 6.6 Other Problems and Remarks.............. ............. 243 References 247