Stochastic Modeling of Microstructures

A major challenge in applied mathematics and mechanics of materials is
to describe various types of material microstructures. The details of
the microstructure of most natural and engineered materials are
usually obscure; uncertainty and randomness are the inherent features.
This complexity due to material heterogeneity has not been A major
challenge in applied mathematics and mechanics of materials is to
describe various types of material microstructures. The details of the
microstructure of most natural and engineered materials are usually
obscure; uncertainty and randomness are the inherent features. This
complexity due to material heterogeneity has not been adequately
described by current classical models and theories. Stochastic
Modeling of Microstructures presents a concise and unified
presentation of the basic principles and tools for the modeling of
real materials, natural and man-made, that possess complex, random
heterogeneity. The book uses the language and methods of random field
theory combined with the basic constructs of stochastic geometry and
geometrical/spatial statistics in order to give the reader the
knowledge necessary to model various types of material
microstructures. The application of the theoretical constructs
reviewed in the first three chapters to the analysis of empirical data
via the tools of statistical inference is also discussed. The final
chapters address practical aspects of specific modeling problems.
Features- ú First comprehensive introduction to the comparatively new
field of stochastic modeling of material microstructures ú
Presentation of basic tools required from the diverse subjects of
random field theory, stochastic geometry and spatial statistics ú
Provides background concepts from probability theory and stochastic
processes are provided ú Applications from various fields are
discussed, including stochastic wave propagation and the mechanics of

1. Probability and Random Variables: A Short Résumé. - 1. 1 Basic Concepts. - 1. 2 Some Probability Distributions. - 1. 3 Convergence of Sequences of Random Variables. - 1. 4 Stochastic Processes. - 2. Continuous Random Fields. - 2. 1 Basic Concepts. - 2. 2 Homogeneous Random Fields. - 2. 3 Isotropic Random Fields. - 2. 4 Locally Homogeneous and Isotropic Random Fields. - 2. 5 Space-Time Random Fields. - 2. 6 Vector-Valued Random Fields. - 2. 7 Tensor-Valued Random Fields. - 2. 8 Markov Random Fields. - 2. 9 Fields Governed by Stochastic Equations. - 3. Random Point Fields. - 3. 1 Basic Properties. - 3. 2 Poisson Random Fields. - 3. 3 Boolean Random Fields. - 3. 4 Markov and Gibbs Fields. - 3. 5 Random Configurations of Objects. - 3. 6 Random Set Patterns. - 4. Statistical Inference. - 4. 1 Introductory Remarks. - 4. 2 Estimation of Mean and Covariance. - 4. 3 Estimation of Spectral Density. - 4. 4 Prediction Problems: Kriging. - 4. 5 Spatial Sampling Design. - 4. 6 Inference for Point Fields. - 4. 7 Stereology. - 4. 8 Simulation and Remarks. - 5. Material Media Microstructure: Modeling Issues. - 5. 1 Basic Characteristics of Microstructure. - 5. 2 Averaging Procedures. - 5. 3 Homogenization and Smoothing. - 5. 4 Random Porous Media. - 5. 5 Spatial Randomness in Solid Materials. - 6. Physical Phenomena in Random Microstructures: Selected Applications. - 6. 1 Introductory Remarks. - 6. 2 Wave Propagation. - 6. 3 Pollution Transport in Groundwater. - 6. 4 Deformation of Random Elastic Materials. - 6. 5 Random Microstructure and Fracture. - 6. 6 Other Problems and Remarks. - References. - Author Index.