7R3. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. - S Torquato (Dept of Chem, Princeton Mat Inst, Princeton Univ, Princeton NJ 08544). Springer-Verlag, New York. 2002. 701 pp. ISBN 0-387-95167-9. $69.95.

Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ of Maryland, College Park MD 20742-3035).

Random heterogeneous materials are those composed of randomly distributed phases or different material domains such as composites or materials with voids. The various domains forming the microstructure have macroscopic properties because they are larger than molecules. A major problem is to develop a rigorous method of characterizing the microstructure. Probability functions are used to describe the various microstructures, whether heterogeneous, homogeneous, isotropic, etc. A canonical n-point correlation function provides a unified means to describe arbitrary microstructures. The presentation is restricted to microstructures which are independent of time.

Several idealized models of the microstructure are presented. In a hard sphere model, the ill-defined idea of randomly closed packed spheres is replaced by the well-defined idea of a maximally random jammed configuration. The degree of penetrability and the distribution of radii of the spheres are adjusted to represent different types of microstructure. Anisotropy is modeled with ellipsoidal or cylindrical inclusions. Some hierarchical laminates have optimal transport properties. Foams or some biological materials are better modeled by the space-filling cell structures described. Percolation is modeled using the idea of clusters, a connected group of the random elements.

The Monte Carlo method is used to produce both equilibrium and non-equilibrium realizations of random media. The idea is to generate a random microstructure and then study its properties. An appendix lists a FORTRAN program to create an equilibrium configuration of uniform hard disks.

The second part of the book studies the relation of the bulk effective properties to the microstructure of a random heterogeneous material. The probability characterizations of the microstructure given in the first part are used to compute the effective bulk properties in the four different areas of conduction, elasticity, trapping, and flow in porous media. The study is generally restricted to ergodic two-phase random media. In each case, a local linear constitutive model is assumed, as is local statistical homogeneity. The focus is on steady-state behavior. The resulting elasticity theory permits definition of linear viscoelasticity through the correspondence principle.

Processes are represented by averaged, or homogenized, partial differential equations. The averaging requires two length scales, one for the microstructure and one for the bulk material. Generally, the response is composed of a slow component due to the global scale including the applied loads and a fast component due to the microstructure. The averaging is carried out subject to conservation relations.

Variational principles are used to bound the values of the properties, which in general cannot be computed exactly. Exceptions are some periodic microstructures, the Hashin-Shtrikman isotropic dispersion of spheres, or the hierarchical laminates. Single inclusion problems are solved and used to estimate the dependence of effective properties on volume fraction and inclusion shape. Trial fields to determine bounds on the properties, using the variational principles, are developed for the four classes of example effective properties. The results of the first part of the book are used to compute the bounds in two-phase random heterogeneous materials. The book ends by examining the problem of determining one effective property from information about others.

This book is part of an Applied Mathematics series, and it primarily concerns the mathematical analysis of random structures. This work can serve as a foundation for those who wish to design random heterogeneous structures with specified bulk properties. A major contribution of the book is to clearly and rigorously explain the statistical characterization of the microstructure of these materials as a foundation for an analysis of effective properties. This theory, at least to the extent developed in this book, only produces linear models for the bulk properties of solids; nonlinear behavior is not covered.

The intended interdisciplinary audience is graduate students and researchers in fields such as applied mathematics, physics, chemistry, materials science, engineering, geology, and biology. The suggested prerequisites are the basics of probability, statistical mechanics, advanced calculus, and continuum mechanics. The author’s stated attempt to avoid technical jargon is largely successful. An extensive reference list is provided. Random Heterogeneous Materials: Microstructure and Macroscopic Properties should become a standard reference for those beginning a study of random heterogeneous media.