**The determination of the transport, electromagnetic
and mechanical properties of heterogeneous materials
has a long and venerable history, attracting
the attention of some of the luminaries of science,
including Maxwell (1873), Lord Rayleigh (1892)
and Einstein (1906). In his Treatise on
Electricity and Magnetism, Maxwell derived
an expression for the effective conductivity
of a dispersion of spheres that is exact for
dilute sphere concentrations. Lord Rayleigh
developed a formalism to compute the effective
conductivity of regular arrays of spheres that
is used to this day. Work on the mechanical
properties of heterogeneous materials began with the famous paper
by Einstein in which he determined
the effective viscosity of a dilute suspension
of spheres. Since the early work on the physical
properties of heterogeneous materials,
there has been an explosion in the literature
on this subject because
of the rich and challenging fundamental problems
it offers and its
manifest technological importance. **

This book is divided into two parts. Part I deals with the quantitative characterization of the microstructure of heterogeneous materials via theoretical, computer-simulation and imaging techniques. Emphasis is placed on theoretical methods. Part II treats a wide variety of effective properties of heterogeneous materials and how they are linked to the microstructure. This is accomplished using rigorous methods. (Readers interested in property prediction can immediately skip to Part II.) Whenever possible, theoretical predictions for the effective properties are compared to available experimental and computer-simulation data. The overall goal of the book is to provide a rigorous means of characterizing the microstructure and properties of heterogeneous materials that can simultaneously yield results of practical utility. A unified treatment of both microstructure and properties is emphasized.

In Chapter 2, the various
microstructural functions that are essential in determining
the effective properties of random heterogeneous materials
are defined.
Chapter 3 provides a review of the statistical mechanics
of particle systems that is particularly germane to the study
of random heterogeneous materials. In Chapter 4,
a unified approach to characterize the microstructure
of a large class of media is developed. This is accomplished
via a canonical *n*-point function *H_n* from which one
can derive exact analytical expressions for any
microstructural function of interest.
Chapters 5, 6 and 7 apply the formalism of Chapter
4 to the case of identical systems of spheres,
spheres with a polydispersivity in size, and
anisotropic particle systems (including laminates),
respectively. In Chapter 8, the methods
of Chapter 4 are extended to quantify the microstructure
of cell models. Here the random-field approach is also
discussed. Chapter 9 reviews the study of *percolation*
and *clustering* on a *lattice* and introduces
*continuum* percolation. Chapter 10 describes
specific developments *continuum* percolation theory.
Chapter 11 describes a means to study microstructural
fluctuations that occur on local length scales.
Finally, Chapter 12 discusses computer-simulation
techniques (primarily Monte Carlo methods) to quantify
microstructure. Moreover, it is shown
how to apply the same methods to compute relevant
microstructural functions from two- and three-dimensional
images of the material.

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