We have seen that random heterogeneous materials exhibit a remarkably broad spectrum of rich and complex microstructures. Our focus in Part I of this book is to develop a machinery to characterize statistically this broad class of microstructures, i.e., to develop a statistical, or stochastic, geometry of heterogeneous materials. How or where does one begin to address this challenging task? The answer, of course, depends on what is the goal of the statistical characterization. Our goal is ultimately the prediction of the macroscopic or effective physical properties of the random heterogeneous material, and thus this determines our starting point. The diverse effective properties that we are concerned with in this book naturally and necessarily lead to a wide variety of microstructural descriptors, generically referred to as microstructural correlation functions. As we noted in Chapter 1, such descriptors have applicability in other seemingly disparate fields, such as cosmology (Peebles 1993, Saslaw 2000) and ecology (Pielou 1977, Diggle 1983, Durrett and Levin 1994).
KeywordsIsotropic Medium Positive Semidefinite Random Medium Complementary Cumulative Distribution Function Exclusion Probability
Unable to display preview. Download preview PDF.