Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties

The purpose of the present article is to review recent advances made in the determination and calculation of improved bounds on the effective properties of random heterogeneous media that depend upon the microstructure via n-point correlation functions. New breakthroughs made in the quantitative characterization of the microstructure of heterogeneous materials are also reviewed. The following four different effective properties shall be studied: (i) effective conductivity tensor (which includes, by mathematical analogy, the dielectric constant, magnetic permeability, and diffusion coefficient); (ii) effective stiffness tensor; (iii) diffusion-controlled trapping constant; and (iv) fluid permeability tensor. It shall be demonstrated that improved upper and lower bounds can provide a relatively sharp estimate of the effective property even when the bounds diverge from one another. Although this article reviews state-of-the-art advances in the field, an attempt will be made to elucidate methods and principles for the nonexpert.