Microstructure and properties of random heterogeneous materials: A review of theoretical results

We present an overview of some theoretical results that have been obtained to rigorously bound the effective conductivity and effective elastic moduli of random heterogeneous materials. These bounds are obtained through the appropriate use of integrals over certain microstructure functions that capture essential morphological information about random materials. Such a characterization has been obtained analytically for several models, including fully penetrable spheres, totally impenetrable spheres, and the “cherry‐pit” model. These analytical techniques—characterizing the microstructure and using this information to bound the effective properties—will undoubtedly be useful in obtaining sharper bounds on the effective properties of polymers. While we present some advanced ideas and methods, an attempt will be been made to elucidate these methods for the nonexpert.