Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran–Molyneux bounds for bulk modulus and the Beran bounds for conductivity are well-known. It is first shown how to make direct use of bounds and spatial correlation information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the Peselnick–Meister bounds (based on Hashin–Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains for either electrical and thermal conductivity) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information—and how accurately that information—needs to be known in order to be useful for estimating material constants in random and heterogeneous media.