Microgeometry of random composites and porous media

Abstract

For practical applications of variational bounds to the effective properties of composite materials, the information available is often not that required by the formulae for the optimal bounds. It is therefore important to determine what can be said rigorously about various unknown material properties when some other properties are known. The key quantities to be analysed are the parameters zeta and eta , depending on the microgeometry through integrals of the three-point correlation functions. The physical significance of these parameters for two-phase composites and porous media is elucidated by examining the various relationships between them and material properties. The bounds on conductivity due to Beran (1965) and the bounds on elastic constants due to Beran and Molyneux (1966) and to McCoy (1970), as well as those of Milton and Phan-Thien (1982), are considered. For the special case of porous media, the formulae simplify greatly and the resulting analytical relationships between transport properties and geometrical parameters are easily interpreted. In particular, it is shown that the microgeometry parameter zeta places limits on the pore space connectivity. Examples of bounds on one effective material property from measurements of another are also derived. These include bounds correlating the effective electrical or thermal conductivities and the effective shear modulus with the effective bulk modulus. These bounds are somewhat more restrictive than the well known bounds of Hashin and Shtrikman (1962). For porous materials, measurements of bulk modulus provide bounds on electrical formation factor and vice versa.