Calculation of bounds for some average bulk properties of composite materials

The problem of calculating the effective bulk dielectric constant εe (or magnetic permeability or conductivity) of a macroscopically homogeneous composite material is shown to require the evaluation of a single characteristic function which includes all the relevant information about the microscopic geometry of the composite. This function is found to have some remarkable convexity properties that allow the construction of various new and improved bounds on εe for two-phase as well as for multiphase systems by incorporating information on other similar effective bulk properties of the same system.