Bounds for mixtures of an arbitrary number of materials

This paper studies the problem of bounding the effective conductivity coefficients of mixtures made of several materials, in given proportions. Lower and upper bounds are obtained, and the optimality of these bounds is proven under certain hypotheses. Also, necessary and sufficient conditions for a mixture to attain the bounds are described. Some of the results were already known, but we give simpler proofs. Copyright © 2001 John Wiley & Sons, Ltd.