The Theory Of Composites

This page intentionally left blank Some of the greatest scientists, including Poisson, Faraday, Maxwell, Rayleigh, and Einstein, have contributed to the theory of composite materials. Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than 100 years, an explosion of ideas in the last four decades (and particularly in the last two decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli that govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results. Graeme W. Milton is a Distinguished Professor in the Mathematics Department at the University of Utah. He has been awarded Sloan and Packard Fellowships and is on the editorial board of the Archive for Rational Mechanics and Analysis. He has published more than 70 papers on the theory of composite materials. CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS Series Editors P. G. CIARLET, A. ISERLES, R. V. KOHN, M. H. WRIGHT 6 The Theory of Composites The Cambridge Monographs on Applied and Computational Mathematics reflects the crucial role of mathematical and computational techniques in contemporary science. The series publishes expos