Composites have been studied for more than 150 years, and interest in their properties has been growing. This classic volume provides the foundations for understanding a broad range of composite properties, including electrical, magnetic, electromagnetic, elastic and viscoelastic, piezoelectric, thermal, fluid flow through porous materials, thermoelectric, pyroelectric, magnetoelectric, and conduction in the presence of a magnetic field (Hall effect). Exact solutions of the PDEs in model geometries provide one avenue of understanding composites; other avenues include microstructure-independent exact relations satisfied by effective moduli, for which the general theory is reviewed; approximation formulae for effective moduli; and series expansions for the fields and effective moduli that are the basis of numerical methods for computing these fields and moduli. The range of properties that composites can exhibit can be explored either through the model geometries or through microstructure-independent bounds on the properties. These bounds are obtained through variational principles, analytic methods, and Hilbert space approaches. Most interesting is when the properties of the composite are unlike those of the constituent materials, and there has been an explosion of interest in such composites, now known as metamaterials. The Theory of Composites surveys these aspects, among others, and complements the new body of literature that has emerged since the book was written. It remains relevant today by providing historical background, a compendium of numerous results, and through elucidating many of the tools still used today in the analysis of composite properties. This book is intended for applied mathematicians, physicists, and electrical and mechanical engineers. It will also be of interest to graduate students.

Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological fluids are all examples of materials where an understanding of the mathematics on the different length scales is a key to interpreting their physical behavior. In their analysis of these media, scientists coming from a number of disciplines have encountered similar mathematical problems, yet it is rare for researchers in the various fields to meet. The 1995-1996 program at the Institute for Mathematics and its Applications was devoted to Mathematical Methods in Material Science, and was attended by materials scientists, physicists, geologists, chemists engineers, and mathematicians. The present volume contains chapters which have emerged from four of the workshops held during the year, focusing on the following areas: Disordered Materials; Interfaces and Thin Films; Mechanical Response of Materials from Angstroms to Meters; and Phase Transformation, Composite Materials and Microstructure. The scales treated in these workshops ranged from the atomic to the microstructural to the macroscopic, the microstructures from ordered to random, and the treatments from "purely" theoretical to the highly applied. Taken together, these works form a compelling and broad account of many aspects of the science of multiscale materials, and will hopefully inspire research across the self-imposed barriers of twentieth century science.

The 1995-1996 program at the Institute for Mathematics and its Applications was devoted to mathematical methods in material science, and was attended by materials scientists, physicists, geologists, chemists engineers, and mathematicians. This volume contains chapters which emerged from four of the workshops, focusing on disordered materials; interfaces and thin films; mechanical response of materials from angstroms to meters; and phase transformation, composite materials and microstructure. The scales treated in these workshops ranged from the atomic to the macroscopic, the microstructures from ordered to random, and the treatments from "purely" theoretical to highly applied. Taken together, these results form a compelling and broad account of many aspects of the science of multi-scale materials, and will hopefully inspire research across the self-imposed barriers of twentieth century science.