Cambridge Monographs on Applied and Computational Mathematics.
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. Althoungh extensively studied for more than 100 years, an explosion og 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 og 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.
1. Introduction
2. Some Equations of interest and numerical approaches to
solving them
3. Duality transformations in two-dimensional media
4. Translations and equivalent media
5. Some microstructure-independent exact relations
6. Exact relations for coupled equations
7. Assemblages of spheres, ellipsoids, and other neutral
inclusions
8. Tricks for generating other exactly solvable microgeometries
9. Laminate materials
10. Approximations and asymptotic formulas
,etc.