The Theory of Composites

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Overview

The theory of composite materials is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the past four decades has dramatically increased our understanding of the relationship among the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective 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.

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Editorial Reviews

From the Publisher
"...there existed no book or review paper that would allow a newcomer to get a general knowledge of the state of the art. The book of Graeme Milton fills this gap, and it does the job in a splendid manner that will make it the reference book on composite materials for a long time." Mathematical Reviews
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Product Details

Table of Contents

1. Introduction; 2. Equations of interest and numerical approaches; 3. Duality transformations; 4. Translations and equivalent media; 5. Microstructure independent exact relations; 6. Exact relations for coupled equations; 7. Assemblages of inclusions; 8. Tricks for exactly solvable microgeometries; 9. Laminate materials; 10. Approximations and asymptotic formulae; 11. Wave propagation in the quasistatic limit; 12. Reformulating the problem; 13. Variational principles and inequalities; 14. Series expansions; 15. Correlation functions and series expansions; 16. Other perturbation solutions; 17. The general theory of exact relations; 18. Analytic properties; 19. Y-tensors; 20. Y-tensors and effective tensors in circuits; 21. Bounds on the properties of composites; 22. Classical variational principle bounds; 23. Hashin-Shtrikman bounds; 24. Translation method bounds; 25. Choosing translations and finding geometries; 26. Bounds incorporating three-point statistics; 27. Bounds using the analytic method; 28. Fractional linear transformations for bounds; 29. The field equation recursion method; 30. G-closure properties and extremal composites; 31. Bounding and quasiconvexification.

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