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Chapitre D'ouvrage Année : 2016

A rigorous approach to the field recursion method for two-component composites with isotropic phases

Maxence Cassier
Aaron Welters
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Résumé

In this chapter we give a rigorous derivation of the eld equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since it has proven to be a powerful tool in developing sharp bounds for the e ective tensor of a composite material. The reason is that the e ective tensor L_* can be interpreted in the general framework of the abstract theory of composites as the Z-operator on a certain orthogonal Z(2) subspace collection. The base case of the recursion starts with an orthogonal Z(2) subspace collection on a Hilbert space H, the Zproblem, and the associated Y -problem. We provide some new conditions for the solvability of both the Z-problem and the associated Y -problem. We also give explicit representations of the associated Z-operator and Y -operator and study their analytical properties. An iteration method is then developed from a hierarchy of subspace collections and their associated operators which leads to a continued fraction representation of the initial e ffctive tensor L_*.
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hal-01967595 , version 1 (31-12-2018)

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  • HAL Id : hal-01967595 , version 1

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Maxence Cassier, Aaron Welters, Graeme W. Milton. A rigorous approach to the field recursion method for two-component composites with isotropic phases. In Extending the theory of composites to other areas of science edited by Graeme W. Milton and reviewed by the Journal of Applied Mechanics and SIAM Review, 2016. ⟨hal-01967595⟩

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