A rigorous approach to the field recursion method for two-component composites with isotropic phases
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 eective tensor of a composite material. The reason is that the eective
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 effctive tensor L_*.