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**Iterated homogenization, differential effective medium theory and applications.**
*(English)*
Zbl 0629.73010

In this paper the author studies the realizability, basic properties and some applications of the theory of composite materials. The differential effective medium theory proposes a class of multiphase materials for which the effective elastic properties can be calculated fairly explicit. In this paper to construct the models and calculate their effective properties, the method of iterated periodic homogenization is considered to be adequate tool for a phenomenological approach to the question of realizability of effective parameters. The microgeometry varies in length scales \(\epsilon\), \(\epsilon^ 2\),..., \(\epsilon^ n\),..., where \(\epsilon\) is a small parameter. The effective property corresponding to a given strain field is defined as the limit of the strain energy as the parameter \(\epsilon\) tends to zero. Eventually two examples are discussed. The first describes the set of all possible laminar mixture of several phases, which depends on several “control variables” governing the choise of phase and direction of lamination used. The second one is the class of differential coated geometry models, for which the effective parameters can be computed exactly.

Reviewer: M.Codegone

### Keywords:

differential effective medium theory; multiphase materials; effective elastic properties; iterated periodic homogenization; phenomenological approach; realizability; laminar mixture of several phases; control variables; differential coated geometry models
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\textit{M. Avellaneda}, Commun. Pure Appl. Math. 40, No. 5, 527--554 (1987; Zbl 0629.73010)

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