H-Measures and Bounds on the Effective Properties of Composite Materials

Gr{é}goire Allaire and Herv{é} Maillot

Abstract: The goal of this paper is to show how the notion of $H$-measure, introduced by G{é}rard and Tartar, can be used to derive optimal bounds on the effective properties of two-phase composite materials. We re-derive the classical optimal bounds of Hashin and Shtrikman in the conductivity or elasticity setting. In an attempt to extend further this method in the context of two different conductivity properties, we obtain coupled bounds, known as Bergman bounds, which are merely sub-optimal in most cases.

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