Is it wise to keep laminating?

¹ Centre de Mathématique, INSA de Rennes & IRMAR, 20 avenue des Buttes de Coësmes, 35043 Rennes Cedex, France; mbriane@insa-rennes.fr.
² Dip. di Mat., Universitá di Roma, La Sapienza P.le A. Moro 2, 00185 Rome, Italy; nesi@mat.uniroma1.it.

Received: 23 July 2002

Abstract

We study the corrector matrix  to the conductivity equations. We show that if  converges weakly to the identity, then for any laminate at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. We use this property of laminates to prove that, in any dimension, the classical Hashin-Shtrikman bounds are not attained by laminates, in certain regimes, when the number of phases is greater than two. In addition we establish new bounds for the effective conductivity, which are asymptotically optimal for mixtures of three isotropic phases among a certain class of microgeometries, including orthogonal laminates, which we then call quasiorthogonal.