We study the corrector matrix P-epsilon to the conductivity equations. We show that if P-epsilon converges weakly to the identity, then for any laminate det P-epsilon >= 0 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.

Is it wise to keep laminating? / Nesi, Vincenzo; Marc, Briane. - In: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS. - ISSN 1262-3377. - 10:4(2004), pp. 452-477. [10.1051/cocv:2004015]

Is it wise to keep laminating?

NESI, Vincenzo;
2004

Abstract

We study the corrector matrix P-epsilon to the conductivity equations. We show that if P-epsilon converges weakly to the identity, then for any laminate det P-epsilon >= 0 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.
2004
bounds; composites; homogenization; laminates
01 Pubblicazione su rivista::01a Articolo in rivista
Is it wise to keep laminating? / Nesi, Vincenzo; Marc, Briane. - In: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS. - ISSN 1262-3377. - 10:4(2004), pp. 452-477. [10.1051/cocv:2004015]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/70665
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