Is it wise to keep laminating?

Marc Briane; Vincenzo Nesi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 452-477
  • ISSN: 1292-8119

Abstract

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We study the corrector matrix P ε  to the conductivity equations. We show that if P ε  converges weakly to the identity, then for any laminate det P ε ≥ 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.

How to cite

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Briane, Marc, and Nesi, Vincenzo. "Is it wise to keep laminating?." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 452-477. <http://eudml.org/doc/90738>.

@article{Briane2010,
abstract = { We study the corrector matrix $P^\{\varepsilon\}$ to the conductivity equations. We show that if $P^\{\varepsilon\}$ converges weakly to the identity, then for any laminate $\det P^\{\varepsilon\}\geq 0$ 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. },
author = {Briane, Marc, Nesi, Vincenzo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; bounds; composites; laminates.; homogenization; Hashin-Shtrikman bounds; effective conductivity; quasiorthogonal laminates},
language = {eng},
month = {3},
number = {4},
pages = {452-477},
publisher = {EDP Sciences},
title = {Is it wise to keep laminating?},
url = {http://eudml.org/doc/90738},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Briane, Marc
AU - Nesi, Vincenzo
TI - Is it wise to keep laminating?
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 452
EP - 477
AB - We study the corrector matrix $P^{\varepsilon}$ to the conductivity equations. We show that if $P^{\varepsilon}$ converges weakly to the identity, then for any laminate $\det P^{\varepsilon}\geq 0$ 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.
LA - eng
KW - Homogenization; bounds; composites; laminates.; homogenization; Hashin-Shtrikman bounds; effective conductivity; quasiorthogonal laminates
UR - http://eudml.org/doc/90738
ER -

References

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