# Is it wise to keep laminating?

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

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

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topBriane, 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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