Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity

Briane, Marc; Milton, Graeme W.; Nesi, Vincenzo

doi:10.1007/s00205-004-0315-8

Cited by 45 publications

(81 citation statements)

References 20 publications

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“…Indeed, we give (see Theorem 4.2) an example of a microstructure for which R ε is positive isotropic a.e., while R * is a constant negative isotropic matrix. This surprising result is linked to the change of sign of the corrector's determinant derived in [5].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient

Briane

Milton

2008

Arch Rational Mech Anal

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The notion of a Hall matrix, associated with a possibly anisotropic conducting material in the presence of a small magnetic field, is introduced. Then, for any material having a microstructure we prove a general homogenization result satisfied by the Hall matrix in the framework of the Hconvergence of Murat-Tartar. Extending a result of Bergman, it is shown that the Hall matrix can be computed from the corrector associated with the homogenization problem when no magnetic field is present. Finally, we give an example of a microstructure for which the Hall matrix is positive isotropic a.e., while the homogenized Hall matrix is negative isotropic.

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Section: Introductionmentioning

confidence: 99%

Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient

Briane

Milton

2008

Arch Rational Mech Anal

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“…In the simplest case, the Hall coefficient A H is equal to the inverse of the charge density, i.e., A H ¼ ρ −1 . A few years ago, building upon earlier work [17][18][19], Marc Briane and Graeme W. Milton predicted theoretically that the sign of the isotropic Hall coefficient can be reversed in chainmail-like three-dimensional metamaterials [20]. Notably, art inspired science: Chainmail artist Dylon Whyte suggested to them the three-dimensional structure [21].…”

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confidence: 99%

Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

2017

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Effectively inverting the sign of material parameters is a striking possibility arising from the concept of metamaterials. Here, we show that the electrical properties of a p-type semiconductor can be mimicked by a metamaterial solely made of an n-type semiconductor. By fabricating and characterizing threedimensional simple-cubic microlattices composed of interlocked hollow semiconducting tori, we demonstrate that sign and magnitude of the effective metamaterial Hall coefficient can be adjusted via a tori separation parameter-in agreement with previous theoretical and numerical predictions.

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“…In an attempt to a crude summary, the present paper studies the correctors to conductivity equations. In conjunction with the work [6], it shows, quite unexpectedly, that even in a simple linear conduction problem, in dimension greater than two, laminates and non laminates can be discriminated by a very simple property which will be stated later in the section. This property is always enjoyed by laminates but (as shown in [6]), not necessarily by non laminates.…”

Section: Introductionmentioning

confidence: 69%

“…Let us briefly digress to comment on the fact that in dimension d ≥ 3 Milton and the authors [6] proved that there exists a (non laminate) periodic two-phase geometry such that the solution to the problem analogue to (1.7) but in dimension d greater or equal to three, satisfies the following properties. It is defined almost everywhere and…”

Section: Dimension D ≥mentioning

confidence: 99%

Is it wise to keep laminating?

Briane

Nesi

2004

ESAIM: COCV

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Abstract. 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.Mathematics Subject Classification. 35B27, 74Q15.

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Change of Sign of the Corrector’s Determinant for Homogenization in Three-Dimensional Conductivity

Cited by 45 publications

References 20 publications

Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient

Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient

Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

Is it wise to keep laminating?

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