Homogenization of the two-dimensional Hall effect

Briane, Marc; Manceau, David; Milton, Graeme W.

doi:10.1016/j.jmaa.2007.07.044

Cited by 16 publications

(30 citation statements)

References 9 publications

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“…We then show that in contrast to the two-dimensional Hall effect studied in [4], the homogenization process from R ε to R * in three dimensions does not preserve any positivity property. 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.…”

Section: Introductionmentioning

confidence: 71%

“…Part (i) of Theorem 2.7 is proved in [4]. It is an extension of Theorem 2.5 in [6] where A ε (h) is of class C n with respect to h and its n+1 derivatives satisfy the Lipschitz condition (2.5).…”

Section: H-convergence With a Parametermentioning

confidence: 88%

“…In dimension two the Bergman approach was rigorously justified and extended in the paper [4] in the H-convergence framework of Murat-Tartar [11], which is not restricted to the periodic case.…”

Section: Introductionmentioning

confidence: 99%

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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: 71%

Section: H-convergence With a Parametermentioning

confidence: 88%

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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].…”

mentioning

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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“…For the bronchial distal part, we may refer to a reduction technique, as the one advocated in the blood circulation [30] and in the bronchial tree [27], or to a more sophisticated approach [24]. Concerning the terminal part, the description of the set of acini may be obtained thanks to the homogenization framework, either fractal homogenization as in [7], or periodic homogenization for the alveola part, involving fluid structure interaction, which is the subject of the present paper.…”

Section: Prefacementioning

confidence: 99%

Homogenization of Elastic Media with Gaseous Inclusions

Baffico¹,

Grandmont²,

Maday³

et al. 2008

Multiscale Model. Simul.

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We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period ε > 0, and a perfect gas placed in each of these perforations, as ε goes to zero. The model we use is linear corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as ε goes to 0. We state that in the limit, we get a two-scale linear elasticity-like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allows us to find the first corrector term. The analysis is performed both in the case of an incompressible and compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we numerically calculate the homogenized coefficients in the incompressible case, for different types of elastic materials.

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Homogenization of the two-dimensional Hall effect

Cited by 16 publications

References 9 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

Homogenization of Elastic Media with Gaseous Inclusions

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