Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three.

Marc Briane (1, 2) , Laurent Pater (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes

Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a magnetic field $h$ are derived in two and three dimensions under the strong field assumption. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on a polynomial of $h$ through some homogenized matrix-valued function obtained in the absence of magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. In dimension three a fiber-reinforced medium shows in contrast a rational dependence on $h$ of the homogenized conductivity due to the high diffusion along the fibers direction.

Mots clés: homogenization; high-contrast conductivity; magneto-transport; strong field; two-phase composites

Domaines: Equations aux dérivées partielles [math.AP]

Marc Briane, Laurent Pater. Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three.. 2011. ⟨hal-00625033v1⟩