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

Marc Briane (1) , Laurent Pater (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a parameter $h$ are derived in two and three dimensions. The parameter $h$ characterizes the antisymmetric part of the conductivity for an idealized model of a conductor in the presence of a magnetic field. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on $h$ through some homogenized matrix-valued function obtained in the absence of a magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. Using other tools, a fiber-reinforced medium in dimension three provides a quite different homogenized conductivity.

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.. Journal of Mathematical Analysis and Applications, 2012, 393 (2), pp.563-589. ⟨10.1016/j.jmaa.2011.12.059⟩. ⟨hal-00625033v2⟩