  This paper deals with bounds satisfied by the effective non-symmetric
conductivity of three-dimensional composites in the presence of a strong
magnetic field. On the one hand, it is shown that for general composites the
antisymmetric part of the effective conductivity cannot be bounded solely in
terms of the antisymmetric part of the local conductivity, contrary to the
columnar case. So, a suitable rank-two laminate the conductivity of which has a
bounded antisymmetric part together with a high-contrast symmetric part, may
generate an arbitrarily large antisymmetric part of the effective conductivity.
On the other hand, bounds are provided which show that the antisymmetric part
of the effective conductivity must go to zero if the upper bound on the
antisymmetric part of the local conductivity goes to zero, and the symmetric
part of the local conductivity remains bounded below and above. Elementary
bounds on the effective moduli are derived assuming the local conductivity and
effective conductivity have transverse isotropy in the plane orthogonal to the
magnetic field. New Hashin-Shtrikman type bounds for two-phase
three-dimensional composites with a non-symmetric conductivity are provided
under geometric isotropy of the microstructure. The derivation of the bounds is
based on a particular variational principle symmetrizing the problem, and the
use of Y-tensors involving the averages of the fields in each phase.
