  It is now well established that the homogenization of a periodic array of
parallel dielectric fibers with suitably scaled high permittivity can lead to a
(possibly) negative frequency-dependent effective permeability. However this
result based on a two-dimensional approach holds merely in the case of linearly
polarized magnetic fields, reducing thus its applications to infinite
cylindrical obstacles. In this paper we consider a dielectric structure placed
in a bounded domain of $\mathbb{R}^3$ and perform a full 3D asymptotic
analysis. The main ingredient is a new averaging method for characterizing the
bulk effective magnetic field in the vanishing-period limit. We evidence a
vectorial spectral problem on the periodic cell which determines
micro-resonances and encodes the oscillating behavior of the magnetic field
from which artificial magnetism arises. At a macroscopic level we deduce an
effective permeability tensor that we can be make explicit as a function of the
frequency. As far as sign-changing permeability are sought after, we may
foresee that periodic bulk dielectric inclusions could be an efficient
alternative to the very popular metallic split-ring structure proposed by
Pendry.
