  In this paper we study the homogenization of a linear elastodynamics system
in an elastic body with soft inclusions, which is embedded in a highly
oscillating magnetic field. We show two limit behaviors according to the
magnetic field. On the one hand, if the magnetic field has two different
directions on the interface between the hard phase and the soft phase, then the
limit of the displacement in the hard phase is independent of time, so that the
magnetic field induces an effective infinite mass. On the other hand, if the
magnetic field has a constant direction $\xi$ on the interface, then the limit
of the displacement in the hard phase and in the direction $\xi$ is solution to
an elastodynamics equation with a memory mass, a memory stress tensor and
memory external forces depending on the initial conditions, which read as time
convolutions with some kernel. When the magnetic has the same direction $\xi$
in the soft phase with smooth inclusions, we prove that the space-average of
the kernel is regular and that the limit of the overall displacement in the
direction $\xi$ is solution to a viscoelasticity equation.
