  In the perspective of developing smart hybrid materials with customized
features, ferrogels and magnetorheological elastomers allow a synergy of
elasticity and magnetism. The interplay between elastic and magnetic properties
gives rise to a unique reversible control of the material behavior by applying
an external magnetic field. Albeit few works have been performed on the
time-dependent properties so far, understanding the dynamic behavior is the key
to model many practical situations, e.g. applications as vibration absorbers.
Here we present a way to calculate the frequency-dependent elastic moduli based
on the decomposition of the linear response to an external stress in normal
modes. We use a minimal three-dimensional dipole-spring model to theoretically
describe the magnetic and elastic interactions on the mesoscopic level.
Specifically, the magnetic particles carry permanent magnetic dipole moments
and are spatially arranged in a prescribed way, before they are linked by
elastic springs. An external magnetic field aligns the magnetic moments. On the
one hand, we study regular lattice-like particle arrangements to compare with
previous results in the literature. On the other hand, we calculate the dynamic
elastic moduli for irregular, more realistic particle distributions. Our
approach measures the tunability of the linear dynamic response as a function
of the particle arrangement, the system orientation with respect to the
external magnetic field, as well as the magnitude of the magnetic interaction
between the particles. The strength of the present approach is that it
explicitly connects the relaxational modes of the system with the rheological
properties as well as with the internal rearrangement of the particles in the
sample, providing new insight into the dynamics of these remarkable materials.
