  Magnetic gels are composite materials, consisting of a polymer matrix and
embedded magnetic particles. Those are mechanically coupled to each other,
giving rise to the magnetostrictive effects as well as to a controllable
overall elasticity responsive to external magnetic fields. Due to their
inherent composite and thereby multiscale nature, a theoretical framework
bridging different levels of description is indispensable for understanding the
magnetomechanical properties of magnetic gels. In this study, we extend a
recently developed density functional approach from two spatial dimensions to
more realistic three-dimensional systems. Along these lines, we connect a
mesoscopic characterization resolving the discrete structure of the magnetic
particles, to macroscopic continuum parameters of magnetic gels. In particular,
we incorporate the long-range nature of the magnetic dipole-dipole interaction,
and consider the approximate incompressibility of the embedding media, and
relative rotations with respect to an external magnetic field breaking
rotational symmetry. We then probe the shape of the model system in its
reference state, confirming the dependence of magnetostrictive effects on the
configuration of the magnetic particles and on the shape of the considered
sample. Moreover, calculating the elastic and rotational coefficients on the
basis of our mesoscopic approach, we examine how the macroscopic types of
behavior are related to the mesoscopic properties. Implications for real
systems of random particle configurations are also discussed.
