  Magnetic gels and elastomers are promising candidates to construct reversibly
excitable soft actuators, triggered from outside by magnetic fields. These
magnetic fields induce or alter the magnetic interactions between discrete
rigid particles embedded in a soft elastic polymeric matrix, leading to overall
deformations. It is a major challenge in theory to correctly predict from the
discrete particle configuration the type of deformation resulting for a
finite-sized system. Considering an elastic sphere, we here present such an
approach. The method is in principle exact, at least within the framework of
linear elasticity theory and for large enough interparticle distances.
Different particle arrangements are considered. We find, for instance, that
regular simple cubic configurations show elongation of the sphere along the
magnetization if oriented along a face or space diagonal of the cubic unit
cell. Contrariwise, with the magnetization along the edge of the cubic unit
cell, they contract. The opposite is true in this geometry for body- and
face-centered configurations. Remarkably, for the latter configurations but the
magnetization along a face or space diagonal of the unit cell, contraction was
observed to revert to expansion with decreasing Poisson ratio of the elastic
material. Randomized configurations were considered as well. They show a
tendency of elongating the sphere along the magnetization, which is more
pronounced for compressible systems. Our results can be tested against actual
experiments for spherical samples. Moreover, our approach shall support the
search of optimal particle distributions for a maximized effect of actuation.
