  Embedding rigid inclusions into elastic matrix materials is a procedure of
high practical relevance, for instance for the fabrication of elastic composite
materials. We theoretically analyze the following situation. Rigid spherical
inclusions are enclosed by a homogeneous elastic medium under stick boundary
conditions. Forces and torques are directly imposed from outside onto the
inclusions, or are externally induced between them. The inclusions respond to
these forces and torques by translations and rotations against the surrounding
elastic matrix. This leads to elastic matrix deformations, and in turn results
in mutual long-ranged matrix-mediated interactions between the inclusions.
Adapting a well-known approach from low-Reynolds-number hydrodynamics, we
explicitly calculate the displacements and rotations of the inclusions from the
externally imposed or induced forces and torques. Analytical expressions are
presented as a function of the inclusion configuration in terms of
displaceability and rotateability matrices. The role of the elastic environment
is implicitly included in these relations. That is, the resulting expressions
allow a calculation of the induced displacements and rotations directly from
the inclusion configuration, without having to explicitly determine the
deformations of the elastic environment. In contrast to the hydrodynamic case,
compressibility of the surrounding medium is readily taken into account. We
present the complete derivation based on the underlying equations of linear
elasticity theory. In the future, the method will, for example, be helpful to
characterize the behavior of externally tunable elastic composite materials, to
accelerate numerical approaches, as well as to improve the quantitative
interpretation of microrheological results.
