To describe many-particle systems suspended in incompressible low-Reynolds-number fluids, effective hydrodynamic interactions can be introduced. Here, we consider particles embedded in elastic media. The effective elastic interactions between spherical particles are calculated analytically, inspired by the approach in the fluid case. Our experiments on interacting magnetic particles confirm the theory. In view of the huge success of the method in hydrodynamics, we similarly expect many future applications in the elastic case, e.g. for elastic composite materials. planes, vehicles, ships, and propellers [4]. All these processes are described by the Navier-Stokes equations [5] or variants thereof. This set of equations typically poses significant challenges during solution due to a convective nonlinearity reflecting inertial effects. Basically, turbulence is driven by the inertial term. It often renders analytical solutions impossible.The situation changes for small dimensions and velocities or high viscosity. Then, the relative strength of inertial effects, measured by the Reynolds number, is low. The nonlinearity can be neglected. A Green's function in terms of the so-called Oseen matrix is then available, which formally solves the problem analytically [6,7]. In this way, semi-dilute colloidal suspensions, i.e. the dispersion of nano-to micrometer-sized particles in a fluid [7,8], or microswimmer suspensions [9] are described effectively. The explicit role of the fluid is eliminated and replaced by effective hydrodynamic interactions between the suspended particles [6,7].Despite the success of this theoretical approach for colloidal suspensions, hardly any investigations consider a surrounding elastic solid instead of a suspending fluid. This is surprising, since, as we show below, the formalism can be adapted straightforwardly to linearly elastic matrices and is confirmed by our experiments. Our approach will, for instance, facilitate describing the response of elastic composite materials to external stimuli. Such materials consist of more or less rigid inclusions embedded in an elastic matrix. They are of growing technological interest and may serve, e.g., as soft actuators or sound attenuation devices [10].In previous theoretical studies, the physics of one single rigid or deformable inclusion was addressed [11,12], also under acoustic irradiation [13]. For more than a single inclusion, mainly the so-called load problem was analyzed theoretically for a pair of rigid inclusions: one prescribes displacements of two rigid inclusions in an elastic matrix, and then determines the forces necessary to achieve these given displacements [14].Here, we take the converse point of view, based on the cause-and-effect chain in our experiments: external forces are imposed onto the inclusions, or mutual forces between the inclusions are induced, for example to actuate the material or to tune its properties. In response to the forces, the inclusions are displaced. Since they cannot penetrate through the surrounding elastic ma...

Soft elastic composite materials containing particulate rigid inclusions in a soft elastic matrix are candidates for developing soft actuators or tunable damping devices. The possibility to reversibly drive the rigid inclusions within such a composite together to a close-to-touching state by an external stimulus would offer important benefits. Then, a significant tuning of the mechanical properties could be achieved due to the resulting mechanical hardening. For a long time, it has been argued whether a virtual touching of the embedded magnetic particles with subsequent detachment can actually be observed in real materials, and if so, whether the process is reversible. Here, we present experimental results that demonstrate this phenomenon in reality. Our system consists of two paramagnetic nickel particles embedded at finite initial distance in a soft elastic polymeric gel matrix. Magnetization in an external magnetic field tunes the magnetic attraction between the particles and drives the process. We quantify our experimental results by different theoretical tools, i.e., explicit analytical calculations in the framework of linear elasticity theory, a projection onto simplified dipole-spring models, as well as detailed finite-element simulations. From these different approaches, we conclude that in our case the cycle of virtual touching and detachment shows hysteretic behavior due to the mutual magnetization between the paramagnetic particles. Our results are important for the design and construction of reversibly tunable mechanical damping devices. Moreover, our projection on dipole-spring models allows the formal connection of our description to various related systems, e.g., magnetosome filaments in magnetotactic bacteria.

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.

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