Forces on Rigid Inclusions in Elastic Media and Resulting Matrix-Mediated Interactions

Abstract

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...

Force-induced elastic matrix-mediated interactions in the presence of a rigid wall

“…As in related hydrodynamic approaches [7,40,41], we only consider the leading-order matrix-mediated couplings between the particles, i.e., couplings to first order in the inverse separation distance between the particles. To this order, our previous comparison with experimental measurements in the bulk confirmed a very good match for moderate particle separation distances [19].…”

“…For this purpose, we specify the forces acting on the particles. For instance, we have recently investigated the behavior of magnetizable paramagnetic Nickel particles embedded in a bulk elastic matrix [19]. There, magnetic interactions between the particles were induced and tuned by applying and rotating an external magnetic field.…”

“…Obviously, this force changes with relative displacements of the particles. In the calculations below, we include these corrections by a simple iterative loop to determine the final displacements and forces [19]. In summary, we work here with pairwise magnetic forces.…”

“…More precisely, if forces are externally imposed on or induced between rigid inclusions in an elastic matrix, they lead to deformations of the matrix. These deformations are long-ranged and affect the positions of other inclusions, leading to matrix-mediated coupled displacements [19].…”

A density functional approach to ferrogels

“…In our one-dimensional set-up, we only consider forces and displacements along the particle centerto-center vector r. To first order in the particle distance, i.e. to order 1/r with r = |r| we then obtain [63][64][65][66]:…”