The interaction between two particles made of an isotropic linearly polarizable magnetic material and embedded in an elastomer matrix is studied. In this case, when an external field is imposed, the magnetic attraction of the particles, contrary to point dipoles, is almost wraparound. The exact solution of the magnetic problem in the linear polarization case, although existing, is not practical; to circumvent its use, an interpolation formula is proposed. One more interpolation expression is developed for the resistance of the elastic matrix to the field-induced particle displacements. Minimization of the total energy of the pair reveals its configurational bistability in a certain field range. One of the possible equilibrium states corresponds to the particles dwelling at a distance, the other—to their collapse in a tight dimer. This mesoscopic bistability causes magnetomechanical hysteresis which has important implications for the macroscopic behavior of magnetorheological elastomers.

Field-induced magnetostatic interaction in a pair of identical particles made of a magnetically soft ferromagnet is studied. It is shown that due to saturation of the ferromagnet magnetization, this case differs significantly from the (super)paramagnetic one. A numerical solution is given, discussed, and compared with that provided by a simpler model (nonlinear mutual dipoles). We show that for multidomain ferromagnetic particles embedded in an elastomer matrix, as for paramagnetic ones in the same environment, pair clusters may form or break by a hysteresis scenario. However, the magnetization saturation brings in important features to this effect. First, the bistability state and the hysteresis take place only in a limited region of the material parameters of the system. Second, along with the hysteresis jumps occurring under the sole influence of the field, the "latent" hysteresis is possible which realizes only if the action of the field is combined with some additional (nonmagnetic) external factor. The obtained conditions, when used to assess the possibility of clustering in real magnetorheological polymers, infer an important role of mesoscopic magnetomechanical hysteresis for the macroscopic properties of these composites.

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This work presents an approach to the macroscopic field-controlled mechanics of magnetoactive elastomers of mixed content, which are a special type of smart materials made of an elastic composite and a combination of two essentially different ferromagnetic fillers. High-coercive particles of NdFeB-alloy powder for the magnetically hard filler and carbonyl iron powder particles with nearly zero coercivity for the magnetically soft filler are usually used. The magnetically hard particles are tens-of-micron in size and impart to the elastomer a remanent magnetisation, whereas due to the magnetically soft particles of several microns in size, the elastomer acquires a high magnetic susceptibility. Since large magnetically hard particles once magnetised in a strong field possess their own fields to which the magnetically soft particles are susceptible, the overall elastomer magnetisation as well as its mechanical response greatly depends on the relative concentration of both fillers. This work particularly studies the bending deformation of horizontally fixed magnetoactive cantilevers with the permanent magnetisation along the length axis under the action of gravity and a vertically applied uniform magnetic field. The cantilevers of the same geometry and fixed NdFeB content but different carbonyl iron concentration are considered. The magnetomechanical model is developed based on the finite-strain theory assuming the plane-stress approximation of the two-dimensional cantilever of infinite width. The magnetic energy comprises two magnetic terms, one of which is qualitatively linear and the other one is quadratic in the applied field strength. The numerically calculated field-programmed equilibrium bending shapes of the cantilevers are compared with the experimentally observed shapes. The model provides good agreement with the experiment up to moderate concentrations of the magnetically soft filler, when the coefficients of customary interpolation formulas for the concentration dependencies of elastic modulus and magnetic susceptibility are properly adjusted.