Mesoscopic models for definition of the large-scale elastic properties of the soft magnetic elastomers

A pair of magnetizing particles embedded in a cylinder made of a high-elasticity (hyperelastic) material is considered as a model of a mesoscopic structure element of a soft magnetic elastomer. In the presence of the magnetic field particles magnetize and the force interaction is arisen between them. Particles change position inside the elastomer matrix as the elastic resistance allows it. Equilibrium position of the particles inside the sample is determined by the balance of magnetic and elastic forces and corresponds to the minimum of total energy of the system. In its calculation both the non-linearity and heterogeneity of the magnetization of the particles and non-linearity of the elastic properties of the elastomer have been taken into account. This brings us to the real magnetorheological composite, that is a soft elastomer filled with a micron ferromagnetic particles. The considered system exhibits bistability: increase and decrease of the applied magnetic field, leads to change of the distance between the particles in hysteretic manner, from a few radii to the tight contact (collapse). This behavior significantly affects the ability of a mesoscopic element to resist external load. Collapse of the particles inside it by a magnetic field or compressive load causes sharp increase of stiffness. The dependence of mechanical characteristics of the system on the strength of an applied magnetic field is studied for the elements of different compliance. This dependence also has a hysteresis. Despite its simplicity, the model in a generally correct way describes the field-induced changes of the internal structure of soft magnetic elastomers. The obtained results are used for qualitative analysis of the macroscopic magnetomechanics of the composite, this is done with the aid of a homogenisation procedure based of Voigt's hypothesis. The obtained dependence of the magnetic stiffness of soft magnetic elastomer on the external magnetic field agrees qualitatively with the published experimental results.