Microscopic theory and analysis of the mechanical properties of magneto-sensitive elastomers in a homogeneous magnetic field

Magneto-sensitive elastomers (MSEs) establish a special class of smart materials, which are able to change their shape and mechanical behavior under external magnetic field. Nowadays, MSEs are one of the most perspective smart materials, since they can be used for design of functionally integrated lightweight structures in sensors, robotics, actuators and damper applications. MSEs typically consist of micron-sized magnetizable particles (e.g. carbonyl iron) dispersed within a non-magnetic elastomeric matrix. The spatial distribution of magnetic particles in MSEs can be either isotropic or anisotropic, depending on whether they have been aligned by an applied magnetic field before the cross-linking of the polymer. Depending on the magnetic properties of the particles, their shape, size and spatial distribution, the MSEs can exhibit different mechanical behavior. Most experimental studies show that MSEs with isotropic distribution of magnetic particles demonstrate a uniaxial expansion along the magnetic field. On the other side, it was shown experimentally that MSEs with anisotropic particle distributions demonstrate a uniaxial contraction along the magnetic field. Also, the experimental works show that the shear moduli of MSEs increase with increasing strength of the magnetic field and depend on the magnetic properties, volume fraction and spatial distribution of particles. Different analytical approaches were used in theoretical studies of the mechanical behavior of MSEs. They can be roughly classified as phenomenological, continuum-mechanics and microscopic approaches. In the phenomenological approaches, the expansion into a series of the shear modulus as a function of the strength of the magnetic field has been proposed, the coefficients of the expansion being considered as phenomenological fitting parameters. In the continuum-mechanics approach, an MSE is considered as continuous magnetic media. It allows us to determine the shape and the change in volume of a spherical MSE sample, placed in a uniform magnetic field. However, this approach is restricted to homogeneous particle distributions. The microscopic approach has a clear advantage, while a discrete particle distribution and pair-wise interactions between induced magnetic dipoles can be considered explicitly. The aim of the present work is to develop a microscopic theory, which properly describes the mechanical behavior of MSEs in the external magnetic field. The theory takes a microscopic structure, finite shape of the samples and magneto-mechanical coupling between particle positions and sample deformation explicitly into account