Multiplicative magneto-elasticity of magnetosensitive polymers incorporating micromechanically-based network kernels

Magnetosensitive elastomers are a class of composite materials whose mechanical response may be altered by application of magnetic fields. Such materials have tunable mechanical properties and find use in controllable stiffness devices and applications for active control of structural components. In this work we present a novel approach to the macroscopic magneto–elastic modeling of magnetorheological elastomers (MREs) at finite strains, whose structure accounts in a modular format for micromechanically-based ingredients. Keeping in mind the composite nature at the microscale, we develop a constitutive formulation that provides two microstuctural-based kernels for (i) the energy of the magnetized rigid iron particles and (ii) the elastic energy of the deformable polymer network. The latter is achieved by a multiplicative elasto-magnetic split of the deformation gradient. Here, a left decomposition is proposed that includes a magnetostrictive deformation in terms of the spatial (true) magnetization vector of the composite material. This approach induces an anisotropic Eulerian metric depending on the magnetization, which is used to map standard isotropic chain statistics into anisotropic ones. As a consequnce, the approach allows to make use of micromechnically-based isotropic network models for polymers in a modular format and extends their application to the anisotropic coupled magnetomechanical response. In particular, such a formulation allows the inclusion of the microsphere model for network-based elasticity, that has been successfully applied to the modeling of rubber-like polymers. We discuss details of a unified modeling structure within a variational formulation of finite magneto-elasticity, and outline ingredients of the finite element implementation of the coupled problem. The modeling capabilities are demonstrated by solving application-oriented boundary value problems.