A finite-strain constitutive model for magnetorheological elastomers: Magnetic torques and fiber rotations

Abstract

This paper is concerned with the development of constitutive models for a class of magnetoelastic composites consisting of stiff, aligned cylindrical fibers of a magnetizable material that are embedded firmly in a soft elastomeric matrix. The fibers have elliptical cross section and their (transverse) in-plane axes are also aligned, but their distribution is random and characterized by “elliptical” two-point correlations. Estimates are obtained for the macroscopic response and stability of this new type of magnetorheological elastomer (MRE) under combined in-plane mechanical and magnetic loading by means of the finite-strain homogenization framework and “partial decoupling approximation” of Ponte Castañeda and Galipeau (2011). The resulting macroscopic magnetoelastic constitutive model accounts for the microstructure of the composite and its evolution under finite strains and rotations, as well as for the nonlinear magnetic behavior of the fibers, including the effect of magnetic saturation. When the loading directions are not aligned with the fiber axes, the model predicts magnetic and mechanical torques on the fibers, leading to their in-plane rotation, which is found to have significant effects on the coupled magnetoelastic response of the composite, including the possible development of macroscopic torques on a given finite-size sample of the composite. To eliminate these macroscopic torques, while maintaining the advantageous effects of the fiber rotations, we also investigate the response of a laminated composite consisting of plus/minus orientations of the fibers relative to the layering direction, and subjected to magnetic and mechanical loadings along the layering direction. The results for the actuation tractions, magnetostrictive strain and magnetoelastic moduli demonstrate that the microstructure of these laminated MRE samples can be designed optimally for significantly enhanced magnetoelastic effects. In particular, the actuation tractions and magnetostrictive strains can be made several times larger than the corresponding tractions and strains for isotropic MREs with spherical (circular) particles.