A comprehensive framework for hard-magnetic beams: Reduced-order theory, 3D simulations, and experiments

Thin beams made of magnetorheological elastomers embedded with hard-magnetic particles (hard-MREs) are capable of large deflections under an applied magnetic field. We propose a comprehensive framework, comprising a beam model and 3D finite element modeling (FEM), to describe the behavior of hard-MRE beams under both uniform and constant gradient magnetic fields. First, based on the Helmholtz free energy of bulk (3D) hard-MREs, we perform dimensional reduction to derive a 1D description and obtain the equilibrium equation of the beam through variational methods. In parallel, we extend the existing 3D continuum theory for hard-MREs to the general case of non-uniform fields by incorporating the magnetic body force induced by the field gradient and implementing it in FEM. Then, we validate the beam model and FEM using experiments on a cantilever beam in either a uniform or a constant gradient field, and identify the dimensionless parameters governing the magneto-elastic coupling. Further, a set of comparative numerical studies for different field configurations and magnetization profiles yields additional insight into the beam response. Our study builds on previous work on hard-MRE beams, while providing a more complete framework, both in terms of the methodologies used and the broader range of configurations considered, serving as a valuable predictive toolbox for the rational design of beam-like hard-magnetic structures.