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. The beam model and FEM are first validated using experiments and then employed to predict the deflection of a cantilever beam in either a uniform or a constant gradient field. The corresponding parameters governing the magneto-elastic coupling are identified. Then, a set of comparative numerical studies for actuation in different configurations 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 configurations considered, to serve as a valuable predictive toolbox for the rational design of beam-like hard-magnetic structures.