  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.
