  Magneto-rheological elastomers (MREs) are functional materials that can be
actuated by applying an external magnetic field. MREs comprise a composite of
hard magnetic particles dispersed into a nonmagnetic elastomeric matrix. By
applying a strong magnetic field, one can magnetize the structure to program
its deformation under the subsequent application of an external field. Hard
MREs, whose coercivities are large, have been receiving particular attention
because the programmed magnetization remains unchanged upon actuation. Hence,
once a structure made of a hard MRE is magnetized, it can be regarded as
magnetized permanently. Motivated by a new realm of applications, there have
been significant theoretical developments in the continuum description of hard
MREs. By reducing the 3D description into 1D or 2D via dimensional reduction,
several theories of hard magnetic slender structures such as linear beams,
elastica, and shells have been recently proposed. In this paper, we derive an
effective theory for MRE rods under geometrically nonlinear 3D deformation. Our
theory is based on reducing the 3D magneto-elastic energy functional for the
hard MREs into a 1D Kirchhoff-like description. Restricting the theory to 2D,
we reproduce previous works on planar deformations. For further validation in
the general case of 3D deformation, we perform precision experiments with both
naturally straight and curved rods under either constant or constant-gradient
magnetic fields. Our theoretical predictions are in excellent agreement with
both discrete simulations and precision-model experiments. Finally, we discuss
some limitations of our framework, as highlighted by the experiments, where
long-range dipole interactions, which are neglected in the theory, can play a
role.
