A Kirchhoff-like theory for hard magnetic rods under geometrically nonlinear deformation in three dimensions. (March 2022) - Sano, Tomohiko G.; Pezzulla, Matteo; Reis, Pedro M.
Abstract: 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 (soft) matrix. Hard MREs have been receiving particular attention because the programmed magnetization remains unchanged upon actuation. Motivated by a new realm of applications, there have been significant theoretical developments in the continuum (three-dimensional) description of hard MREs. In this paper, we derive an effective theory for MRE rods (slender, mono-dimensional structures) under geometrically nonlinear deformation in three dimensions. Our theory is based on reducing the three-dimensional magneto-elastic energy functional for the hard MREs into an one-dimensional Kirchhoff-like description (centerline-based). Restricting the theory to two dimensions, we reproduce previous works on planar deformations. For further validation in the general case of three-dimensional 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–dipole interactions, which are neglected in the theory, can play a role. Graphical abstract: Highlights: We propose a reduced-order theory for the 3D deformation of a magneto-elastic rod. The theory is validated experimentally in different magnetic-field conditions. The theory is in excellent agreement with discrete simulations and experiments. Dipole–dipole interactions are not included in the theory but studied experimentally.