Reduced theory for hard magnetic rods with dipole-dipole interactions

Hard magnetic elastomers are composites of soft elastic foundations and magnetic particles with high coercivity. We formulate a theoretical framework to predict the large deformation of a hard magnetic elastomeric rod. In the previous work, the magnetic Kirchhoff rod equations, which constitute a framework for analyzing instabilities for hard magnetic rods, have been developed and validated experimentally for negligible dipole-dipole interactions. Building on previous studies, we derive the magnetic Kirchhoff rod equations with dipole-dipole interactions. The derived equations are integro-differential equations, representing the force and moment balance along the rod centerline that include long-ranged dipole-magnetic force and torque. On the basis of its discrete numerical simulation, we systematically study the effect of the the dipole-dipole interactions strength on the large deformation of hard magnetic rods. In addition, we find that our theory can predict previous experimental results without any adjustable parameters.