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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.