Deformation behavior of hard-magnetic soft material beams under combined magnetic and mechanical forces

Abstract

This paper presents a theoretical model for a two-dimensional hard-magnetic soft material (HMSM) beam under combined mechanical loads and magnetic fields, based on Euler–Bernoulli beam theory. The model is validated by comparison with existing literature. Numerical simulations show that the deformation of HMSM beams is highly sensitive to the magnitude and direction of the applied mechanical and magnetic fields. Small variations in these parameters lead to significant changes in the beam's shape and response. When the external force and magnetic field are of similar magnitude, strong magneto-mechanical coupling results in pronounced bending deformation. In contrast, when the external force is much smaller than the magnetic field, the magnetic field dominates the overall deformation, while the external force subtly adjusts the bending angle. These findings provide valuable insights for optimizing HMSM-based materials in adaptive and multifunctional applications. For instance, in HMSM-based soft robots operating in confined spaces (e.g., blood vessels or pipelines), the model helps predict deformation behavior while accounting for mechanical interactions with the surrounding environment, such as friction and normal forces from vessel or pipeline walls.