MagCBCM: Chained Beam-Constraint-Model of Magneto-Mechanical Response in Hard-Magnetic Compliant Beams

Hard-magnetic soft materials (HMSMs) have emerged as pivotal components in the development of soft robotics, biomedical devices, and flexible electronics. However, existing modeling frameworks for HMSM-based compliant structures are often computationally intensive due to inherent geometric nonlinearities and are typically limited to pure magnetic actuation, offering insufficient consideration of coupled magneto-mechanical loading. To address these challenges, this study develops a magneto-mechanical chained beam-constraint-model (MagCBCM) derived from Euler–Bernoulli beam theory and the principle of virtual work. The proposed model facilitates the comprehensive analysis of diverse structural deformations under combined mechanical and magnetic loading, encompassing magnetic soft continuum robots with external contact, thin-walled architectures, and the snap-buckling behavior of bistable hard-magnetic beams. Validation against established analytical models and experimental results confirms the accuracy and effectiveness of the framework. Generally, MagCBCM can serve as a versatile parameterized modeling tool for the rational design and optimization of advanced hard-magnetic functional structures.