The quaternion beam model for hard-magnetic flexible cantilevers

The recently developed hard-magnetic soft (HMS) materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields, e.g., soft robotics, flexible electronics, and biomedicine. Theoretical investigations on large deformations of HMS structures are significant foundations of their applications. This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams. A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact (GE) beam model. The singularity-free governing equations for the three-dimensional (3D) large deformations of an HMS beam are first derived, and then solved with the Galerkin discretization method and the trust-region-dogleg iterative algorithm. The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones. The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples. It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field. The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.