A model of hard-magnetic soft curved beams at large deformation

Abstract

<p indent="0mm">Magnetic-activated soft materials have shown their considerable advantages such as quick response, remote and reversible control, and programmable shape-shifting, making them promising in functional designs of bionic robots, bionic sensors, grippers, valves and metamaterials. Magnetoactive elastomers (MAEs) (also referred to as magnetic polymer composites (MPCs) or magnetorheological elastomers) are a kind of soft active composites that can respond to magnetic field excitation, which are generally composed of micrometer-scale magnetizable particles (e.g., NdFeB particles) embedded in a soft matrix (e.g., silica gel, hydrogel and shape memory polymer). The matrix acts as a continuum medium transmitting the internal force, while the magnetic particles have the interaction among themselves and with the external magnetic field. Compared with soft magnetic materials which are difficult to achieve robust programming, hard-magnetic materials hold residual magnetic flux density vector that does not change with the external magnetic field. After saturation of magnetization, the interaction with the external magnetic field produces micro torque, and programming the micro torque distribution generated by hard-magnetic particles can further form complex shapes. Owing to the intricate interplay between inherent electromagnetic coupling and geometric nonlinearity, it is still a challenge to model and predict the mechanical behaviors of hard-magnetic soft materials. Most of existing models of hard-magnetic elastomers are based on the assumption that the remanence direction is along the axis, which limits the diversification degree of deformation to some extent. In addition, most hard-magnetic beam models are based on Euler-Bernoulli hypothesis (normal remains normal) by neglecting shear deformations. When the beam becomes thick, the effect of shear deformation may turn out to be significant and cannot be ignored. Here, we develop a nonlinear curved beam model to predict large deformations of hard-magnetic soft elastica. The model follows Timoshenko’s first-order shear deformation theory, by accounting for axis elongation and the constitutive law of hard-magnetic elastomers. A set of ordinary differential governing equations containing seven basic unknowns are deduced theoretically. The model can thus describe shear deformation, axis elongation and curvature change of hard-magnetic soft beams upon large deformation. Combined with boundary conditions, the model can be efficiently solved by using shooting methods. In order to verify the accuracy of our hard-magnetic soft beam model, we compare it with the experiments, finite element simulations, and the Euler-Bernoulli hypothesis-based beam model in the literature, for different directions of applied magnetic field. The results verify and highlight the accuracy and universality of our hard-magnetic soft curved beam model which has higher accuracy and wider application range (applicable to thick beams, thin beams and curved beams) compared to the hard-magnetic thin beam model based on Euler-Bernoulli hypothesis (only considering bending deformations). Besides, we explore the bending and buckling deformations of hard-magnetic soft beams with different length-thickness ratios, under different applied magnetic fields. We reveal that the decrease of length-thickness ratio and of applied magnetic fields can enhance shear effects on deformations of hard-magnetic soft beams. Our model could be used not only to analyze geometrically nonlinear behaviors of hard-magnetic elastomers, but also to guide rational designs of ferromagnetic slender structures and soft continuum robots.