Recently developed hard-magnetic soft (HMS) materials/structures are very attractive for applications in the fields of soft robots, biomedical devices and stretchable electronics, etc. In general, the mechanical responses of HMS structures strongly depend on the volume fraction of the embedded magnetic particles in the HMS materials. In this work, we focus on the mechanics of functionally graded hard-magnetic soft (FGHMS) beams, and demonstrate that the mechanical properties of HMS structures can be significantly affected by the volume fraction variation of the embedded magnetic particles. In particular, we consider a slender cantilevered beam made of FGHMS materials actuated by a uniform external magnetic field, and both linear and exponential variations of particle volume fraction are adopted in the investigation. By taking into account the exact geometric nonlinearity due to bending deformation, the governing equations for the mechanical responses of the FGHMS beam are derived, which are then discretized by Galerkin's method and solved by an iterative algorithm. Numerical results present the buckling, post-buckling, small and large bending behaviors of the FGHMS beam with different key parameters. It is found that the magnetic sensibility of the cantilevered HMS beam can be effectively improved by tuning the volume fraction of magnetic particles, thus a remarkable advantage of functionally graded materials is manifest. Furthermore, the underlying mechanism of the post-buckling behavior and bending deformations of the HMS beam is revealed and it is shown the mechanical responses of the FGHMS beam with linear and exponential variations of magnetization distribution have much difference. The results obtained in this work are expected to be useful for the design of FGHMS structures in the future.

最近开发的硬磁软（HMS）材料/结构对于软机器人，生物医学设备和可伸缩电子学等领域的应用非常有吸引力。通常，HMS结构的机械响应强烈取决于嵌入物的体积分数。 HMS材料中的磁性颗粒。在这项工作中，我们专注于功能梯度硬磁软（FGHMS）梁的力学，并证明HMS结构的机械性能会受到嵌入的磁性颗粒的体积分数变化的显着影响。特别地，我们考虑了由均匀的外部磁场驱动的由FGHMS材料制成的细长悬臂梁，并且在研究中采用了颗粒体积分数的线性和指数变化。通过考虑弯曲变形引起的精确几何非线性，导出了FGHMS梁机械响应的控制方程，然后通过Galerkin方法将其离散化，并通过迭代算法进行求解。数值结果显示了具有不同关键参数的FGHMS梁的屈曲，后屈曲，大小弯曲行为。发现通过调节磁性颗粒的体积分数可以有效地改善悬臂式HMS光束的磁敏性，因此，功能梯度材料具有明显的优势。此外，揭示了HMS梁屈曲后行为和弯曲变形的潜在机理，表明FGHMS梁的机械响应随磁化强度分布的线性和指数变化而有很大差异。这项工作中获得的结果有望在将来对FGHMS结构的设计有用。