We present a theory for finite and spatial elastic deformation of rods under the influence of arbitrary magnetic field and boundary condition. The rod is modeled as a Kirchhoff rod and is assumed to have uniformly distributed array of uniaxial spheroidal paramagnetic inclusions embedded in it all pointing in the same direction in the undeformed state. The governing equations of the magnetoelastic rod are derived which are further non-dimensionalized and linearized to investigate buckling in such rods. Analytical expressions for the onset of buckling from the rod’s trivial state are obtained in terms of loading parameters (applied magnetic field, axial load, torque) as well as geometric (inclusion orientation in the undeformed state) and material (ratio of bending and torsional stiffnesses) parameters for different combinations of boundary conditions. The buckled shape of the rod at the onset of buckling is also examined. The presented work can be useful in simulation and design of magnetic soft continuum robots.

我们提出了在任意磁场和边界条件影响下杆的有限和空间弹性变形的理论。该杆被建模为基尔霍夫杆，并假定其中嵌入了均匀分布的单轴球状顺磁性夹杂物阵列，所有夹杂物在未变形状态下都指向同一方向。推导出磁弹性杆的控制方程，将其进一步无量纲化和线性化以研究此类杆的屈曲。根据载荷参数（外加磁场、轴向载荷、扭矩）以及几何（未变形状态下的夹杂物方向）和材料（弯曲和扭转刚度的比率）参数，用于不同边界条件组合。还检查了杆在屈曲开始时的屈曲形状。所提出的工作可用于磁软连续体机器人的模拟和设计。