In this paper, we propose a plate model, within the framework of finite-strain elasticity, to study the magneto-mechanical behaviors of hard-magnetic soft material plates. The 2D vector plate equation is derived from the 3D governing equations through a series expansion and truncation approach. The displacement and traction boundary conditions on the edges of plate samples are also proposed. Compared with the other plate or shell theories, the current plate model naturally incorporates the elastic incompressibility and the magneto-mechanical coupling effect of the material, which doesn't involve any hypotheses on the geometry and deformation of the plates. Thus, it is applicable for modeling the large deformations of hard-magnetic soft material plates. To show the efficiency of the plate model, we study a benchmark problem, i.e., the plane-strain bending deformations of plate samples induced by magnetic and mechanical loads. Some analytical solutions of the plate equation system are derived, which provide accurate predictions on the magneto-mechanical response of the plate samples with different thickness-length ratios and magnetization directions. To promote applications of the plate model, we further derive a simplified plate equation system and write out its weak form, which is then implemented into the finite element software COMSOL Multiphysics. The efficiency of the simplified plate equation system is also verified through some typical examples. It is found that the numerical results obtained from the 2D plate model show good consistency with those of the 3D volume model, while the plate model exhibits obvious advantages in the aspects of numerical efficiency.

中文翻译：

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硬磁软材料板磁力行为的有限应变板模型


在本文中，我们提出了一种在有限应变弹性框架内的板模型，以研究硬磁软材料板的磁机械行为。 2D 矢量板方程是通过级数展开和截断方法从 3D 控制方程导出的。还提出了板样品边缘的位移和牵引边界条件。与其他板或壳理论相比，当前的板模型自然地考虑了材料的弹性不可压缩性和磁力耦合效应，不涉及对板的几何形状和变形的任何假设。因此，它适用于模拟硬磁软材料板的大变形。为了显示板模型的效率，我们研究了一个基准问题，即磁力和机械载荷引起的板样品的平面应变弯曲变形。推导了板方程组的一些解析解，为不同厚长比和磁化方向的板样品的磁机械响应提供了准确的预测。为了促进板模型的应用，我们进一步推导了简化的板方程组并写出了其弱形式，然后将其实现到有限元软件COMSOL Multiphysics中。并通过一些典型算例验证了简化板方程组的有效性。研究发现，2D板块模型得到的数值结果与3D体积模型的数值结果具有良好的一致性，同时板块模型在数值效率方面表现出明显的优势。