Numerical research on magnetostrictive deformation of hard magnetic soft materials
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摘要: 光滑有限元基于光滑应变技术，在进行数值积分时避免使用等参变换，在模拟软材料大变形问题具有一定的优势。建立了基于光滑应变技术的硬磁软材料大变形模拟的数值格式，给出了数值格式中所需要的应力张量和本构张量。采用该数值格式研究了不同长高比下的硬磁软材料梁在外部磁场激励下的弯曲特性, 得到的磁载荷-位移曲线和试验结果进行了对比；模拟了含有不同残余磁场方向分布的硬磁软材料结构在外磁场作用下的形态演化过程，计算的最终变形形态与试验结果进行比较。计算结果表明：采用该数值格式得到的结果与试验结果吻合较好；在不受约束的前提下，硬磁软材料内部残余磁场方向突变处变形较大。研究结果可为由硬磁软材料组成的软体机器人和智能柔性结构的力学分析与变形调控设计提供参考。Abstract: Smoothed finite element method is based on strain-smoothed technology; it avoids using the isoparametric transformations during numerical integration, and has certain advantages in simulating large deformation problems of soft materials. A numerical format for simulating large deformation of hard magnetic soft materials based on strain-smoothed technology has been established, and the necessary stress tensors and constitutive tensors have been provided. The bending characteristics of hard magnetic soft material beams with different aspect ratios under external magnetic field were studied, and the magnetic load displacement curves obtained were compared with experimental results; evolution process of the morphological of hard magnetic soft material structures with different directions of residual magnetic field under the action of an external magnetic field was simulated, and the calculated final deformation morphology was compared with experimental results. The numerical results indicate that the results obtained by using this numerical format are in good agreement with the experimental results; there is significant deformation at the sudden change in the direction of the residual magnetic field inside hard magnetic soft materials. The research results can provide reference for the mechanical analysis and deformation control design of soft robots and intelligent flexible structures composed of hard magnetic soft materials.
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图 2 硬磁软材料梁的几何尺寸、外磁场和残余磁场方向
Figure 2. Geometric size, direction of external magnetic field, and residual magnetic field of hard magnetic soft material beams
$ {{\boldsymbol{B}}^{{\text{ext}}}} $and$ {{\boldsymbol{\tilde B}}^{\text{r}}} $denote external magnetic field and residual magnetic field respectively
图 3 Cell-based光滑有限元和有限元的网格收敛性分析
Figure 3. Grid convergence analysis of CS-FEM and FEM
$ {{{{\left( {{u_z}} \right)}_{\max }}} \mathord{\left/ {\vphantom {{{{\left( {{u_z}} \right)}_{\max }}} L}} \right. } L} $—Ratio of deflection-to-span
图 4 硬磁软材料梁的畸变网格模型和变形云图
Figure 4. Mesh model and deformation of hard magnetic soft material beam
图 5 硬磁软材料梁在在外部磁通量为50 mT的挠曲变形
Figure 5. The bending deformation of a hard magnetic soft material beam with an external magnetic flux of 50 mT
图 6 两种硬磁软材料梁的磁载荷-位移曲线
Figure 6. Magnetic loading-displacement curves of two kinds of hard magnetic soft material beams
(uz)max/L stands for the ratio of deflection-to-span; $ {{\boldsymbol{B}}^{{\text{ext}}}} $ and $ {{\boldsymbol{\tilde B}}^{\text{r}}} $ denote external magnetic field and residual magnetic field respectively; $ \mu $is the shear modulus; $ {\mu _0} $ is permeability
图 7 交叉结构的几何尺寸、残余磁通量方向
Figure 7. Geometric size and direction of residual magnetic flux of the cross structure.
图 8 H形结构在不同外磁通量下的变形响应和最终试验结果[27]
Figure 8. Deformation response of H-shaped structure under different external magnetic fluxes and final experimental result[27]
(a) $ {{\boldsymbol{B}}^{ext}} $=3 mT (b) $ {{\boldsymbol{B}}^{ext}} $=15 mT (c) $ {{\boldsymbol{B}}^{ext}} $=35 mT (d) Experimental result [27]
图 9 十字形结构在不同外磁通量下的变形响应和最终试验结果[27]
Figure 9. Deformation response of cross shaped structure under different external magnetic fluxes and final experimental result[27]
(a) $ {{\boldsymbol{B}}^{ext}} $=5 mT (b) $ {{\boldsymbol{B}}^{ext}} $=15 mT (c) $ {{\boldsymbol{B}}^{ext}} $=40 mT (d) Experimental result [27]
Figure 10. Geometric size and direction of residual magnetic flux of the hollow structure
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