Due to the dual characteristics of material nonlinearity and geometric nonlinearity exhibited by silicone rubber-like hyperelastic incompressible materials, the dynamic problems involving such materials become complex and challenging. In previous research, the Absolute Nodal Coordinate Formulation (ANCF) has demonstrated its effectiveness in addressing geometric nonlinearities during large deformations. However, ANCF tends to suffer from mesh distortion and configuration distortion issues. On the other hand, the Moving Least Squares Method (MLS) from meshfree methods uses a substantial number of nodes when constructing shape functions, which effectively improves mesh distortion problems in finite element methods when dealing with large deformations. Therefore, this paper employs Hermite-type MLS approximation functions to construct three-dimensional interpolation shape functions that replace the finite element shape function used in the traditional ANCF, thus creating an MLS-ANCF(Absolute node coordinate method based on the moving least square method) approach. Additionally, three nonlinear material models are introduced to tackle the material nonlinearity of hyperelastic beams. Moreover, Lagrange multipliers and Hamilton's principle are used to derive the static and dynamic equations for the hyperelastic beams system. To further validate the correctness of the MLS-ANCF method, this study first compares its results with those obtained from commercial software ABAQUS and static equilibrium experiments, thereby demonstrating the accuracy and effectiveness of MLS-ANCF; Next, dynamic analysis of a cantilevered silicone rubber beam under gravity alone is conducted to show the advantages of MLS-ANCF over other methods and effectively solve the issue of geometric configuration distortion caused by meshing; Furthermore, this paper also investigates the influencing factor of dynamics analysis, such as the incompressibility constant , weight function, damping coefficient, number of elements, and different nonlinear material models; Ultimately, a comparison with experimental data reveals that MLS-ANCF outperforms conventional ANCF beam elements in terms of agreement with experimental data. This demonstrates the significant role of MLS-ANCF in analyzing the dynamic characteristics of nonlinear hyperelastic beams.

中文翻译：

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基于MLS-ANCF的超弹性柔性梁动力特性分析


由于硅橡胶类超弹性不可压缩材料表现出材料非线性和几何非线性的双重特性，涉及此类材料的动力学问题变得复杂且具有挑战性。在之前的研究中，绝对节点坐标公式（ANCF）已经证明了其在解决大变形期间几何非线性方面的有效性。然而，ANCF 往往会遇到网格失真和配置失真问题。另一方面，无网格方法中的移动最小二乘法（MLS）在构造形状函数时使用了大量的节点，有效改善了有限元方法在处理大变形时的网格畸变问题。因此，本文采用Hermite型MLS逼近函数构造三维插值形函数来代替传统ANCF中使用的有限元形函数，从而创建了MLS-ANCF（基于移动最小二乘法的绝对节点坐标法） ） 方法。此外，还引入了三种非线性材料模型来解决超弹性梁的材料非线性问题。此外，拉格朗日乘子和汉密尔顿原理被用来推导超弹性梁系统的静态和动态方程。 为了进一步验证MLS-ANCF方法的正确性，本研究首先将其结果与商业软件ABAQUS和静力平衡实验得到的结果进行比较，从而证明MLS-ANCF的准确性和有效性；接下来，对悬臂硅橡胶梁在重力作用下进行动力分析，展示了MLS-ANCF相对于其他方法的优势，有效解决了网格划分带来的几何构形畸变问题；此外，本文还研究了动力学分析的影响因素，如不可压缩常数、权重函数、阻尼系数、单元数量以及不同的非线性材料模型；最终，与实验数据的比较表明，在与实验数据的一致性方面，MLS-ANCF 优于传统的 ANCF 梁单元。这证明了MLS-ANCF在分析非线性超弹梁动态特性方面的重要作用。