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 引用本文: 徐兵兵, 彭帆. 基于新型虚单元法的超弹性材料变形研究. 力学学报, 2024, 56(9): 2635-2645.
Xu Bingbing, Peng Fan. Research on deformation of hyperelastic materials based on a new virtual element method. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(9): 2635-2645.
 Citation: Xu Bingbing, Peng Fan. Research on deformation of hyperelastic materials based on a new virtual element method. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(9): 2635-2645.

## RESEARCH ON DEFORMATION OF HYPERELASTIC MATERIALS BASED ON A NEW VIRTUAL ELEMENT METHOD

• 摘要: 虚单元法是一种先进的求解固体力学问题的数值方法, 在过去10年间, 该算法在线弹性问题中得到了较为广泛地开发和应用. 文章尝试给出一种通用的高阶虚单元法格式, 可用于计算超弹性问题以及更普遍的非线性问题. 与传统求解力学问题的虚单元法的思想不同, 其主要思想是对泊松方程求解映射算子, 并将该映射算子直接用于位移场的近似, 从而可求解众多非线性力学问题. 由于采用了标量场的映射算子来近似矢量场, 因此该算法格式简单, 并且可以轻易扩展到高阶格式或者三维问题求解. 将从泊松方程出发, 介绍虚单元法中椭圆映射算子的计算方法, 在此基础上, 详细推导虚单元法在求解超弹性问题时的具体格式, 并给出切线刚度矩阵的计算方法. 最后, 给出了几个典型的超弹性数值算例, 从而证明该虚单元法格式的有效性.

Abstract: The virtual element method is an advanced numerical method for solving solid mechanics problems. In the past ten years, the numerical method has been widely developed and applied in linear elasticity problems. This work attempts to give a general high-order virtual element method format that can be used to calculate hyperelastic problems and more general nonlinear problems. Different from the traditional virtual element method for solving mechanical problems, its main idea is to solve the projection operator for the Poisson equation and use the projection operator directly for the approximation of the displacement field, so that it can solve many nonlinear mechanical problems. Since the projection operator of the scalar field is used to approximate the vector field, the method has a simple format and can be easily extended to high-order formats or three-dimensional problems. This work will start from the Poisson equation and introduce the calculation method of the elliptical projection operator in the virtual element method. On this basis, the specific format of the virtual element method in solving hyperelastic problems will be derived in detail, and the calculation of the tangent stiffness matrix will be given. Finally, this paper gives several typical hyperelastic numerical examples to prove the effectiveness of the virtual element method format.

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