弹塑性力学问题的虚单元法  

Elastoplastic Mechanical Analysis Based on the Virtual Element Method

作　　者：林姗 杨永涛[2] 孙冠华[2] 张国华[2] 石露[2] 李伟 Shan Lin;Yongtao Yang;Guanhua Sun;Guohua Zhang;Lu Shi;Wei Li(Key Laboratory of Urban Security and Disaster Engineering,Ministry of Education,Beijing University of Technology,Beijing,100124;State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan,430071;School of Civil Engineering and Architecture,Linyi University,Linyi,276000)

机构地区：[1]北京工业大学城市与工程安全减灾教育部重点实验室,北京100124 [2]岩土力学与工程国家重点实验室,中国科学院武汉岩土力学研究所,武汉430071 [3]临沂大学土木工程与建筑学院,临沂276000

出　　处：《固体力学学报》2020年第1期30-40,共11页Chinese Journal of Solid Mechanics

基　　金：国家重点研发计划项目(2018YFE0100100);国家自然科学基金青年科学基金项目(11902134)资助。

摘　　要：具有有限差分法特征的虚单元法,可视为是有限元法向任意多边形单元的扩展.对于颗粒增强复合材料等非均质材料,在材料细观力学性能表征、非均质材料力学分析等非线性问题方面,传统的弹塑性有限元具有网格数目多、效率低下等不足之处,而虚单元法使网格划分更加灵活,更能细观反应材料真实结构,为材料的弹塑性力学分析等非线性问题提供了新的思路.基于增量法弹塑性力学原理和双线性投影算子,建立了弹塑性力学问题的虚单元法求解技术,提出了弹塑性力学问题虚单元法的应力更新方案,研究了弹性力学问题虚单元法的精度和收敛性,讨论了虚单元法求解弹塑性力学问题的网格依赖性.同时,开展了任意多边形和凹多边形单元的数值试验研究,结果表明,虚单元法无须分割多边形,仅需节点自由度便可求得单元刚度矩阵和应力等效荷载,程序实现简单,计算精度高,改善了传统有限元的网格依赖性和塑性区的网格奇异性.The virtual element method(VEM) with the features of finite difference method(FDM) can be regarded as an extension procedure of the finite element method(FEM) to arbitrary polygonal elements. Similar to the FEM, the VEM is also a Galerkin method that discretizes the entire physical domain into polygonal meshes, including non-convex shapes. The difference is that there is no need to calculate the interpolation function inside the element. In terms of the nonlinear properties such as meso-mechanical properties and mechanical analysis of heterogeneous materials, e.g., the particle-reinforced composite materials, traditional elastoplastic finite elements have many deficiencies, such as the great number of meshes and low efficiency. The virtual element method makes the mesh division more flexible, and can observe the real structure of the reaction material more closely, which provides a new idea for nonlinear problems such as the elastoplastic analysis of materials. In this research, a new procedure for elastoplastic mechanical problems with the VEM is proposed by considering the specialties of incremental elastoplastic calculation and bilinear projection operator. Then the scheme for updating the stress of elastoplastic mechanical problems with the VEM is provided. Based on the above-mentioned techniques, the accuracy and convergency of VEM for elastoplastic mechanical problems are studied. The mesh dependence of VEM for elastoplastic mechanical problems are discussed as well. Finally, some numerical experiments of arbitrary polygonal and concave polygonal elements are carried out. The triangle or quadrilateral mesh is not necessary in the VEM, and the stiffness matrix is constructed by the projection operator on the element. Only the degree of freedom of nodes is employed to build up the element stiffness matrix and the stress equivalent load. The research shows that the procedure of VEM is simple to implement and has high accuracy. At the same time, the mesh dependence and singularity of the plastic region are impr

关 键 词：数值分析 虚单元法 有限元法 弹塑性分析 屈服准则 

分 类 号：O344.3[理学—固体力学] TB115[理学—力学]