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Abstract 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 improved accordingly.
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Received: 25 July 2019
Published: 14 April 2020
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2020, Vol. 41 
