Численная реализация метода переменных параметров при решении упругопластических задач на основе графовой модели упругого тела

Abstract

The article highlights the results of using a graph model of a continuous medium for solving elastic-plastic problems. The stress-strain state is determined by the method of variable parameters based on the deformation diagram of the medium material. The method is based on the representation of the defining elastic plasticity relations in the form of equations of linear elasticity theory, but with variable elasticity parameters. The computational process is an iterative procedure in which each successive approximation is reduced to solving a linear elastic problem. The problem is solved by the graph method. The stress-strain state is found by a non-standard numerical method in which a solid body is represented by a discrete model in the form of an oriented graph. Concrete examples show the high efficiency of the method in comparison with the traditional finite element method. The increased accuracy of calculations, even when using coarse grids, is ensured by the fact that: 1) the vertex and contour laws of graph theory implement the equations of equilibrium and compatibility of deformations for the element as a whole; 2) the equilibrium equations are performed locally by the volume of the element. The problems of the elastic-plastic bending of the console and the elastic-plastic state of a plate with a circular hole are solved as examples. Comparison of the obtained results with the solutions of these problems by other methods showed a good match. The high accuracy of calculations makes it possible to use the iterative procedure of the method of variable elasticity parameters as a subroutine in the application package created for the graph method.