Solution Of Elasto-plastic Plane Stress Problems Based On The Quadrature Element Method

The nonlinear problem of materials is the key point in the analysis of engineering structures.The integral of constitutive relation is also an important link in the process of solving nonlinear problems of materials.The quadrature element method is a numerical method of high precision and high order interpolation based on the weak form description of the region to be solved.The quadrature element method allows different material properties of different interpolation points in the element.In this paper,a high order quadrature element is constructed for calculating elastic-plastic plane stress problems,and a new method for analyzing elastic-plastic plane stress problems is developed by combining the quadrature element method with the incremental variational principle and using the integral constitutive relation of implicit return mapping algorithm.The main contents and conclusions include:1.The quadrature element method is introduced for numerical discretization by treating the incremental energy functional of the solution problem,and the incremental quadrature element scheme is derived by combining with the incremental variational principle.A new method for simulating elastic-plastic plane stress problems is developed by combining with the implicit return mapping algorithm.The proposed method can be used to solve the elastic-plastic problems of ideal elastic-plastic,strain-hardening and mix-hardening constitutive relations.2.Fortran language programming is used to implement the algorithm in this paper,and it is applied to solve the elastic limit load of the structure.Through calculating five examples,the influence of factors such as the division form of the quadrature element,the interpolation number of quadrature element,Poisson's ratio and the structure size on the calculation accuracy of the elastic limit load of the structure is studied.The numerical results of the elastoplastic limit load are compared with the theoretical analytical results,and the results show that the proposed method is accurate and reliable for elastoplastic limit load calculation.3.On the basis of high precision elastic limit load,the proposed algorithm was applied to simulate the plastic zone expansion of three numerical examples.The results obtained after processing were compared with the finite element simulation results,which verified the effectiveness of the proposed algorithm in the study of plastic zone expansion.When the stress concentration phenomenon exists in the simulation structure,the quadrature element method only needs a few and high order elements to achieve the calculation accuracy of the finite element method under the very fine mesh.4.Based on the incremental theory of plastic mechanics,the displacement-type quadrature element constructed in this paper shows strong computing power when it is used for nonlinear analysis of materials.In order to obtain a higher precision solution,it only needs to increase the interpolation points in the quadrature unit.Moreover,the elastic and plastic behaviors of different interpolation points in the same quadrature element can coexist.5.The results of numerical examples show that the quadrature element method can be used to calculate the elastic-plastic plane stress problems,and the results are quite reliable,which provides a new idea and method for the nonlinear analysis of engineering structure materials.