Numerical analysis of two-dimensional elastoplastic problems based on zonal free element method

In this paper, a weak-form numerical method, the zonal free element method (ZFrEM) is introduced to solve two-dimensional elastoplastic problems. In ZFrEM, the whole domain is divided into several sub-domains, and then a series of points are used to discretize each sub-domain. The Lagrange isoparametric element concept in the finite element method (FEM) is employed to form the collocation element for each collocation node with the neighboring points, and the system equations are generated with the point-by-point. The continuous model separates strain into elastic and plastic components, ensuring that stress variation is solely dependent on the elastic component of the strain, which is consistent with classical elasticity theory, and assuming that plastic strain is independent of stress increments, which results in a nonlinear system of equations with the coefficient matrix dependent on the current incremental stress state. For the nodes in the plastic state, a stress regression technique aligns stresses with defined yield surfaces. Three numerical examples are given to verify the accuracy and convergence of the present method for solving the elastoplastic problems.