Simultaneously iterative procedure based on block Newton method for elastoplastic problems

In this article, the authors formulate elastoplastic problems as a coupled problem of the equilibrium equation and the yield condition at each material point, and develop a numerical procedure based on the block Newton method to solve the overall structure using the finite element discretization. For the integration of stress, the backward difference scheme is employed. In the conventional return mapping algorithm, the algorithmic tangent moduli are derived analytically so that it is consistent with local iterative calculation to determine internal variables. On the other hand, in the proposed block Newton method, the tangent moduli can be obtained algebraically by eliminating the internal variables, and the internal variables are also updated algebraically without local iterative calculation. The residual of the yield condition is incorporated into the linearized equilibrium equation. The proposed approach enables the errors of the equilibrium equation and the yield condition to decrease simultaneously. Some numerical examples show the validity and effectiveness of the proposed approach by comparing the results of the proposed approach with those of the conventional return mapping algorithm.