Improvement of the return mapping algorithm based on the implicit function theorem with application to ductile fracture analysis using the GTN model

Elastic-plastic analyses based on finite element method are widely applied to simulate the nonlinear behavior of materials. When an analysis is conducted by implicit method, the stress values are generally updated with a time increment by using the so-called return mapping algorithm. This algorithm requires solving simultaneous nonlinear equations related to a constitutive model by an iterative solution scheme. In this study, we proposed a general method to reduce the number of unknowns in the return mapping algorithm based on the implicit function theorem. In addition, the proposed method was applied to the Gurson-Tvergaard-Needleman (GTN) model that considers the influence of damage due to nucleation and growth of microscopic void in materials in the simulation of the nonlinear behavior. After the GTN model with the proposed method was verified, elasticplastic analysis was performed by the implicit method for a 4-point bending test of pipe with a through-wall crack. The numerical solution agreed well with the experimental result. In addition, the number of iterations to obtain converged solution was reduced. Thus, we concluded that the proposed method is useful for simulating nonlinear behavior, including void nucleation and growth in materials.