Improvement of return mapping algorithm in GTN model by reducing the number of equations to be solved

Mano, Akihiro ; Imai, Ryuta*; Miyamoto, Yuhei*; Katsuyama, Jinya  ; Li, Y. 

Finite element analysis (FEA) based on Gurson-Tvergaard-Needleman (GTN) model is utilized for the simulation of the non-linear plastic behavior of a material. In the analysis, the variation of the size of yield surface based on the void nucleation, growth and coalescence in a material is considered. When the analysis is conducted by an implicit method, the stress values are generally updated with a time increment considering the size of yield surface by using the so-called return mapping algorithm. This algorithm requires solving simultaneous nonlinear equations related to a constitutive model. Thus, their solutions are usually obtained by an iterative calculation scheme. However, when the number of equations is large as with the GTN model, it becomes difficult to obtain the converged solutions in the iterative calculation scheme. In the commercial FEA software Abaqus version 2018, the analysis based on the GTN model considering the void coalescence cannot be performed by the implicit method, in which the converged solutions are difficult to obtain. In this research, the number of equations in return mapping algorithm of the GTN model was reduced by applying the implicit function theorem and the convergence of the analysis was improved. Based on the return mapping algorithm improved regarding the number of equations, the analysis based on GTN model was conducted by the implicit method in Abaqus taking the 4-point pipe bending test as an example. As the result, it was confirmed that the variation in load against displacement in the test could be simulated.