Return mapping with a line search method for integrating stress of the distortional hardening law with differential softening

Abstract

A stress-update algorithm for the recently proposed distortional anisotropic hardening law is developed based on the Newton–Raphson (N–R) algorithm and line search method. The investigated yield function enables the modeling of complex path-dependent flow stress evolutions, particularly the Bauschinger effect, latent hardening/softening, and differential permanent softening. Computationally, the continuous distortion of the yield function under strain-path changes leads to numerical instability, which is overcome by a newly reformulated step-size control method in the line search algorithm. The developed algorithms were implemented in ABAQUS using the closest-point projection method and validated for extra-deep drawing quality and DP780 steel sheets in terms of numerical accuracy and stability under reverse- and cross-loading path changes. The results show that the return-mapping algorithm significantly improves the accuracy and speed of convergence when the proposed line search method is incorporated. In contrast, the conventional N–R based algorithm fails to obtain converged stresses under abrupt loading path changes owing to the sharp corners introduced by the distorted yield surface.