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

• Published in: Computers & structures Vol. 257; p. 106652
• Main Authors: Lee, Jinwoo, Bong, Hyuk Jong, Lee, Myoung-Gyu
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.12.2021; Elsevier BV
• Subjects: Accuracy; Algorithms; Bauschinger effect; Control methods; Control stability; Convergence; Deep drawing; Finite element; Finite element method; Hardening; Line search; Mapping; Metal sheets; Newton–Raphson; Search algorithms; Search methods; Softening; Strain-path change; Stress-update algorithm; Yield strength; Line search; Strain-path change; Stress-update algorithm; Newton–Raphson; Finite element
• ISSN: 0045-7949; 1879-2243
• DOI: 10.1016/j.compstruc.2021.106652

•Stress-update algorithms are provided for the anisotropic hardening model.•Comparative study of the N–R and the line search methods is performed.•The numerical instability problem is solved using the newly reformulated algorithm.•The accuracy and the speed of convergence are evaluated.•Significant improvement in the accuracy and the convergent speed is attained. 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.