Choi, K. and Pan, J., "A Generalized Anisotropic Hardening Rule Based on the Mroz Multi-Yield-Surface Model and Various Classical Yield Functions," SAE Int. J. Mater. Manf. 2(1):537-546, 2009, doi:10.4271/2009-01-1392.
In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model is derived. The evolution equation for the active yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker-Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress-plastic strain curves are plotted for materials under cyclic loading conditions based on the three yield functions. For materials based on the Mises and the Hill anisotropic yield functions, the stress-plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker-Prager yield function, the stress-plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. A user material subroutine based on the Mises yield function, the anisotropic hardening rule and the constitutive relation was written and implemented into ABAQUS. Computations were conducted for a simple plane strain finite element model under uniaxial monotonic and cyclic loading conditions based on the anisotropic hardening rule, the isotropic and nonlinear kinematic hardening rules of ABAQUS. The results indicate that the plastic response of the material follows the intended input stress-strain data for the anisotropic hardening rule whereas the plastic response depends upon the input strain ranges of the stress-strain data for the nonlinear kinematic hardening rule.