Consistency condition of isotropic–kinematic hardening of anisotropic yield functions with full isotropic hardening under monotonously proportional loading

For the combined type isotropic–kinematic hardening law to account for the Bauschinger, transient and permanent softening behaviors observed in reverse loading, formulations have been initially developed for isotropic yield functions, mainly based on the von Mises criterion, and then later extended for anisotropic yield functions. Among the efforts to introduce anisotropic yield functions to the combined type hardening formulation, however, some inconsistency has been encountered in manipulating the kinematic hardening law, especially for the nonlinear type law, even though their von Mises yield function versions have been consistent. Therefore, theoretical clarification and clearance of such inconsistency were attempted in this work by imposing the following consistency condition: the combined type hardening law is expected to behave the same as the full isotropic hardening for monotonously proportional loading, regardless of anisotropic yield functions which are coupled with the combined type hardening law. An example to account for the anisotropic hardening of an anisotropic yield function utilizing the combined type hardening law, but for which the consistency condition was partially released, was also demonstrated.

► The consistency condition for the anisotropic combined type hardening law.
► Proper definition of the anisotropic equivalent plastic strain increment.
► Proper anisotropic normalization of stress quantities for the back stress evolution.
► Introduction of anisotropic pseudo-equivalent quantities.
► Anisotropic hardening based on the relaxed combined type hardening law.