A simple scalar directional hardening model for the Bauschinger effect compared with a tensorial model

Modeling the Bauschinger effect is usually accomplished by introducing a second-order back-stress or directional hardening tensor. The objective of this paper is to propose a simpler scalar model of the Bauschinger effect based on a scalar directional hardening parameter that is determined by integration of an evolution equation. The behavior of this scalar model is compared to a tensorial model for a number of load cases. Strongly objective numerical algorithms are developed for integrating the evolution equations for both the tensorial and scalar models. Also, a consistent tangent is developed for both models. Obviously, the numerical implementation of the scalar model is significantly less complicated than for the tensorial model. Examples show that the tensorial and scalar models predict the same results for cyclic proportional triaxial extension and triaxial compression loadings. In contrast, the tensorial model predicts a Bauschinger effect for cyclic proportional pure torsion loading which is not predicted by the scalar model. More complicated examples with nonproportional loading paths and inhomogeneous deformations indicate that, relative to the tensorial model, the scalar model accounts for directional hardening fairly well and the simplicity of the model makes it an attractive option to add to isotropic hardening models.