Non-proportional hardening models for predicting mean and peak stress evolution in multiaxial fatigue using Tanaka’s incremental plasticity concepts

Abstract Non-proportional (NP) hardening can have a significant effect on multiaxial fatigue lives due to the increase in mean and peak stresses it can cause when compared to proportional loads of similar range. Its effect depends both on the material and on the load history path, quantified respectively by the additional NP hardening coefficient α NP and the NP factor F NP . However, since F NP depends on the load path, it is not easy to evaluate under complex multiaxial loading conditions. So, several estimates for the steady-state value of F NP have been proposed to deal with such cases, based on enclosing ellipses or on integrals calculated along the stress, strain, or plastic strain path. Tanaka’s incremental plasticity model, on the other hand, is able to predict the NP hardening evolution as a function of the accumulated plastic strain p along any load history, based on the concept of a polarization tensor [ P T ] that can be correlated to an internal dislocation structure. In this work, it is shown that the eigenvectors of [ P T ] represent principal directions along which dislocation structures may form, whose intensity is quantified by the respective eigenvalues. Two new integral estimates for the steady-state F NP of periodic load histories are proposed, one that exactly reproduces the incremental plasticity predictions from Tanaka’s model, and a simpler version calculated solely as a function of the eigenvalues of [ P T ]. Both can be readily used in fatigue design to correctly account for the additional mean or peak stress effects induced by NP periodic histories. For general non-periodic multiaxial histories, it is also shown that Tanaka’s model can be expressed as an evolution equation for F NP ( p ). Tension–torsion experiments with 316L steel tubular specimens under especially selected discriminating square NP strain paths are conducted to validate the proposed models.