Assessment of third invariant elasto-plastic models: Mathematical aspects, numerical strategies and comparative results

This contribution presents a study of Hosford's and Gao et al.'s elasto-plastic models each influenced by the third invariant of the deviatoric stress tensor (J3) in their formulations. The first stage of this work is a review of the concepts of Mechanics of Materials and Constitutive Modeling Theory. Subsequently, Hosford's and Gao et al.'s models with isotropic hardening are assessed taking into account their mathematical and numerical aspects. Their yield surface's shape and consequent convexity issues are investigated through a preliminary analysis of the effects that the third invariant (J3) has on their formulations. A return mapping algorithm is then proposed and tested for Hosford's and Gao et al.'s elasto-plastic models based on the operator splitting methodology. The proposed algorithm was implemented through an implicit numerical integration method in an academic finite element environment, along with its consistent tangent matrix. Finally, the numerical results obtained from the proposed models are compared to experimental data available in the literature. This research assesses the performance and precision of the proposed constitutive formulations to correctly describe the mechanical behavior of an aircraft aluminum alloy and 1045 alloy steel under realistic stress and strain fields. In order to do so, different specimens capable of generating distinct stress states within high and low stress triaxiality regions are taken into account. As a result, this work demonstrates that Gao et al.'s yield criterion may generate non-convex yield surfaces when the influence of the third invariant is especially strong. In addition, it is observed that Hosford's constitutive model presents better agreements between the obtained numerical results and the experimental data collected from the literature. The most important finding however is that the third invariant should be essential to any formulation.