On the numerical implementation of thermomechanically coupled distortional hardening

In contrast to the by now classic isotropic and kinematic hardening, the more general framework of distortional hardening characterised by an evolution of the yield surface's shape can capture the effect of texture evolution on the macroscopic response. Although different distortional hardening models can indeed be found in the literature, efficient numerical implementations of such models are still missing. This statement proves particularly true within a thermomechanically coupled framework which is important for most technologically relevant processes - such as for deep drawing. Accordingly, this paper deals with an efficient finite element formulation for distortional hardening within a thermomechanically coupled framework. As a first step towards this target, the recently advocated isothermal distortional hardening framework Shi et al. (2014) is extended to the thermomechanically coupled setting. In order to avoid an over-estimation of the temperature increase due to plastic deformation, the initial yield stress is decomposed into a classic dissipative part and a non-classic energetic part. By doing so, the restrictions imposed by thermodynamical principles are fulfilled and simultaneously realistic temperature predictions are obtained. For the resulting model, an efficient numerical implementation is proposed. By developing a suitable time integration scheme for the evolution equations of the fourth-order tensor describing the distortional hardening, a return-mapping scheme for updating the internal variables is derived which shows the same numerical complexity as a return-mapping scheme for purely isotropic hardening. This efficient return-mapping scheme is finally incorporated into a thermomechanically coupled finite element formulation, and the resulting set of equations is fully implicitly and monolithically solved by means of a Newton-type iteration. Several numerical complex examples demonstrate the capabilities of the distortional hardening model as well as the robustness and efficiency of the numerical formulation.