On the theoretical and numerical modelling of Armstrong–Frederick kinematic hardening in the finite strain regime Original Research Article

On the theoretical level, the present paper presents a detailed comparison of recent finite strain models for Armstrong–Frederick kinematic hardening. Thereby two strategies are discussed: (1) “Chaboche-type” concepts, considering the back stress as internal variable, (2) continuum mechanical extensions of the classical rheological model, using only strain-like internal variables. It is shown in the paper that models of the second kind can be recast in the format of anisotropic inelasticity with structure tensors. Second, the work focuses on the algorithmic treatment of the kinematic hardening concepts presented before. This problem has been tackled up to now only in the context of linearized models. In contrast to isotropic finite elastoplasticity, the integration cannot be carried out with respect to principal axes. Therefore, a new integration algorithm is developed which is suitable for the anisotropic case but still retains plastic incompressibility. In the case of small elastic deformation, the algorithm reduces to a system of only one non-linear equation and twelve linear equations. In general, the computational effort of the new scheme does not exceed the one of the backward Euler scheme which has the disadvantage that plastic incompressibility is not fulfilled automatically. Several numerical examples show that the representatives of both approaches, (1) and (2), yield similar results, if physically reasonable material parameters are chosen.