Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening

Variational constitutive updates provide a physically and mathematically sound framework for the numerical implementation of material models. In contrast to conventional schemes such as the return-mapping algorithm, they are directly and naturally based on the underlying variational principle. Hence, the resulting numerical scheme inherits all properties of that principle. In the present paper, focus is on a certain class of those variational methods which relies on energy minimization. Consequently, the algorithmic formulation is governed by energy minimization as well. Accordingly, standard optimization algorithms can be applied to solve the numerical problem. A further advantage compared to conventional approaches is the existence of a natural distance (semi metric) induced by the minimization principle. Such a distance is the foundation for error estimation and as a result, for adaptive finite elements methods. Though variational constitutive updates are relatively well developed for so-called standard dissipative solids, i.e., solids characterized by the normality rule, the more general case, i.e., generalized standard materials, is far from being understood. More precisely, (Int. J. Sol. Struct. 2009, 46:1676–1684) represents the first step towards this goal. In the present paper, a variational constitutive update suitable for a class of nonlinear kinematic hardening models at finite strains is presented. Two different prototypes of Armstrong–Frederick-type are re-formulated into the aforementioned variationally consistent framework. Numerical tests demonstrate the consistency of the resulting implementation.