Incremental variational homogenization of elastoplastic composites with isotropic and Armstrong-Frederick type nonlinear kinematic hardening

In order to investigate the behavior of elastoplastic composites exhibiting both isotropic and nonlinear kinematic hardening, we extend the Double Incremental Variational (DIV) formulation of Lucchetta et al. (2019), based on both the incremental variational principles introduced by Lahellec and Suquet (2007) and the formulation proposed by Agoras et al. (2016). However, the Armstrong-Frederick model (Armstrong and Frederick), which is very often used to describe nonlinear kinematic hardening and refers to the framework of non-associated plasticity (Chaboche, 1977), cannot be handled within the framework of generalized standard materials as required by the incremental variational principles on which the DIV formulation relies. That is why we work with an approximation of this model, namely the modified Chaboche model (Chaboche, 1983). As the dissipation potential associated with this model depends on internal variables, we have to extend the incremental variational principles of Lahellec and Suquet to such a situation. Then, we apply twice the variational procedure of Ponte Castañeda (1991), first to linearize the local behavior and then to deal with the intraphase heterogeneity of the thermoelastic Linear Comparison Composite (LCC) induced by the linearisation step. The resulting thermoelastic LCC with per-phase homogeneous properties is homogenized by classical linear schemes. We develop and implement this new incremental variational procedure for two-phase matrix-inclusions composites with an isotropic elastoplastic matrix exhibiting combined isotropic and nonlinear kinematic hardening. For various cyclic loadings, the predictions of the proposed DIV formulation compare favorably with Finite Element simulations based on the Armstrong-Frederick model.