Advanced modeling of higher-order kinematic hardening in strain gradient crystal plasticity based on discrete dislocation dynamics

An extensive study of size effects on the small-scale behavior of crystalline materials is carried out through discrete dislocation dynamics (DDD) simulations, intended to enrich strain gradient crystal plasticity (SGCP) theories. These simulations include cyclic shearing and tension-compression tests on two-dimensional (2D) constrained crystalline plates, with single- and double-slip systems. The results show significant material strengthening and pronounced kinematic hardening effects. DDD modeling allows for a detailed examination of the physical origin of the strengthening. The stress–strain responses show a two-stage behavior, starting with a micro-plasticity regime with a steep hardening slope leading to strengthening, and followed by a well-established hardening stage. The scaling exponent between the apparent (higher-order) yield stress and the geometrical size $h$ varies depending on the test type. Scaling relationships of $h^{-0.2}$ and $h^{-0.3}$ are obtained for respectively constrained shearing and constrained tension-compression, aligning with some experimental observations. Notably, the DDD simulations reveal the occurrence of the uncommon type III (KIII) kinematic hardening of Asaro in both single- and double-slip cases, emphasizing the relevance of this hardening type in the realm of small-scale plasticity. Inspired by insights from DDD, two advanced SGCP models incorporating alternative descriptions of higher-order kinematic hardening mechanisms are proposed. The first model uses a Prager-type higher-order kinematic hardening formulation, and the second employs a Chaboche-type (multi-kinematic) formulation. Comparison of these models with DDD simulation results underscores their ability to effectively capture the observed strengthening and hardening effects. The multi-kinematic model, through the use of quadratic and non-quadratic higher-order potentials, shows a notably better qualitative congruence with DDD findings. This represents a significant step towards accurate modeling of small-scale material behaviors. However, it is noted that the proposed models still have limitations, especially in matching the DDD scaling exponents, with both models producing $h^{-1}$ scaling relationships (i.e., Orowan relationship for precipitate size effects). This indicates the need for further improvements in gradient-enhanced theories in order to guarantee their suitability for practical engineering applications.