Strain gradient crystal plasticity model with strengthening and kinematic hardening due to plastic slip gradient

Abstract

A strain-gradient single-crystal plasticity framework is developed to capture size-dependent strengthening and gradient-induced hardening effects. The constitutive equations are derived from a constrained minimization of a dual dissipative potential, with positive plastic dissipation imposed to ensure thermodynamic consistency. The plastic slip gradient is decomposed into recoverable and unrecoverable components, rather than decomposing the higher-order stresses. The constrained minimization results in the higher-order stress of each slip plane evolving nonlinearly, similar to the Armstrong-Frederick type backstress model. The evolution equation includes strain gradient hardening along with a relaxation term. In the absence of the relaxation term, the formulation produces purely gradient-induced linear kinematic hardening without additional plastic dissipation. The inclusion of the relaxation term enhances dissipation and gives rise to an higher-order isotropic-type hardening effect associated with the plastic slip gradient. As cumulative plastic flow progresses due to evolution, the higher-order stress attains saturation. Both size-dependent kinematic and isotropic hardening also reach saturation when the recoverable part of the slip gradient saturates. Conversely, the unrecoverable slip gradient continues to rise with the plastic flow. Numerical simulations are performed to assess the effect of the relaxation coefficient on a single-crystal infinite shear layer subjected to monotonic, cyclic, and non-proportional loading conditions, with responses compared to the dislocation dynamic study. Two-dimensional polycrystalline tension with a hard interface illustrates the effect of grain size on macroscopic yield stress. It is observed that size-dependent long-range interactions are active near the grain interface and exhibit a saturating behavior. Finally, the proposed methodology is assessed against recent experimental investigations.