A mesoscale numerical study on the geometrically necessary dislocations at grain boundaries and the back stress in polycrystalline grains

Elucidating the relationship between geometrically necessary dislocations (GNDs) and back stress is essential for modeling the strain hardening behavior of polycrystalline materials. This study employs dislocation dynamics simulations to quantitatively assess the impact of GND distributions on the associated back stress at the mesoscale. In a simple cubic lattice, the stress fields generated by elementary GND boundaries, including variations in boundary sizes, dislocation types, and distribution patterns, are systematically analyzed. By taking into account the fluctuation of surface GND density, the calculation of back stress is established using the elasticity theory of dislocations combined with scaling functions. It has been demonstrated that the surface GND density is a critical parameter that controls the amplitude of back stress. Subsequently, the prediction of back stress in face-centered cubic crystalline grains is validated with more realistic GND distributions. Considering identical initial Frank-Read sources, dislocation pile-ups are predominantly formed in coarse grains, yet the resulting surface GND density remains comparable to that observed in smaller grains. This phenomenon is responsible for the similar back stress values in grains of varying sizes. Finally, the activation of cross-slip inhibits the formation of dislocation pile-ups, leading to a linear decrease in back stress with increasing plastic strain.