On dislocation pileups and stress-gradient dependent plastic flow

In strain-gradient plasticity, the length scale controlling size effect has been attributed to so-called geometrically necessary dislocations. This size dependency in plasticity can also be attributed to dislocation pileups in source-obstacle configurations. This has led to the development of stress-gradient plasticity models in the presence of stress gradients. In this work, we re-examine this pileup problem by investigating the double pileup of dislocations emitted from two sources in an inhomogeneous state of stress using both discrete dislocation dynamics and a continuum method. We developed a generalized solution for dislocation distribution with higher-order stress gradients, based on a continuum method using the Hilbert transform. We qualitatively verified the analytical solution for the spatial distribution of dislocations using the discrete dislocation dynamic. Based on these results, we developed a dislocation-based stress-gradient plasticity model, leading to an explicit expression for flow stress. Findings show that this expression depends on obstacle spacing, as in the Hall-Petch effect, as well as higher-order stress gradients. Finally, we compared the model with recently developed models and experimental results in the literature to assess the utility of this method.