Mesoscale modelling of dislocation pile-up based on discrete interactions

Conventional crystal plasticity formulations fail to correctly predict dislocation pile-ups against e.g. hard particles because they cannot capture the effect of short-range dislocation interactions. We illustrate this limitation by studying an idealised pile-up of infinite edge dislocation walls. In the limit where all discreteness is averaged out, as is done in crystal plasticity, the pile-up collapses into a continuous tilt wall and its finite width is lost. As a consequence, size effects such as the Hall-Petch effect can no longer be captured. Based on an analysis of the short-range interactions in the same idealised pile-up, we derive a back-stress which can be used to repair the conventional theory so that it predicts pile-ups of a finite width.