PPT Slide
In a continuous medium, the stresses on the center would build up and become infinite in the absence of a hole; in real dislocations the crystalline lattice is periodic, and this does not occur.
In mechanics terminology, this is called singularity. A “singularity” is a spike, or a single event. For instance, the Kilimanjaro is a singularity in the African plans.
Therefore, we “drill out” the central core, which is a way of reconciling the continuous-medium hypothesis with the periodic nature of the structure.