Continuum modeling of dislocation interactions: why discreteness matters?

Continuum frameworks of dislocation-based plasticity theories are gaining prominence in the research community. In these theories, the underlying discrete lattice defects are represented by an averaged continuous description of a signed dislocation density. The long-range stress fields are accurately characterized but the short-range interactions are modeled phenomenologically. In this paper, we demonstrate by a rigorous analysis that short-range interactions resulting from certain aspects of the underlying discreteness cannot be neglected. An idealized problem of dislocation pile-ups against a hard obstacle is used to illustrate this observation. It is also demonstrated that the modeling of short-range interactions by a local gradient of dislocation distribution has limitations. It is realized that even though the stress contribution for distant dislocations is relatively small, it is the accumulation of these stress contributions from numerous such dislocations which culminates in substantial contributions. It would be inaccurate to neglect these effects. Our benchmark problem can be used for calibration of current and future theories of plasticity that attempt to accurately model short-range interactions.