Derivation of strain-gradient plasticity from a generalized Peierls–Nabarro model

We derive strain-gradient plasticity from a nonlocal phase-field model of dislocations in a plane. After scaling, from the nonlocal elastic interaction we derive a continuous energy with linear growth depending on a measure which characterizes the macroscopic dislocation density as well as a nonlocal effective energy representing the far-field interaction between dislocations. Relaxation and formation of microstructures at intermediate scales are automatically incorporated in the limiting procedure based on Gamma-convergence.