Modelling the strengthening effect of grain boundaries (Hall–Petch effect) in metallic polycrystals in a physically consistent way, and without invoking arbitrary length scales, was a long-standing, unsolved problem. A two-scale method to treat the interactions of large numbers of dislocations with grain boundaries was developed, implemented, and tested. At the first scale, a standard grain-scale simulation based upon a finite element formulation made use of recently proposed dislocation-density-based single-crystal constitutive equations to determine local stresses, strains, and slip magnitudes. At the second scale, a novel meso-scale simulation redistributes the mobile part of the dislocation density within grains consistent with the plastic strain, computes the associated inter-dislocation back stress, and enforces local slip transmission criteria at grain boundaries.

Simulation of Polycrystal Deformation with Grain and Grain Boundary Effects. H.Lim, M.G.Lee, J.H.Kim, B.L.Adams, R.H.Wagoner: International Journal of Plasticity, 2011, 27[9], 1328-54