Inverse Hall-Petch Effect (2D): Grain-Boundary Area Model

Shrinking a metal's grains normally makes it stronger — dislocations pile up against more grain boundaries per unit area of cross-section, the classical Hall-Petch relation. But push the grain size below a few nanometres and the trend reverses: so much of the cross-section is now disordered grain-boundary rim that boundary sliding takes over from dislocation motion, and strength falls again. This 2D companion to the 3D nanograin simulator renders a real square lattice of grains sized directly from the slider, computes the grain-boundary area fraction from actual 2D geometry (exponent 2, not the 3D model's volume exponent 3), derives the Hall-Petch coefficient from a discrete Eshelby-Frank-Nabarro dislocation pile-up equilibrium rather than a fitted constant, and animates the two competing deformation mechanisms — dislocation glide inside large grains at their computed pile-up spacing, grain-boundary shear in the nanocrystalline regime — so the crossover is visible, not just numeric.