Crystal Plasticity — Dislocation Pile-Up

Edge dislocations driven by applied shear stress pile up against a grain boundary — the Hall-Petch strengthening mechanism visualized.

Hall-Petch relation: σ_y = σ₀ + k/√d. A pile-up of n dislocations concentrates stress at the grain boundary by factor n. When this stress exceeds the boundary activation stress, slip transfers to the next grain. Smaller grains → shorter pile-ups → harder to activate the boundary.