Dislocation Pile-Up

In crystal plasticity, dislocation pile-ups form when mobile dislocations are blocked by grain boundaries or precipitates. The Peach-Köhler force F = τb drives each dislocation forward while mutual elastic repulsion (1/r force between edge dislocations) balances. The equilibrium positions are the zeros of Chebyshev polynomials. The stress at the tip scales as τ_tip = N·τ_app, explaining the Hall-Petch relation: σ_yield ~ d^{-1/2} (grain size d controls N ~ L/d).