A1. The Hall–Petch relationship states that yield strength increases with decreasing grain size: 𝜎 𝑦 = 𝜎 0 + 𝑘 𝑑 − 1 / 2 σ y ​ =σ 0 ​ +kd −1/2 where 𝑑 d = grain size, 𝑘 k = strengthening constant. Smaller grains provide more grain boundaries, which act as barriers to dislocation motion. However, below ~10–20 nm, the relationship breaks down because grain boundaries themselves become active sites for deformation (grain boundary sliding), reducing strength.