Which equation expresses the Hall-Petch relation for yield strength, and what does the parameter d represent?

• A. σ_y = σ_0 + k_y d^(-1/2); d is the average grain diameter.
• B. σ_y = σ_0 + k_y d; d is diffusion distance.
• C. σ_y = σ_0 d^(-1/2) + k_y; d is lattice parameter.
• D. σ_y = σ_0 + k_y √d; d is grain boundary area.

Answer: (A)

Grain size controls how easily dislocations move, because grain boundaries act as barriers to dislocation motion. The Hall-Petch relation adds a term to the intrinsic lattice resistance that scales with the inverse square root of the average grain diameter: σ_y = σ_0 + k_y d^(-1/2). Here σ_0 is the baseline stress needed to start plastic flow in a single crystal, and k_y is a material-dependent constant describing how strongly grain boundaries impede dislocations. The parameter d represents the average grain diameter, i.e., the mean size of the grains in the polycrystal.

The d^(-1/2) dependence captures why smaller grains strengthen the material: increasing the number of grain boundaries raises the obstacles dislocations must overcome, and the strengthening effect grows as grain size decreases. If d were anything other than the average grain diameter or if the relation used a different dependence on d, the equation would not match the observed trend that yield strength rises as grains get finer.