The yield stress of a steel follows the Hall–Petch relation σy = σ₀ + k/√d with σ₀ = 70 MPa and k = 0.74 MPa·√m. Which value is closest to the yield stress for an average grain size d = 25 µm?

Answer: C. About 218 MPa

Converting the grain size to meters gives d = 25×10⁻⁶ m and √d = 5.0×10⁻³ √m, so k/√d = 0.74/(5.0×10⁻³) = 148 MPa. Hence σy = 70 + 148 = 218 MPa. About 75 MPa results from leaving the grain size in mm as d = 0.025, so the units of k (MPa·√m) and of the grain size do not match. About 148 MPa is only the grain boundary strengthening contribution, forgetting to add σ₀. Grain refinement is one of the few strengthening methods that raises strength and toughness at the same time, and it is exploited in processes such as controlled rolling of steel. When the grain size shrinks to the nanometer scale, the relation can break down.