The yield stress of a metal was 150 MPa at an average grain size of 100 µm and 250 MPa at 25 µm. Assuming the Hall–Petch relation σy = σ₀ + k/√d, which value is closest to the yield stress at a grain size of 4 µm?

Answer: B. About 550 MPa

Converting the grain sizes to meters and computing 1/√d gives 100 for d = 100 µm, 200 for 25 µm and 500 for 4 µm (all in 1/√m). From the two data points, k = (250 − 150)/(200 − 100) = 1.0 MPa·√m and σ₀ = 150 − 1.0 × 100 = 50 MPa, so at 4 µm, σy = 50 + 1.0 × 500 = 550 MPa. About 950 MPa is the mistake of drawing a straight line assuming the yield stress is proportional to 1/d, and about 278 MPa extrapolates as if the yield stress varied linearly with grain size d. The strengthening is inversely proportional to the square root of the grain size, so each time the grain size is reduced to 1/4, the grain boundary contribution doubles.