The relation σy = σ₀ + k/√d in which the yield stress increases in inverse proportion to the square root of the grain size d. Grain boundaries obstruct dislocation motion, so finer grains give higher strength.
Japanese name: ホール・ペッチの関係
Hall–Petch relationship appears in 3 practice questions across 2 topics, including Heat treatment and metallic materials, High-temperature materials and creep.
The relation in which yield stress increases in inverse proportion to the square root of grain size was shown for steel by Hall in 1951 and by Petch in 1953. Crystal orientation changes at a grain boundary, so dislocations gliding toward it stop in front of it and form a line (a dislocation pile-up). A large stress concentrates at the head of the pile-up, and when it activates dislocation sources in the neighboring grain, yielding spreads. The larger the grain, the longer the pile-up and the greater the stress concentration, so yielding occurs at a lower stress; this picture leads to the d^(−1/2) dependence.
σ₀ is the friction stress without grain boundaries, and k is a coefficient expressing how effectively grain boundaries strengthen, in units of MPa·√m. k differs greatly between materials: it is large for BCC iron and steel and small for FCC aluminum and copper, so grain refinement is especially effective in steel. The grain boundary contribution is proportional to 1/√d, so reducing the grain size to 1/100, for example from 100 µm to 1 µm, increases it tenfold. In calculations, remember to convert grain size to meters to match the units of k.
When the grain size falls below about 10–20 nm, dislocation pile-ups can no longer form within grains and other deformation such as grain boundary sliding takes over, so strength levels off or even decreases (the inverse Hall–Petch effect). At high temperature, finer grains also make diffusion creep and grain boundary sliding easier, so refinement becomes a disadvantage. Relations of the same form are used for hardness, the block size of martensite and the lamellar spacing of pearlite, and grain refinement also lowers the ductile-to-brittle transition temperature.
Grain refinement is one of the few strengthening methods that raise strength and toughness together, and because it does not require more alloying elements, it does not impair weldability. In steel, TMCP, which combines controlled rolling with accelerated cooling, makes ferrite grains a few µm to about 10 µm in size to produce strong, tough heavy plate. In aluminum castings, Al–Ti–B grain refiners are added to increase the number of solidification nuclei. Grain size is evaluated with JIS or ASTM grain size numbers or with the intercept method, which counts the grain boundaries crossed by a line.
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.
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.
Grain boundaries obstruct dislocation motion and raise the yield stress, while a cleavage crack must change direction at each grain boundary and propagates with more difficulty, increasing the energy needed for fracture. As a result, the ductile-to-brittle transition temperature also falls.
MPa·√m (MPa·m^(1/2)). Grain size d is expressed in meters so that k/√d is a stress. Substituting the grain size in µm or mm leads to errors of orders of magnitude.
No. Below a few tens of nanometers the gain in strength slows, and below about 10–20 nm strength can level off or even decrease, depending on the material and how it is made. At high temperature, finer grains also creep more readily, so a grain size suited to the application must be chosen.
Diffusion creep Columnar grains Young's modulus Hardenability Martensite Precipitation hardening Tempering