Hall-Petch Calculator
Mild steel with 25 µm grains has a yield strength of about 218 MPa: 148 MPa of it from grain boundaries.
Mild steel about 0.74 MPa·m^½; aluminium about 0.07.
218 MPa
- Grain-boundary strengthening k_y d^(−½)
- 148 MPa
- Yield strength at half the grain size
- 279.3036072 MPa
Step-by-step solution
- σ_y = σ₀ + k_y d^(−½) = 70 + 0.74 ÷ √0.000025
- σ_y = 218 MPa
Hall-Petch Formula
σ_y = σ₀ + k_y / √d
σ₀ lattice friction stress, k_y Hall–Petch coefficient, d grain diameter
Worked Example
Mild steel (σ₀ = 70 MPa, k_y = 0.74 MPa·m^½) with 25 µm grains:
- σ_y = σ₀ + k_y d^(−½) = 70 + 0.74 ÷ √0.000025
- σ_y = 218 MPa
Yield strength at half the grain size: 279.3036072 MPa
GATE tip
Given yield strengths at two grain sizes, solve two linear equations in d^(−½) for σ₀ and k_y, then predict the third grain size.
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Frequently asked questions
Why do smaller grains strengthen?
Grain boundaries block dislocations; smaller grains mean shorter pile-ups and less stress concentration at the next grain, so more applied stress is needed.
Does Hall–Petch always hold?
It breaks down below roughly 10–20 nm, where grain-boundary sliding takes over (the inverse Hall–Petch effect).
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Hall-Petch Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/hall-petch-calculator/
IEEE
Scientific Converters, "Hall-Petch Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/hall-petch-calculator/