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Grain size and Hall–Petch

Metals consist of many small crystals (grains) with different orientations, separated by grain boundaries. Grain boundaries hinder dislocation motion, so a fine-grained material is stronger than a coarse-grained one of the same composition. This is the Hall–Petch relationship.

\sigma_y = \sigma_0 + \frac{k}{\sqrt{d}}

Hall–Petch: yield strength vs. grain size

Symbols

$\sigma_y$	yield strength	MPa
$d$	average grain size	µm
$\sigma_0,\ k$	material constants

Example

Reducing the grain size $d$ increases $1/\sqrt d$ and thus the yield strength $\sigma_y$ .

A fine-grained structure gives higher strength without changing the alloy's composition.

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Part of Materials Technology: Structure.