Grain boundaries impede the collective motion of dislocations, so reducing grain size often raises the yield strength of a polycrystalline metal.

A common empirical relation is the Hall-Petch equation

$$\sigma_y=\sigma_0+k_y d^{-1/2},$$

where $d$ is a characteristic grain diameter, $\sigma_0$ is an intercept representing other resistance to slip, and $k_y$ measures the sensitivity to grain size.

Example

Suppose $\sigma_0=100,\mathrm{MPa}$ and $k_y=0.6,\mathrm{MPa\sqrt m}$. For $d=100,\mu\mathrm m=10^{-4},\mathrm m$,

$$\sigma_y=100+0.6(10^{-4})^{-1/2}=160,\mathrm{MPa}.$$

Reducing grain size to $25,\mu\mathrm m$ doubles $d^{-1/2}$ and raises the predicted yield strength to

$$220,\mathrm{MPa}.$$

The relation is not a fundamental law valid at every scale. Very coarse grains, unusual textures, multiphase materials and extremely fine nanocrystalline structures can deviate from a simple Hall-Petch fit.

Grain refinement can therefore strengthen a material without changing its chemical composition, but its effects on toughness, diffusion, creep and phase transformations must also be considered.