The Hall-Petch relationship is mathematically expressed as:
σ_y = σ_0 + k_y * d^(-1/2)
Where:
σ_y is the yield strength. σ_0 is the friction stress. k_y is the Hall-Petch coefficient. d is the average grain diameter.
As the grain size (d) decreases, the term d^(-1/2) increases, leading to a higher yield strength (σ_y). This is particularly significant for nanocrystalline materials where d is extremely small.
