A dislocation density approximation for the flow stress—grain size relation of polycrystals

Abstract

Based on the dislocation pile-up theory and the Ashby model of polycrystalline deformation, an expression for flow stress was derived, which shows that the flow stress depends on the densities of statistically stored dislocation ρ s and geometrically necessary dislocation ρ g. The contribution of ρ g to the flow stress is related to the grain boundary region fraction 1 − x which was derived to be the functions of ρ s and ρ g. This expression does not predict a precise Hall—Petch relation. If ρ s dominates, the expression reduces to a linear σ− d −1 relation: if ρ g dominates, the expression reduces to a linear σ− d −1/2 relation, i.e. the Hall—Petch relation. The expression was applied to calculate the σ− d −1/2 curves of three typical polycrystalline materials, aluminium, copper and brass in the literatures and very good agreements between the calculations and experiments were obtained. By incorporating the experimental data, the contributions of ρ s and ρ g to flow stress was analysed and the Hall—Petch parameters were discussed in terms of ρ s and ρ g.