On the basis of dislocation pile-up theory, and the Ashby model for polycrystalline deformation, an expression was derived for flow stresses. This showed that the flow stress depended upon the densities of statistically stored dislocations and geometrically necessary dislocations. The contribution of the latter to the flow stress was related to the fraction of grain boundary region. This expression did not predict a precise Hall-Petch relationship. If the statistically stored dislocation density predominated, the expression reduced to a linear function. If the geometrically necessary dislocation density predominated, the expression reduced to the Hall-Petch relationship. The results were used to calculate the Hall-Petch curves of 3 typical polycrystalline materials: Al, Cu and brass. Very good agreement was found between calculation and experiment. By incorporating the experimental data, the contributions of statistically stored dislocation densities and geometrically necessary dislocation densities to the flow stress were analyzed.

A Dislocation Density Approximation for the Flow-Stress versus Grain-Size Relationship of Polycrystals. Z.Jiang, J.Lian, B.Baudelet: Acta Metallurgica et Materialia, 1995, 43[9], 3349-60