Calculating the grain size effect during strain hardening through a probabilistic analysis of the mean slip distance in polycrystals

Abstract

Grain refinement is an important mechanism to produce stronger alloys. Strain hardening is an essential phenomenon in metal forming processes. The interaction between grain size and strain hardening is evident: a decrease in grain size $\left(d_g\right)$causes an increase in ultimate tensile strength but a decrease in uniform elongation. The Kocks-Mecking (KM) model for strain hardening is based on the relationship between shear strain and the path length for dislocation slip. It provides good general estimates for stress-strain curves, and empirical modifications have been made to include $d_g$. Here, the empirical approach is substituted by theoretical probability calculations, accounting for the fact that the grain size imposes a bound on the mean slip distance, while strain compatibility defines a relationship between grain boundary-dislocation interaction and bulk storage and annihilation. The resulting differential only uses the two parameters inherent to KM. Fitting to published tensile curves for Al, Cu, and Ni produces excellent results. The fitting parameters allow to predict the tensile strength as a function of $d_g$to good approximation, for $d_g>1\mu m$. Below this limit, fundamental changes in dislocation statistics impose the activation of grain boundary dislocation sources and may induce dislocation density gradients, which seem to determine the flow stress in the sub-$\mu m$ range.