A probabilistic derivation of the effect of grain size on the dislocation free path in a deforming polycrystal

The mean free path length for dislocation movement during deformation of a polycrystal determines the storage term during strain hardening. The classical Mecking–Estrin (ME) model postulates additivity of the effects of substructure and grain boundaries. The present work considers the probability distribution of obstacles along the path and uses the laws of probability to derive an expression of which the ME model is a good approximation. The physical implications of the approach and the extension of this method to particle hardening are briefly discussed.