The Hall-Petch relation [1,2] linking the yield strength to the mean grain size has been found to match quantitatively the grain size effect on the overall behaviour of polycrystalline steels even though the physical mechanisms responsible for this effect are local. It has been shown recently that the Hall-Petch behaviour continues to be valid until a very fine grain regime (on the order of 100 nm) following Masumura et al. [3] who explored a lot of experimental data for many materials. A first micro-macro modelling considering grain size effect has been elaborated by Weng [4] who considered a Hall-Petch type equation with a single valued grain size at the scale of the slip systems and derived the homogenized behaviour of polycrystailine metals which leaded to the Hall-Petch type behaviour as well. Since the grain size distribution in steels provides heterogeneity, it appears fundamental to get an accurate description of the effect of grain size on the local interactions and behaviours, and also, a relevant mathematical description of the grain size statistics inherent to the processing route (prior working, annealing etc.). Advanced homogenization techniques developed these last decades such that the self-consistent procedure did not focused on the effect of grain size distribution on the behaviour of heterogeneous materials. The objective of the present paper is to study in a systematic statistical way grain size effects on the macroscopic plastic flow stress of steels assuming a given grain size distribution with higher moments than the mean grain size, Especially, the role of the grain size dispersion is underscored.
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