Impact of the grain size distribution on the yield stress of heterogeneous materials

When the mean grain size is larger than ∼100 nm, it is well accepted for different heterogeneous materials that the macroscopic yield stress follows the so-called Hall–Petch relation. However, in this classic formalism, only the mean grain size is considered in a semi-phenomenological way and the fact that the grains form a population of stochastic nature with different sizes and shapes is not stated. Moreover, an efficient homogenization procedure leading to the aforementioned grain size dependent behaviour from the individual properties of grains has not yet been reported in the literature mainly due to a lack of statistical description. In this paper, a recently developed self-consistent scheme making use of the “translated fields” technique for elastic–viscoplastic materials is used as micro–macro scale transition. The representative volume element is composed of grains supposed to be spherical and randomly distributed with a grain size distribution following a log-normal statistical function. The viscoplastic strain rate of the grains depends on their individual size. Intragranular plastic anisotropy and strain hardening are not considered in order to focus this work on grain size heterogeneities only. Numerical results with unimodal log-normal distributions firstly display that the overall yield stress depends not only on the mean grain size but also on the dispersion of the distribution. A decrease of the yield stress with an increase of the dispersion occurs and is more important when the mean grain size is on the order of the μm. Secondly, prediction of the evolution of the internal structure indicates an increase of the internal stresses when the dispersion is increased. Lastly, numerical results concerning bimodal grain size distributions, considered as mixtures of unimodal log-normal distributions, are discussed.