Micro–macro modelling of the effects of the grain size distribution on the plastic flow stress of heterogeneous materials

When the mean grain size of polycrystalline materials is larger than ∼100 nm, it is commonly accepted for metals, intermetallics or ceramics that the plastic flow stress scales linearly with the inverse square root of the mean grain size (the so-called Hall–Petch behaviour). 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. Here, a new self-consistent model making use of the “translated fields” technique for elastic–viscoplastic materials is developed as micro–macro scale transition scheme, and, the aggregate is composed of spherical randomly distributed grains with a grain size distribution following a log-normal statistical function. The constitutive behaviour of each grain is described by a partitioned strain rate into an elastic part and a viscoplastic part. The viscoplastic strain rate is described by an isotropic power law including the grain diameter through the reference stress. Numerical results firstly display that the plastic flow stress of the material depends on both the mean grain size and the grain size dispersion of the distribution. Besides, the role of the dispersion is more important when the mean grain size is on the order of the μm and the trend is a decrease of the flow stress with an increase of the dispersion. Secondly, predictions of second order internal stresses within the material indicate an increase in the internal stresses when grain size dispersion is increased, so that the plastic flow stress of the material depends on a competition between the respective distributions of internal stresses and individual flow stresses of the grains.