A continuum model describing the reverse grain-size dependence of the strength of nanocrystalline metals

The breakdown of the Hall–Petch relation in the grain-size strength dependence of nanocrystalline metals has been rationalized by the activation of deformation mechanisms taking place at the grain boundary which compete with crystal plasticity and become dominant when grain sizes are sufficiently small. In this work, we develop a continuum description of the effective response of nanocrystalline metals. The model is based on a finite element formulation of the continuum three-dimensional problem describing the deformation of polycrystal grains explicitly and on the consideration of grain boundaries as surfaces of discontinuity with finite thickness embedded in the continuum. A phenomenological model formulated within the framework of variational constitutive updates is proposed to describe the operative grain-boundary deformation mechanisms of sliding and opening accommodation. The model parameters are fitted to atomistic results for copper at very high loading rate. Tensile test simulations using this model reproduce the inverse grain-size dependency of the macroscopic yield stress predicted by atomistic simulations. In particular, the model predicts that the grain-size dependency of the yield stress shows a linear relation to the inverse square root of the grain size, as in the traditional Hall–Petch law, but with a negative coefficient. The results are in good agreement with the atomistic simulations and also with quasi-static experimental tests. The ability to model the effective response of nanocrystals without the need of explicitly accounting for each individual atom opens the way for the analysis of nanocrystalline materials in sample sizes and under strain rates of technological significance.