Continuum modeling of the reverse Hall–Petch effect in nanocrystalline metals under uniaxial tension: how many grains in a finite element model?

Abstract

Owing to their very high strength, nanocrystalline metals have been extensively studied over the recent years. The direct Hall–Petch law, empirically proportioning the material strength to the inverse square root of its grain size has been shown to break down below a grain size of the order of tenths of nanometers. This phenomenon has been widely rationalized as a gradual switch from intragrain mediated deformation mechanisms to grain boundary mediated deformation mechanisms. This transition has been observed in many finite element simulations, despite the intrinsic restriction of necessarily limiting the nanocrystalline representative assembly to only a few grains. Such a limitation is generally overlooked, and its influence on an uniaxial tension test – when compared to a complete sample of millions of grains – ignored. We propose here to quantify the approximation done by considering a finite number of grains by means of a simple analytical model based on the early work of Stevens [R.N. Stevens, Philos. Mag. 23 (1971) p. 265]. The finite element approximation is demonstrated to be relatively good, even down to only three grains in width, and a method to “correct” the stress-strain curves of small representative volumes is proposed.