Discussion about Scaling Effects in Metals with a Continuum Mechanics Approach

Abstract

The mechanical behavior of metals is well known to display scaling effects (e.g. Hall-Petch law). The scale that is usually referred to is the grain size, however, a new scale, larger than the grain size, is expected to also play a major role on the deformation mechanisms of a polycrystal. Actually, in elasticity, a load percolation network forms through the polycrystal. This network has its own scale which can be much larger than the grain size. The load percolation effect, through the polycrystal, is inherited from the variability of the mechanical response of individual grains with respect to a given load direction. This variability and, consequently, the intensity of the percolation network, are all the higher that the elastic anisotropy of the grains is strong. As a result, before the yield point, the stress distribution within the polycrystal is heterogeneous (which is well-known) and self-organized at the scale of this load percolation network (which is a new result). After the yield point, if the material displays plastic softening, this self-organization controls the development of plasticity and lead to early plastic strain localization. Now the propensity for a metal, to display hardening or softening after the yield point, was found by numerous authors to be size dependent. For instance, metallic materials exhibit a general trend for softening during nanoindentation tests and bulk nanocrystalline materials, for example, nanosteel for tire cords, often display a large softening effect at the onset of plasticity. That trend for softening is explained by the theory of geometrically necessary dislocations and predicted by atomic scale simulations for a few grains. However, the analysis of the collective behavior of a large number of grains is not yet possible with such approaches. A continuum approach would therefore be useful in order to identify the effects of the elastic percolation network on plastic flow but, classical constitutive models for continuum mechanics are unable to predict any grain size effect. The missing ingredients that should be included, in order to close the gap between continuum mechanics and the dislocation theory, are reviewed and discussed briefly. Strain gradient plasticity models are expected to be suitable to solve this type of problems.