Multiscale Modeling of Nanocrystalline Materials: A Variational Approach

Abstract

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior (GI) phase and a grain boundary (GB) phase. A rate-independent isotropic porous plasticity model is employed to describe the GB phase, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the GI phase. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization. It is shown that the proposed model is able to capture the inverse Hall-Petch effect. Description of Multiscale Model for Nanocrystalline MetalsIn the sequel the proposed multiscale model is briefly outlined. We refer to [3] for further details. Following the classical multiplicative decomposition framework, the deformation gradient F = F e F p is assumed to decompose into an elastic part F e and a plastic part F p . Treating a particular grain as a composite material composed of a grain interior (GI) phase and a grain boundary (GB) phase, the free energy function can be expressed as a simple volume average W = ξW gi + (1 − ξ)W gb , where ξ is the volume fraction of the GI phase, W gi and W gb denote the free energies of the GI and GB phases, respectively. The average first Piola-Kirchhoff stress P reads similarly P = ξP gi + (1 − ξ)P gb . The volume average stress P, the GI stress P gi and the GB stress P gb are computed from Coleman's relations by evaluating the partial derivatives of corresponding energies with respect to F. Assuming cubical grains and a constant thickness d gb of the GB phase, the volume fraction is evaluated as Following [3], the flow rule for the GB phase readṡ