Softening in Nanocrystalline Metals: Modeling of Grain Growth Induced Creep and Relaxation

Abstract

A variational multiscale model is presented for grain growth in face-centered cubic nanocrystalline (nc) metals. In particular, grain-growth-induced stress softening and the resulting relaxation and creep phenomena are addressed. The behavior of the polycrystal is described by a conventional Taylor-type averaging scheme in which the grains are treated as two-phase composites consisting of a grain interior phase and a grain boundary affected zone. Furthermore, a grain growth law that captures the experimentally observed characteristics of the grain coarsening phenomena is proposed. The model is shown to provide a good description of the experimentally observed grain-growth-induced relaxation in nc-copper. Description of Grain Growth ModelFollowing 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 partTreating each 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 grain core region, W gi and W gb denote the free energies of the grain interior and boundary 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 ξ = (d − d gb ) 3 /d 3 . Following [2], the flow rule for the GB phase is assumed to bė