Theoretical and computational comparison of models for dislocation dissociation and stacking fault/core formation in fcc crystals

Abstract

The purpose of the current work is the theoretical and computational comparison of selected models for the energetics of dislocation dissociation resulting in stacking fault and partial dislocation (core) formation in fcc crystals as based on the (generalized) Peierls–Nabarro (GPN: e.g., Xiang et al., 2008; Shen et al., 2014), and phase-field (PF: e.g., Shen and Wang, 2004; Hunter et al., 2011, 2013; Mianroodi and Svendsen, 2015), methodologies (e.g., Wang and Li, 2010). More specifically, in the current work, the GPN-based model of Xiang et al. (2008) is compared theoretically with the PF-based models of Shen and Wang (2004), Hunter et al. (2011, 2013), and Mianroodi and Svendsen (2015). This is carried out here with the help of a unified formulation for these models via a generalization of the approach of Cahn and Hilliard (1958) to mechanics. Differences among these include the model forms for the free energy density ψela of the lattice and the free energy density ψsli associated with dislocation slip. In the PF-based models, for example, ψela is formulated with respect to the residual distortion HR due to dislocation slip (e.g., Khachaturyan, 1983; Mura, 1987), and with respect to the dislocation tensor curlHR in the GPN model (e.g., Xiang et al., 2008). As shown here, both model forms for ψela are in fact mathematically equal and so physically equivalent. On the other hand, model forms for ψsli differ in the assumed dependence on the phase or disregistry fields ϕ, whose spatial variation represents the transition from unslipped to slipped regions in the crystal. In particular, Xiang et al. (2008) and Hunter et al. (2011, 2013) work with ψsli(ϕ). On the other hand, Shen and Wang (2004) and Mianroodi and Svendsen (2015) employ ψsli(ϕ,∇ϕ). To investigate the consequences of these differences for the modeling of the dislocation core, dissociation, and stacking fault formation, predictions from the models of Hunter et al. (2011, 2013) and Mianroodi and Svendsen (2015) are compared with results from molecular statics (MS) for the deformation field of dissociated edge and screw dipoles in Al and Au. Particularly notable is the agreement of the MS and PF strain field results for the case of perfect screw dissociation which, in contrast to the edge case, are characterized by asymmetric displacement and strain fields. The degree of this asymmetry is apparently related to the corresponding anisotropy ratio. As well, comparison of MS and PF disregistry fields implies that the gradient dependence of ψsli results in a broadening of the (otherwise too narrow) disregistry profile to the form predicted by MS.