The line bundle regime and the scale-dependence of continuum dislocation dynamics

TL;DR

This work develops a scale-aware framework for continuum dislocation dynamics by formalizing local orientation distributions $g^{(L)}$ and global fluctuation distributions $f^{(L)}$ to study the line-bundle vs. higher-order descriptions across coarse-graining lengths $L$. It introduces a novel line bundle closure and juxtaposes it with maximum entropy closure, testing both against discrete dislocation data to assess their validity. The results show that line bundle closure is accurate up to about half the mean dislocation spacing, while maximum entropy closure consistently underpredicts key higher-order moments; the global fluctuations are well described by a wrapped Cauchy form with a pronounced zero-fluctuation spike. These findings support a hybrid modeling strategy that uses vector-density closures at small scales and higher-order closures at larger scales, with implications for evolution equations and orientation-dependent dislocation reactions at intermediate mesoscales.

Abstract

Continuum dislocation dynamics (CDD) has become the state-of-the-art theoretical approach for mesoscale dislocation plasticity of metals. Within this approach, there are multiple CDD theories that can all be derived from the principles of statistical mechanics. In these theories density-based measures are used to represent dislocation lines. Establishing these density measures requires some level of coarse graining with the result of losing track of some parts of the dislocation population due to cancellation in the tangent vectors of unaligned dislocations. The leading CDD theories either treat dislocations as nearly parallel or distributed locally over orientation space. The difference between these theories is a matter of the spatial resolution at which the definition of the relevant dislocation density field holds: for fine resolutions, single dislocations are resolved and there is no cancellation; for coarse resolutions, whole dislocation loops could contribute at a single point and there is complete cancellation. In the current work, a formulation of the resolution-dependent transition between these limits is presented in terms of the statistics of dislocation line orientation fluctuations about a local average line direction. From this formulation, a study of the orientation fluctuation behavior in intermediate resolution regimes is conducted. Two possible closure equations for truncating the moment sequence of the fluctuation distributions relating the two theories mentioned above are evaluated from data, the newly introduced line bundle closure and the previous standard maximum entropy closure relations. The line bundle closure relation is shown to be accurate for coarse-graining lengths up to half the dislocation spacing and the maximum entropy closure is found to poorly agree with the data at all coarse-graining lengths.

Figures (6)

• Figure 1: Typical dislocation densities and mean dislocation spacing from the discrete dislocation configurations. Because of the asymmetry of dislocation multiplication and thus dislocation density on active and inactive slip systems, the calculations of the densities and corresponding average spacings are shown for the cases where these calculations are restricted to active and inactive slip systems as well as the unrestricted average. Error bars represent one standard deviation (over the set of 45 simulations).
• Figure 2: Typical global orientation fluctuation distributions from the dataset. Both figures show a selection of the orientation fluctuation distribution functions at various coarse-graining lengths. Fig. 2a shows the sharp peak at zero fluctuation angle coming from single-crossing events, and Fig. 2b shows the same functions with the zero fluctuation point shielded from view. Cauchy fits are shown on the latter figure to demonstrate the shape of the distributions at non-zero fluctuation angle.
• Figure 3: Trend of distributions with coarse-graining length. The two parameters chosen to describe the shape of the distributions are shown as coarse-graining length is increased: Dirac mass weight at zero fluctuation--the prevalence of zero crossing events--is shown in (a) while the Cauchy parameter--the half-width-at-half-maximum--is shown in (b).
• Figure 4: Characteristic sequence components. The first- (a), second- (b), and third-order (c) characteristic components are plotted against coarse-graining length.
• Figure 5: Fractional closure errors associated with the line bundle and maximum entropy closure forms. These show the fractional error in closing the orientation description at first order (approximating $\beta_2$ as a function of $\beta_1$) and at second order (approximating $\beta_3$ as a function of $\beta_1,\beta_2$) as a function of course graining length.