Dislocation correlations and the continuum dynamics of the weak line bundle ensemble

Progress toward a first-principles theory of plasticity and work hardening is currently impeded by an insufficient description of dislocation kinetics, that is, of the dynamic effect of driving forces in a given dislocation theory. Current continuum theories of dislocation kinetics are often incapable of treating the short-range interaction of dislocations. This work presents a kinetic theory of continuum dislocation dynamics in a vector density framework which takes into account the short-range interactions by means of suitably defined correlation functions. The weak line bundle ensemble of dislocations is defined, whereby the treatment of dislocations by a vector density is justified. It is then found by direct averaging of the dislocation transport equation that additional driving forces arise which are dependent on the dislocation correlations. A combination of spatial coarse graining and statistical averaging of discrete dislocation systems is then used to evaluate the various classes of tensorial dislocation correlations which arise in the line bundle kinetic theory. A chiral classification of slip-system interactions in face-centered-cubic crystals is introduced in order to define proper and improper rotations by which correlation functions corresponding to six interaction classifications can be evaluated. The full set of these six dislocation correlations are evaluated from discrete data. Only the self-correlations (for densities of like slip system) are found to be highly anisotropic. All six classes of correlation functions are found to be of moderate range, decaying within two to four times the coarse-graining distance. The correlations corresponding to the coplanar interactions are found to be negligible. Implications of the evaluated correlations for the implementation of vector density continuum dislocation dynamics are discussed, especially in terms of an additional correlation component of the driving force and a gesture toward a coarse-grained dislocation mobility.

Figure 1

The slip-system graph structure. Nodes represent slip systems; nodes connected by an undirected edge share a Burgers vector, while nodes connected by a directed edge share a slip-plane normal vector. The direction of the edge between two nodes $1\to 2$ is chosen such that $b^{\left[1\right]}\wedge b^{\left[2\right]}=n^{\left[1\right]}=n^{\left[2\right]}$. The Thompson tetrahedron is shown faintly superimposed on the graph for reference.

Figure 2

Classes of interactions on the Thompson tetrahedron. Different classes of paths along the graph denote different slip-system interactions. The first slip system of a given interaction is marked by a diamond, the second by a square. Shown are (a) self-, (b) cross-slip, (c) coplanar, (d) glissile junction, (e) Lomer-type, and (f) Hirth-type interactions. Right- and left-handed interactions are marked.

Figure 3

Scatter plots of discrete and continuum density product data. (a) Volumetric representation of the edge-edge self-correlation function given for reference of separation space locations. The slip plane (hexagon), Burgers direction (blue line), and coarse-graining volume (cube) are shown for reference. In this figure, blue regions are anticorrelated regions ($d<1$), and red regions are positively correlated ($d>1$). (b)–(f) Scatter plots of the underlying discrete and continuum density products, and the linear relationship which the correlation function describes. In (a), blue regions correspond to anticorrelated regions ($g<1$) and red to correlated regions ($g>1)$), while transparent regions correspond to uncorrelated regions ($g=1$). Orienting features of (a) include the Burgers vector direction (blue line), slip plane (hexagon), and the coarse-graining volume (cube). The points in separation space examined in (b)–(f) are shown as well. In (b)–(f), the dotted line represents the relation that would obtain in the perfectly uncorrelated case (a slope of unity). The solid lines show the principal components of the covariance matrix. The slope of the larger component is taken as the correlation value and reported in each plot. Points in (b)–(f) are colored according to the effective polarization [Eq. (65)], with blue denoting low polarization ($\ll 1$) and red denoting high polarization ($\approx 1$) according to the color bar shown on the right.

Figure 4

Various fits for the self-correlation function in the slip plane. Self-correlation functions are shown for separation vectors in the slip plane, where $\mathrm{\Delta }{r}_b$ and $\mathrm{\Delta }{r}_a$ are the separation in the Burgers vector direction and the line direction of an edge dislocation, respectively. The three methods of fitting the correlation are compared, namely, ordinary least squares with continuum (OLSC) and discrete (OLSD) residuals, and the principal component (PC, cf. Fig. 3). These are shown for the screw-screw, screw-edge, and edge-screw components of the self-correlation function. The morphology of the correlation function is unaffected by the choice of fitting method, although the magnitude of the respective maxima differs slightly between the three methods. The faint hexagons represent the boundaries of the fundamental and coarse-graining volumes (cube of side length $2L^{*}$ and $2L$, respectively).

Figure 5

Radial plots for all correlation functions. (a) Cylindrical correlation function [Eq. (67)] for the self-correlation. (b)–(g) Spherical correlation function [Eq. (66)] from the correlation functions corresponding to each interaction class.

Figure 6

Volumetric plots of all correlation functions. Volumetric isosurfaces of the correlation functions for all interaction classes are colored according to the value of the correlation function. The color scale, denoting the value of the correlation, is constant across all plots. The transparency scale of each plot ranges from the maximum deviation from unit correlation (opaque) to uncorrelated (unit correlation, transparent). For spatial reference, several relevant spatial features are shown. Cubes show the coarse-graining volume (cube of side length $2L$). Hexagons show the intersection of the slip planes of the respective slip system with the cube of side length $6L$. Dotted line shows the intersection of the two slip planes if applicable. Blue and orange lines show the Burgers vector of the first and second slip systems, respectively.

Figure 7

Dependencies of screw-screw self-correlation on various calculation parameters. (a) Self-correlations calculated with various coarse-graining lengths and a fixed fundamental length. (b) Self-correlations calculated with various fundamental lengths and a fixed coarse-graining length. (c) Self-correlations calculated from configurations with various values of plastic strain. Note, all separation distances scaled by the coarse-graining length.

Figure 8

Fluctuation variances of the self-correlation calculation. Displayed are the fluctuation variance field for the self-correlation calculation as a fraction of the total variance in the product density data at each point [Eq. (64)] for separation vectors in the slip plane.