Abstract
Coarse-grained descriptions of dislocation motion in crystalline metals inherently represent a loss of information regarding dislocation-dislocation interactions. In the present work, we consider a coarse-graining framework capable of re-capturing these interactions by means of the dislocation-dislocation correlation functions. The framework depends on a convolution length to define slip-system-specific dislocation densities. Following a statistical definition of this coarse-graining process, we define a spatial correlation function which will allow the arrangement of the discrete line system at two points—and thus the strength of their interactions at short range—to be recaptured into a mean field description of dislocation dynamics. Through a statistical homogeneity argument, we present a method of evaluating this correlation function from discrete dislocation dynamics simulations. Finally, results of this evaluation are shown in the form of the correlation of dislocation densities on the same slip-system. These correlation functions are seen to depend weakly on plastic strain, and in turn, the dislocation density, but are seen to depend strongly on the convolution length. Implications of these correlation functions in regard to continuum dislocation dynamics as well as future directions of investigation are also discussed.
Highlights
The dislocation-dislocation correlations represent an important link between the continuum and discrete descriptions of the dislocation dynamics
In the present section we present a scheme by which we may evaluate the expression for the correlation function seen in Eq (24)
We would like to discuss some of the preliminary implications of the findings with respect to incorporation into continuum dislocation dynamics schemes based on a vector density approach
Summary
The dislocation-dislocation correlations represent an important link between the continuum and discrete descriptions of the dislocation dynamics. Many views on what this correlation represents, how to evaluate it, and what kinetically-relevant information it contains have been presented in recent years. The present work puts forward a clear and robust definition of the dislocation-dislocation correlation functions and presents a methodology for their computation using simulations of discrete dislocation systems. One may think of correlation functions as a certain error estimate on mean field representations of discrete systems (cf self-consistent field theories, Hartree-type theories of electronic systems (Hartree 1928)). In our case, the dislocation-dislocation correlation functions represent an error estimate on mean dislocation density field theories (El-Azab and Po 2018). To even define a correlation, we must first have some idea of what we are referring to as our mean dislocation density field. One which considers a single-valued vector density of dislocations at every point in space (Xia 2016)
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