Dislocation-density kinematics: a simple evolution equation for dislocation density involving movement and tilting of di
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Research Letter
Dislocation-density kinematics: a simple evolution equation for dislocation density involving movement and tilting of dislocations A.H.W. Ngan, Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, People’s Republic of China Address all correspondence to A.H.W. Ngan at [email protected] (Received 24 April 2017; accepted 1 August 2017)
Abstract In this paper, a simple evolution equation for dislocation densities moving on a slip plane is proven. This equation gives the time evolution of dislocation density at a general field point on the slip plane, due to the approach of new dislocations and tilting of dislocations already at the field point. This equation is fully consistent with Acharya’s evolution equation and Hochrainer et al.’s “continuous dislocation dynamics” (CDD) theory. However, it is shown that the variable of dislocation curvature in CDD is unnecessary if one considers one-dimensional flux divergence along the dislocation velocity direction.
Introduction In the past two decades, tremendous progress has been made on developing theories for the kinematics of dislocation densities.[1–4] Of special importance is the equation × (a × v) for the evolution of the Nye tensor[5] α proa˙ = −∇ posed initially by Kröner[6] and further developed by Acharya,[1] and a “continuous dislocation dynamics” (CDD) theory proposed by Hochrainer et al.,[4] in which the key concept is the involvement of dislocation curvature as a second field variable in addition to dislocation density itself. However, there has been no documented attempt to reconcile these two important theories. Recently, the present author has also proposed an evolution scheme for dislocation densities based on consideration of the approach and tilting of single dislocations forming the dislocation densities.[7] In this paper, we show that Leung and Ngan’s scheme[7] is fully consistent with Acharya’s equation[1] and the CDD theory.[4] Moreover, we show that the curvature variable in CDD is unnecessary if divergence of dislocation flux is counted one-dimensionally along the dislocation velocity direction, rather than twodimensionally over the slip plane as in the CDD. Furthermore, we also present a clear delineation of how discrete dislocations would form coarse-grained densities, and how the evolution of densities can be derived from the movement of the comprising individual dislocations—this is an aspect which is not clear in Acharya’s equation, which deals only with geometrically necessary dislocations. Before we begin, a note on nomenclature will be useful. First, by a “discrete” dislocation representation, we mean a single dislocation without any description of the evolution of the dislocation core itself. A “discrete” dislocation therefore has its contents represented by a δ-like function that does not
change shape with space or time as the entire dislocation moves, and is therefore different from an “intensive” dislocation representation in which the Burgers vector distribution in the dislocation core
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