On the microscopic definitions of the dislocation density tensor

Abstract

The dislocation density tensor at the macroscale may be obtained by using two seemingly disparate definitions given by Nye and Arsenlis and Parks. Nye’s definition depends on counting the dislocations crossing a Burgers circuit of infinitesimal area at the macroscale, where as Arsenlis and Parks’s definition is defined as an integrated property of dislocations within an infinitesimal volume. In this paper, it is shown that Arsenlis and Parks’ and Nye’s definitions for the dislocation density tensor are equivalent when conditions on the length scales of the spacing and curvature of the dislocation lines are obeyed. It is also shown that the definition by Arsenlis and Parks, which can be easily employed in microscopic dislocation dynamics simulations, follows the fundamental extensive property of the Burgers vector, namely, the total Burgers vector of a Burgers circuit is the sum of Burgers vectors of individual dislocation lines intersecting the circuit.