In order to understand the kinematic and thermodynamic effects of representing discrete dislocations as continuous distributions in their slip planes, stacked double-ended pile-ups of edge and screw dislocations were considered and their distributions and microstructural energies, i.e., the elastic interaction energies of geometrically necessary dislocations were computed. In general, three kinds of representation of geometrically necessary dislocations were used: discrete, semi-discrete and, continuous. The discrete representations were closest to reality. Therefore, the corresponding solutions were considered exact. In the semi-discrete representation, the discrete dislocations were smeared out into continuous planar distributions within discrete slip planes. The solutions to problems formulated using different descriptions were different. Consideration was given to the errors in: dislocation distributions (number of dislocations), and, microstructural energies; when the discrete description was replaced by the semi-discrete one. Asymptotic expressions were derived for: number of dislocations, maximum slip, and, microstructural energy density. They provide a powerful insight into the behavior of the system, and were accurate for a wide range of parameters. Two characteristic lengths emerge from the analysis: the ratio of pile-up length to slip plane spacing (lambda), and, the ratio of slip plane spacing to the Burgers vector. For large enough lambda and large enough number of dislocations, both the discrete and semi-discrete solutions were well-approximated by asymptotic solutions. Results of a comprehensive numerical study were presented.

Energies and Distributions of Dislocations in Stacked Pile-Ups. R.Baskaran, S.Akarapu, S.D.Mesarovic, H.M.Zbib: International Journal of Solids and Structures, 2010, 47[9], 1144-53