Energies and distributions of dislocations in stacked pile-ups

To understand the kinematic and thermodynamic effects of representing discrete dislocations as continuous distributions in their slip planes, we consider stacked double ended pile-ups of edge and screw dislocations and compute their distributions and microstructural energies, i.e., the elastic interaction energies of geometrically necessary dislocations (GNDs).In general, three kinds of representations of GNDs are used: discrete, semi-discrete, and, continuous representation. The discrete representations are closest to reality. Therefore, the corresponding solutions are considered exact. In the semi-discrete representation, the discrete dislocations are smeared out into continuous planar distributions within discrete slip planes. The solutions to problems formulated using different descriptions are different. We consider the errors in: dislocation distributions (number of dislocations), and, microstructural energies; when the discrete description is replaced by the semi-discrete one.Asymptotic expressions are derived for: number of dislocations, maximum slip, and, microstructural energy density. They provide a powerful insight into the behavior of the system, and are 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 are well-approximated by asymptotic solutions. Results of a comprehensive numerical study are presented.