A three-dimensional mesoscopic model, for the simulation of the collective dynamic behavior of a large number of curved dislocations of finite length, was developed for the purpose of analyzing deformation patterns and instabilities; including the formation of dislocation cell structures. Each curved dislocation was approximated by a piecewise continuous array of straight line segments. The interactions among the segments, including line-tension and self-interaction, were treated explicitly. For longer-range interactions, the space was divided into a regular cellular array and the elastic fields of the dislocations in a remote cell were approximated by a multipolar expansion; leading to an order-N algorithm for the description of a cell containing N dislocations. For large arrays, the simulation volume was divided into cubic cells. A discrete random starting array was selected for the master cell and its nearest neighbors, which constituted an order-2 cell. Reflection boundary conditions were imposed for near-neighbor order-2 cells, and so on, thus creating an NaCl-type lattice array. The boundaries between the cells were considered to be relaxed grain boundaries. That is, recovery within the boundaries and rotation across them were considered to occur so that the boundaries had no associated elastic fields. Such a cell hierarchy, coupled with the multipole expansion, was suitable for treatment using parallel computation, in which each individual cells had a separate processor.

On Plastic Deformation and the Dynamics of 3D Dislocations. H.M.Zbib, M.Rhee, J.P.Hirth: International Journal of Mechanical Sciences, 1998, 40[2-3], 113-27