The treatment of traction-free boundary condition in three-dimensional dislocation dynamics using generalized image stress analysis

Recent attention has been given to the proper treatment of the planar traction-free surfaces which typically bound a computational box in three-dimensional dislocation dynamics. This paper presents an alternative to the use of the finite-element method for this purpose. Here, to annul the tractions produced by a sub-surface dislocation segment on a finite-area free surface S, a combination of an image dislocation segment, and a distribution of N prismatic rectangular Volterra dislocation loops meshing S is utilized. The image dislocation segment, with the proper sign selection of the Burgers vector components, annuls the shear stresses, and the normal stress component is annulled discretely at N collocation points representing the centers of the loops. The unknowns in this problem are the magnitudes of the N Burgers vectors for the loops. Once these are determined, one can back calculate the Peach–Koehler force acting on the sub-surface segment and representing the effect of the free surface. As expected, the accuracy of the method improves as the loops continuously decrease in size.