Free-Surface Effects in 3D Dislocation Dynamics: Formulation and Modeling

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Abstract
Recent advances in 3-D dislocation dynamics include the proper treatment of free surfaces in the simulations. Dislocation interaction and slip is treated as a boundary-value problem for which a zero-traction condition is enforced at the external surfaces of the simulation box. Here, a new rigorous method is presented to handle such a treatment. The method is semi-analytical/numerical in nature in which we enforce a zero traction condition at select collocation points on a surface. The accuracy can be improved by increasing the number of collocation points. In this method, the image stress-field of a subsurface dislocation segment near a free surface is obtained by an image segment and by a distribution of prismatic rectangular dislocation loops padding the surface. The loop centers are chosen to be the collocation points of the problem. The image segment, with proper selection of its Burgers vector components, annuls the undesired shear stresses on the surface. The distributed loops annul the undesired normal stress component at the collocation points, and in the process create no undesirable shear stresses. The method derives from crack theory and falls under “generalized image stress analysis” whereby a distribution of dislocation geometries or entities (in this case closed rectangular loops), and not just simple mirror images, are used to satisfy the problem’s boundary conditions (BCs). Such BCs can, in a very general treatment, concern either stress traction or displacements.