The treatment of traction-free boundary condition in three-dimensional dislocation dynamics using generalized image stress analysis
Introduction
Dislocation dynamics, or DD, has recently emerged as a powerful method for predicting the plastic deformation of defected crystalline materials [1], [2]. DD deals with the interaction of three-dimensional dislocation curves located in a simulation box (representing a single crystal) and approximated as a set of connected dislocation line segments each. The self-stress of a straight-line dislocation segment in an infinite medium can be found from the literature [3], [4]. This self-stress represents the main ingredient in dislocation dynamics codes, needed in order to capture the mutual interaction of the segments.
When the dislocation segments are close to external free surfaces (e.g. the computational domain boundaries), the self-stress formula used previously is no longer valid by itself and should instead be augmented with auxiliary terms. Such terms arise, e.g. from image stresses when dealing with infinitely long dislocation lines near a free surface. These terms represent the effect of the free surface on the nearby dislocation segment. Thus, current implementations of dislocation dynamics are not completely accurate since they consider the dislocations to lie in an infinite medium. Instead, the effect of a finite domain, through the computational cell surfaces, needs to be examined.
The objective of this study, thus, is to rigorously treat the traction-free condition imposed on the boundaries of a DD box. The study provides for theoretical and numerical treatments of the condition. It is based on image stress analysis (from dislocation theory) and derives from crack theory. Its approach can be categorized under “generalized image stress analysis,” which here refers to the careful distribution of dislocation entities (lines, loops, etc. that are sources of stress), at or near the boundary of interest such that the zero-traction condition is satisfied. The proper treatment of this boundary condition should improve the DD prediction of macroscopic material behavior, and eliminate any hidden or not readily apparent artifact or bias in the results.
With regard to the effect of free surfaces on sub-surface dislocations, several researchers have investigated different aspects of the problem. Initially, [5] determined the elastic fields of a dislocation half-line terminating at a free surface of an isotropic elastic body for any angle of incidence and Burgers vector. The displacements of an infinitesimal dislocation loop of arbitrary orientation, and residing in a semi-infinite isotropic elastic medium, were obtained by [6]. The elastic field of a closed finite, or semi-infinite, dislocation loop can, thus, be obtained by means of area integration using the results for the infinitesimal loop. Using results from [6], Maurissen and Capella [7], [8] derived the stress fields of a dislocation half-line (and segment) parallel and perpendicular to a free surface of a semi-infinite isotropic medium. Concurrently, with the works of [7], [8], Comninou and Dunders [9] found the displacement field associated with an angular dislocation in an elastic isotropic half space. For a dislocation half-line in an isotropic half-space, [10] have obtained closed form solutions for the stresses, which can be used to find the stresses of a line segment. Fivel et al. [11] treated the image stress problem using the Boussinesq solution for a point load in a half space. Such a treatment is more suitable for thin layers. Finally, Lothe et al. [12] developed an integral expression for the case of a dislocation terminating at the free surface of an anisotropic half-space. They solved the problem using a planar fan-shaped distribution of infinite straight dislocations. This idea is similar to solving crack problems by the proper distribution of stress sources (i.e. dislocation entities).
Section snippets
Treatment of the traction-free surface problem
Consider Fig. 1 below showing a sub-surface dislocation segment A1B1. Here, b is the Burgers vector of the segment and t1 is its line sense. The condition of zero traction requires that T=σn=0 at any surface point, P, which translates to σxz=σyz=σzz=0. To annul the shear stresses on the plane, one can place a mirror segment A2B2 such that . This can be proven after some careful analysis of the segments’ stress fields (available in [3], [4]). This
Results and discussion
Once the Burgers vectors of the surface loops are known, one can compute the Peach–Koehler force (PKF) on a sub-surface segment. The PKF acting on the segment tends to pull it towards the surface to minimize the crystal energy. Due to elasticity theory limits, the force on a segment can only be calculated to within a core distance or depth (i.e. z-depth=0.5b–4b) from the surface. This is not a serious limitation and is in harmony with other DD calculations which take this limitation into
Acknowledgements
This work was performed in part under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract W-7405-Eng-48.
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