Dislocation Image Stresses at Free Surfaces by the Finite Element Method
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Dislocation Image Stresses at Free Surfaces by the Finite Element Method Meijie Tang, Guanshui Xu*, Wei Cai, Vasily Bulatov, Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551 * Dept. of Mechanical Engineering, Univ. of California at Riverside, Riverside, CA 92521 ABSTRACT The finite element method has been routinely used to calculate the image stresses of dislocation segments. When these segments intersect with surfaces, the image stresses at the surfaces diverge singularly. At the presence of these singularities, both convergence and accuracy of using the finite element method need to be examined critically. This article addresses these issues with the aim toward the application of dislocation dynamics simulations in thin films. INTRODUCTION Dislocations in a finite medium have image stresses [1]. Unlike electrostatic systems, the image stress of dislocations can not be straightforwardly obtained by simple image superposition. It is generally recognized that these image stresses play important roles in the dislocation behavior in finite sized systems such as thin films. In recent years, the development of discrete dislocation dynamics simulation [2-3] has made the simulation of thin film plasticity possible, presuming the image stresses can be calculated accurately and efficiently. Typically, the image stresses are calculated using the finite element method (FEM), which is then coupled with a dislocation dynamics method [4-7]. When dislocations intersect with surfaces, the image stresses at the surfaces diverge singularly. At the presence of these singularities, both convergence and accuracy of using the FEM need to be examined critically. This article addresses these issues by performing systematic FEM calculations in simplified systems and to compare with analytical solutions. METHODOLOGY The FEM is a standard method to solve boundary value problems [8]. The problem being concerned here is the following: given the elastic stress field of a dislocation segment in an infinite medium, find the elastic field of the dislocation segment in a finite body with a ∞ stress-free boundary. The solution takes the form σ ij + σ ijimg , where the first term is the stress in the infinite medium, and the second term is the image stress due to the boundary. The image stress can be obtained by FEM through solving the boundary value problem so that the surface traction σ ijimg n j cancels the original traction σ ij∞ n j on the surface, where n j is the component of the normal vector to the surface.
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Figure 1. A schematic view of the setup of the calculations. The thin slab has one free surface at the top perpendicular to the z direction. The infinite dislocation is perpendicular to the free surface. The system being studied is a thin slab with one free surface at the top perpendicular to the z direction, as shown in figure 1. All calculations are done with the dimensions of the slab being 300x300x60 angstroms along x, y, and z directions respectively. An infinite dislocation is located near
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