A general numerical method to solve for dislocation configurations
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I. INTRODUCTION
THE shape of a mechanically equilibrated dislocation or dislocation array is of considerable interest in modeling the deformation behavior of metals and alloys. Solving for such configurations makes it possible to analyze a wide range of phenomena including dislocation multiplication and annihilation; formation of junctions, kinks, and jogs; and strain hardening. The line-tension model,[1] in which a dislocation is treated as a string with a line tension dependent upon the dislocation local curvature, provides a means to estimate the dislocation energy and dislocation equilibrium position and shape. Although simple and often physically intuitive, the linetension model is inherently approximate because dislocation interaction is essentially “nonlocal,” and equilibrium is affected not only by the local curvature, but also by the remote parts of dislocation (for example, refer to the discussion in Hirth and Lothe[2]). For problems where dislocation self-interaction is strong, such as a dislocation bowing out against a periodic row of impenetrable obstacles, the linetension technique is inaccurate.[3] The Brown formula[4] for the stress field generated by the dislocation and the Peach–Koehler formula[5] may be used to establish the equilibrium equation in differential-integral form. Because the final relaxed, equilibrated dislocation shape depends on the combined effect of the external stress field and dislocation self-stress via the dislocation shape, the domain over which the line integral takes place is usually unknown before equilibrium is reached. This introduces daunting complexity in the closed-form equilibrium analysis of dislocations of general shapes in arbitrary stress fields. One way to overcome the problem of unknown shape X.J. XIN, formerly Research Associate, Department of Materials Science and Engineering, Ohio State University, is Assistant Professor, Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506. R.H. WAGONER, Distinguished Professor of Engineering, and G.S. DAEHN, Professor, are with the Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210. Manuscript submitted November 20, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
is to assume, by physical reasoning, that the dislocation approximately adopts a simple geometry. The system energy may then be minimized with respect to configurational parameters (radius, aspect ratio, etc.).[6,7] The mathematics involved may become formidable when the assumed shape is complicated. The method may yield misleading results when the assumed shape is significantly different from the equilibrated one. A more-satisfactory numerical method approximates the dislocation by a series of finite straight-line segments for which the energy or forces can be calculated. A force balance or energy minimization can then be used to obtain the segment positions. With decreasing segment size, the solution approaches the continuous one. In past implementations, the focus has been on se
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