Dislocation Equilibrium in a Uniformly Varying Stress Field

Abstract

The elegant analysis of De Wit and Koehler on the interaction of dislocations with an applied stress in anisotropic crystals is extended to the case of a uniformly varying stress field. As in Reference (1), the energy of a segment of the curved dislocations considered here is taken to be that for a long straight dislocation of the same orientation. This assumption is valid if (a) the orientation dependance of the core energy is negligible or it is the same as that of the elastic energy, and (b) the radius of curvature of the dislocation is large compared to the distance to a free surface which is parallel to the slip plane. The analysis was developed to investigate the case of dislocations near a free surface under conditions where (b) is satisfied. A detailed verification of (a) must await further analysis, but it is clear that unless the dislocation approaches the free surface to within a few core radii, the elastic energy will dominate. The differential equation for the equilibrium shape of a dislocation in a linearly varying stress is first presented. The equilibrium shape of a dislocation in its slip plane is then found. Unlike the solutions for a uniform stress field, the equilibrium curve does not close on itself. The equilibrium shapes are shown to imply critical source lengths between pinning points, beyond which the dislocation is unstable and acts as a Frank-Read source. The critical length is found to be dependant on the direction of the Burgers vector (or the direction of bowing) when the Burgers vector is not perpendicular to the stress gradient.

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