Dislocation Pile‐ups in Anisotropic Crystals

The positions of the n dislocations in a piled‐up group under a constant applied stress have been determined using the analysis of Eshelby, Frank and Nabarro [1]. The positions are related to the zeros of the first derivative of the n‐th Laguerre polynomial which have been computed for n = 2 to 100. The relation between the zeros of the polynomials and the positions of the dislocations involves an effective elastic constant μeff which is a function of the anisotropic elastic constants and the orientation of the dislocations. μeff has been calculated for glide dislocations lying along a number of symmetry directions in various hexagonal, face‐centred cubic and body‐centred cubic crystals. In some b.c.c. crystals it has been noted that screw or mostly screw dislocations repel each other more strongly than dislocations with more edge character, contrary to the predictions of isotropic elasticity. Analysis of the stress‐fields around a pile‐up has shown the total stress (applied plus internal) is negative in small regions behind each individual dislocation. These negative stress regions are associated with individual dislocations, rather than the group as a whole, and they do not extend very far in a direction normal to the slip plane. It is concluded that isolated dislocation pile‐ups are unlikely to be very effective in impeding other dislocations on parallel slip planes.