Institute of Metals Division - On the Growth of Helical Dislocations

Conclusions reached in a paper by weertmanl are amplified in a mathematical and graphical way. It is shown that in a stressed crystal a straight dis-location may be in a position of unstable equilibrium with respect to helix development, and how a small perturbation may start the dislocation off on its helical path. Helix development occurs by diffision; bulk diffusion is necessary for development from a screw, but core diffision is sufficient for development from an edge dislocation. For a mixed dislocation helix development can take place by core diffusion if the helix also climbs; a diffusion reversal takes place when the slope of the helix equals the slope of the Burgers vector. The work done on a crystal by the external forces is graphically shown to be consistent with the direction of motion of the dislocation in helix development. Finally a detailed mechanism for tangle formation from helices is presented. In the course of helix development a glide situation may be reached where a segment of each helix loop lies in a slip plane in which it can expand by glide. Thus is it visualized how a helix can deteriorate into a tangle. A paper by weertmanl proposes dislocation-tangle formation by a helical dislocation mechanism. The present paper amplifies Weertman's arguments in a mathematical and graphical way and presents some further ramifications of the proposed model. The importance of dislocation tangles is discussed in Weertman's paper and the references cited therein. The mechanism of their formation is not yet understood. Weertman proposes that in cold-worked crystals conditions exist suitable for the conversion of straight dislocations into helices and subsequently into tangles. Helix formation occurs by a diffusion process and Weertman proposes that at low temperatures only core diffusion is important. The present paper necessarily repeats some of Weertman's work. Since the mathematical development and notation used are slightly different from his, this was thought necessary for clarity. GEOMETRY OF THE HELIX weertman2 has shown that the equilibrium form of any dislocation line is a helix if it is assumed that the energy per unit length (line tension) of the dis- location is constant. For the special case that no external forces act on the dislocation, the helix reduces to a straight line. We shall assume in this paper that the dislocation has a helical form even when it is not in equilibrium. The parametric equation of a helix is, see Fig. l: where a is the radius of the cylinder tangent to the helix, and 270 the pitch of the helix. The pitch is related to v, the number of turns per unit length (along the axis) by the relation: When a = 0 Eq. [I] gives a straight line in the z direction. The growth of a straight dislocation into a helix is now described mathematically by letting a vary from zero to its final value. The tangent to the helix is given by where is the path length along the helix. This gives the direction of t and since its magnitude is unity we must have So that dcp/ds = (a2 +p2)-"'. The curvature of the helix is given by where n = -(cos + sin) is a unit vector normal to the helix. Note that the vectors k and n always point toward the axis of the helix, Fig. 1. FORCE ON THE HELIX The external force which makes the straight dis-