The self energy of a helical dislocation

A review of dislocation theory is given, which includes Kroner's “continuum theory of dislocations and internal stresses.” This theory leads to an energy expression for a dislocation in terms of a double line integral along the dislocation line For the isotropic, case this expression is used to find the self energy of straight, circular, cylindrical and helical dislocations. The connection with magnetostatics is pointed out. The helical dislocation is assumed to have a uniform shape with the Burgers vector along its axis. The axial length of the helix is large compared to its radius and the radius is large compared to the dislocation "cross section", which .is of the order of a Burgers vector. For a helix of many turns and arbitrary pitch an expansion in a Fourier cosine series is used. The self energy is found in terms of elementary functions and Kapteyn series of Bessel functions. In the limiting cases of a tightly wound helix (small pitch) and a nearly straight helix (large pitch) simple expressions result, which have a plausible physical explanation. For a tightly wound helix the dominant term represents the contribution from the cylindrical part of the helix g the first order terms represent the influence of the size of the dislocation cross section and the second order terms represent the effect of the axial component of the helix. For the nearly straight helix the dominant terms represent the contribution from the straight screw part of the helix and the second order terms give the effect of the interaction between the turns of the helix. The change in self energy when a return loop is present is also considered. Finally a simple expression is obtained for the case of a helix of less than one turn.published or submitted for publicationU of I Onlythesi