The elastic energy of a parallelogram-shaped dislocation loop

The stresses associated with angular dislocations, which are required when evaluating the elastic strain energy of a dislocation loop, are presented in simplified form, and used to obtain the self-energy of a parallelogram-shaped loop of arbitrary Burgers vector. The resulting expression is used to determine the variation of energy with orientation for a dislocation loop on a glide prism of rhombus cross section formed by two pairs of {111} planes in f.c.c. metals. Several surprising features of the results of Bullough and Foreman (1964), who in a similar analysis restricted the loop to orientations in which it has rhombus shape, are explained. The ways in which a loop, initially on a {111} plane, can rotate to its orientation of minimum energy are computed, and it is shown that small loops do not pass through the pure edge {011} orientation. A similar analysis is presented for a glide prism of square cross section, which is a case relevant to loops in b.c.c. metals. It has higher symmetry than the {111} prism, but the loop is again shown to have an asymmetric minimum energy orientation. Finally, the validity of the assumptions and approximations made in determining elastic energies of dislocation configurations are questioned and discussed.