The enumeration and transformation of dislocation dipoles I. The dipole strengths of closed and open dislocation arrays

Abstract

Brown's demonstration in 1977 of a dislocation array in which an interstitial dipole is converted into a vacancy dipole by dislocation glide without climb is paradoxical, because it appears to produce non-conservation of point defects by a conservative process. The paradox is addressed by showing that the formula provides a consistent measure of the dipole strength of a closed dislocation array that can be resolved into a number of loops labelled α. The line integral is taken over each loop, for which b_αi is the Burgers vector of a dislocation vector line element dℓ _αk located at the point r_αj . The formula gives the volume of the total vacancy content of the array and is unchanged by glide motion of the dislocations, provided that no dislocations are lost to the surface. It is shown that the same formula can be used for dislocations that penetrate through the volume under consideration, and for those that extend from the closed array to the surface.

Acknowledgement

We are grateful to Professor O. Penrose and Professor M. Finnis for their criticisms of an earlier version of §§ 3 and 4.

Notes

†More precisely stated, because the N equations relating Burgers vectors at the nodes are a set of homogeneous equations, the determinant of the coefficients must vanish, leaving N−1 independent constraints.

†In equation (7), the sense of the line integral is a circulating vector whose sign depends on the handedness of the axes. Also ϵ_ijk and b_αi change sign on inversion of the axes. Therefore m is a pseudoscalar that is the volume, itself a pseudoscalar, of the missing atoms. This contrasts with equation (5), where ℓ _αk does not change sign, making m a true scalar.

†Note when reading this paper that the actual dislocation arrangement in cyclically deformed metals is an array of dipoles in which the height of a dipole is much less than the separation between dipoles and so differs from a Taylor lattice in that the spacings between adjacent glide planes are alternately large and small.