Theory of Dislocation Cells. II. Dislocation Multipoles

Abstract

The stresses of infinite edge dislocation multipoles are investigated with the aid of a B5500 computer, and contour maps are obtained using the line printer. These multipoles consist of infinitely long, parallel edge dislocations of same strength, arranged so as to define a prism whose cross section is a regular polygon and with their Burgers vectors in radial orientation. As will be shown in Part III, multipoles are the prototypes of dislocation cells containing the same Burgers vectors, independent of cell shape, in the same way that the quadrupole, discussed in Part I, is the prototype of all cells composed of dislocations involving two mutually perpendicular Burgers vectors. It is found that the stress fields of τ_rr, τ_φφ, and τ_rφ all exhibit 2N similar leaves of alternating sign for a multipole of Nth order, except that τ_φφ of the dipole has eight rather than the expected four leaves. The magnitude of the stresses falls as 1/r^N at large distances, except that τ_φφ of the dipole falls as 1/r⁴. Lastly, there exists a ``conservation of zero line'' rule to the effect that a fixed number of contours of vanishing stress passes each dislocation, independent of N, the order of the multipole, namely, six zero contours of τ_φφ and τ_rφ, and two of τ_rr, respectively.