Stresses of dislocation cells

Abstract

The stresses of dislocation cells, i.e. dislocation boundaries enclosing a volume of material rigidly rotated with respect to the surroundings, have been investigated. The study has been restricted to cells of generally prismatic shape, such that the edge dislocation segments forming the sides are all of same length and parallel to each other, and such that the terminating end-faces are composed of pure screw dislocations and are normal to the sides of the cell. Such cells were shown to have stress fields that are closely related to the corresponding dislocation “multipoles”, i.e. regularly spaced sets of infinitely long, parallel edge dislocations of same strength, arranged so as to define a cylinder and with all their Burgers vectors pointing towards the cylinder axis. With the sole exception of τ ΦΦ of the dipole, the stresses of multipoles were shown to fall off with radial distance as r − N , where N is the “order” of the multipole, defined as the number of dislocations and, hence, the number of Burgers vectors in the multipole. The short-range stresses of all cells investigated were very closely the same as those of the corresponding infinitely extended sub-boundaries up to within about one inter-dislocation distance from adjoining walls. The long-range stresses of “unitary” cells, i.e., cells whose cross section can be thought of as composed of one type of multipole only, show the same symmetry as that of the corresponding multipole, independent of their shape, and also fall off as r − N . Moreover, the magnitude of long-range cell stresses, keeping cell size and shape constant, decreases rapidly with decreasing h, the distance between adjacent dislocations, so that for h → 0 all stresses vanish identically. “Hybrid” cells, defined as cells whose construction requires more than one kind of multipole, exhibit long-range stresses which are dominated, in turn, by the contributing multipoles in descending order when examining zones at increasing distance from the cell walls. The results obtained are consistent with, but go well beyond, hypotheses regarding the stresses of dislocation cells that had been made earlier in connection with dislocation behavior in deformed crystals. Particularly important in this respect is the new light being thrown on the role of “forest” dislocations by the result that the range of stresses, and thus the cell energy, is strongly decreased as N, the number of participating Burgers vectors, increases.