Continuous distributions of dislocations IV. Single glide and plane strain

Abstract

The theory of the continuously dislocated crystal is applied to discuss the behaviour of a crystal which has deformed by single glide. When the single glide is also in one crystallographic direction only the deformation is two-dimensional, and this situation is treated in detail. The analysis assumes that the dislocations arrange themselves so that they cause no far-reaching stress. In single glide, the lattice lines originally normal to the slip plane are the orthogonal trajectories of a family of surfaces, the glide surfaces. It is shown that these surfaces are developable, and that this imposes additional restrictions on the dislocation density, which are not considered by Nye (1953) and which provide a basis for the classification of the possible states of dislocation. A convenient analytical solution is given of the general equations for single glide when the lattice rotations are everywhere small. The discussion of plane strain begins by treating the lattice deformation. The general equation governing the lattice rotation is given, and the necessary boundary conditions and methods of solution are discussed. The analysis is compared with that of Nye and illustrated by some examples of practical interest. More realistic problems are posed when the boundary conditions specify changes of shape of the specimens. To discuss these the relationship between the lattice, shape and dislocation deformations is considered, and it is shown how all these deformations can be found from the necessary boundary conditions on the shape deformation. The general solution, which is valid for large strains, is applied to a general class of bending problem, and equations relating the geometrical features of the lattice, and of lines scribed on the crystal are derived. The general analysis is illustrated by examples emphasizing certain confusing degeneracies arising in the very symmetrical situation of uniform plane bending (Nye 1953), and revealing the meaning of many valued solutions for the lattice rotation.