Continuous distributions of dislocations VI. Non-metric connexions

Abstract

The quasi-static plastic-elastic deformation of a crystalline solid containing a continuous distribution of both dislocations and extra-matter is reexamined, retaining the assumption that the lattice structure is everywhere uniquely defined. The various distortions are considered to occur in a formal sequence and the solid is assumed to pass as a continuous whole through a series of generalized spaces as the distortions occur in succession. The principles used in associating the appropriate differentiable manifold with each state of the solid are indicated, and it is emphasized that the theory of non-metric connexions is required if the dislocation density is always to be measured by the torsion of the manifold. The importance of this association is emphasized, since real crystalline matter does in fact deform plastically by the motion of dislocations. It is also shown that the theory of non-metric linear connexions is especially suitable for describing the lattice geometry, and particularly the spatial increments of pure strain and rotation. The governing equations are formulated in such a way that any of the manifolds may be regarded as primary, and the relation of the analysis to treatments of other workers using different coordinate systems is discussed.