A view of the relation between the continuum theory of lattice defects and non-euclidean geometry in the linear approximation

Abstract

A view is presented of the relation between the continuum theory of defects in crystals and the mathematical theory of non-metric, non-Riemannian geometry. Both theories are treated in the linear approximation. The lattice defects consist of disclinations, dislocations, and extra-matter, which are identified with the following three important tensors from non-Euclidean geometry: the Riemann-Christoffel curvature tensor, the Cartan torsion tensor and the non-metric Q-tensor. The correspondence between the two theories is established by finding a relation between the coefficients of linear connection of non-Euclidean geometry and the elastic strain, bend-twist, and quasi-plastic strain of defect theory. The definitions of the important tensors from non-Euclidean geometry then generally correspond to the field equations of defect theory. The identities for the curvature tensor generally correspond to the continuity equations of defect theory. The relation to the conventional formulation of defect theory is pointed out. Two examples are given to illustrate the concepts of the paper. One example is related to the deformations associated with constant dislocation distribution and the other to the deformations of a constant disclination distribution.