A four-dimensional nonlinear geometric theory of defect continuum

Abstract

A nonlinear theory of kinematics for continuum with defects (dislocations and disclinations) is formulated within the framework of a 4-dimensional non-relativistic space-time material manifold M 4 ∗ . Two kinds of Cartan connections are defined on M 4 ∗ , one is Eulidean and the other non-Euclidean, to provide a right mathematical device of delineating the deformation of macroscopic continuum and microscopic defects in it. Densities and current densities of defects are defined in terms of torsion and curvature tensors respectively with respect to vierbein and non-Euclidean connections. A set of total covariant non-linear continuity equations of defect dynamics is derived from Bianchi identities. They can be simplified to the commonly used linear equations under a proper approximation. Two different ways of defining disclination density and current density tensors are given, and it is found that the disclination is of source free for the first definition while source terms appear for the second definition. The source terms come from interaction between dislocations and disclinations.