Affine development of closed curves in Weitzenböck manifolds and the Burgers vector of dislocation mechanics

Abstract

In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure. As opposed to some other approaches to the continuum definition of the Burgers vector, our definition is completely geometric, in the sense that it involves no ambiguous operations such as the integration of a vector field: when we integrate a vector field, it is a vector field living in the tangent space at a given point in the manifold. For a body with distributed dislocations, the material manifold, which describes the geometry of the stress-free state of the body, is commonly taken to be a Weitzenböck manifold, i.e. a manifold with a metric-compatible, flat connection with torsion. We show that for such a manifold, the density of the Burgers vector calculated according to our definition reproduces the commonly stated relation between the density of dislocations and the torsion tensor.