On the geometry of continuous and singular distributions of defects
Reuven Segev (Ben-Gurion University), Marcelo Epstein (University of Calgary)
SES Medal Symposium in honor of D.J. Steigmann
Wed 9:00 - 10:30
MacMillan 115
We present a mathematical description of dislocations in crystalline solids having the following features: It specifies the geometry of dislocations independently of the configuration of the body in space. It applied to both continuous distributions of dislocations as well as discrete dislocations. A continuous body B is modeled as an n-dimensional differentiable manifold. We conceive a layering structure in the body which is analogous to the crystallographic planes structure. In the continuous case, the layering structure is specified by a differential 1-form, the layering form, in B. The value of the layering form at a point in the body is a co-vector which is analogous to the Miller indices for the corresponding crystallographic planes. If the layering form is the gradient of a real valued function u on the body, the function u can be used to label the various crystallographic (n - 1) dimensional surfaces that serve as level sets for u. The condition for such integrability of the layering form is that its exterior derivative vanishes. Thus, the exterior derivative of the layering form, describes the obstructions to integrability---the dislocations in the body. Using the theory of de Rham currents, we take as the basic object is an (n – 1)-current. If a layering differential form is given, such a current T is induced as follows. T acts on an (n – 1) form by integrating over B the exterior product of the layering form and the (n – 1)-form. However, we present examples of currents that are not induced by smooth forms, e.g., non-coherent interfaces, edge dislocations and screw dislocations. We will refer to such currents as layering currents. The generalized integrability condition for a layering current is that its boundary vanishes. Thus, the boundary of the layering current---the dislocation current---describes the geometry of singular dislocations.