Fundamentals of a continuum theory of dislocations

In the context of recent proposals to use statistical mechanics methods for building a continuum theory of dislocation lines, mathematical modelling has to answer three essential questions: (i) What is the mathematical object representing the single dislocation as basic "particle"? (ii) What is the law of motion of this object? (iii) What is the mathematical nature of a dislocation density built of such objects? If a mathematically rigorous answer to these questions can be given, one may expect to derive the kinetic evolution equation for such a density solely from its definition and a conservation law. We present a method for deriving classical and non-classical dislocation density measures as well as their evolution equations from the properties of single dislocations, using the close connection between differential forms and geometrical objects such as dislocation lines. Several dislocation density measures are compared in view of their ability to represent vital aspects of the statics and dynamics of discrete dislocation configurations. A dislocation density measure which considers line directions and curvatures is defined as differential form, and it is shown that its evolution correctly represents the essential features of dislocation motion.