The ‘moving frameʹ, and defects in crystals

I outline a special theory of defective crystals where the microstructure is represented by ®elds of lattice vectors and their spatial derivatives. Since these ®elds are assumed to be smooth, in the model, it is a central issue to be precise about the sense in which such ®elds can represent defects, and that is done by introducing elastic invariant integrals (of which the most elementary examples are the Burgerʹs integrals and the dislocation density). It turns out that the notion of slip has a natural place in the analysis of these elastic invariant integrals, and moreover that the formulation invites one to draw results from Cartanʹs theory of `equivalenceʹ of vector ®elds, and also from the theory of Lie groups. Indeed, it is a remarkable fact that one can identify material points in a crystal that has constant dislocation density tensor with an appropriate Lie group, as a consequence of which one ®nds that such crystals have a self-similarity which generalises the classic idea of generating a perfect crystal (lattice) by translation of a particular unit cell. Generally, it seems that there is much to be done in adapting known mathematical results to this context