Remarks on variational problems for defective crystals

Abstract: "In the context of a continuum theory of crystals with defects, one can define a particular list of tensors which remain unchnaged [sic] when the crystal is deformed elastically. In our model, the defect notion relies on the assumption that deformations that leave these elements invariant do not change the defects. This class of deformations strictly includes the elastic deformations; nonelastic defect-preserving deformations are called neutral and generally involve some kind of rearrangement, or slip, of the crystal lattice. Here we deal with slightly defective crystals, i.e. where defects are so few that a lattice is distinguishable at the microscopic scale. We factor neutral deformations into components which are exclusively elastic at the macroscopic level or exclusively slip at the microscopic level. Using direct methods of the calculus of variations we determine equalibrium [sic] configurations for defective crystals. As in Chipot & Kinderlehrer, we study the behavior of minimizing sequences and their state functions, and in order to take into account possible oscillations of the minimizing sequences we assume that solutions may be measure-valued."