The development of a discrete theory of crystal elasticity and dislocations in crystals was considered. The theory was based upon suitable adaptations, to crystal lattices, of certain elements of algebraic topology and differential calculus; such as chain complexes and homology groups, differential forms and operators and a theory of integration of forms. In particular, the lattice complex of a number of commonly encountered lattices, including body-centered cubic and face-centered cubic, was defined,. It was shown that material frame-indifference led naturally to discrete notions of stress and strain in lattices. Lattice defects, such as dislocations, were introduced by means of locally lattice-invariant (but globally incompatible) eigen-deformations. The geometrical framework furnished discrete analogues of the fundamental objects and relationships of the theory of linear elastic dislocations: such as, the dislocation density tensor, the equation of conservation of Burgers vector, Kröner’s relation and Mura’s formula for stored energy. Conditions were also supplied for the existence of equilibrium displacement fields. It was shown that linear elasticity was recovered, as the Γ-limit of harmonic lattice statics, as the lattice parameter became vanishingly small. The Γ-limit of dilute dislocation distributions were computed, and it was shown that the theory of continuously distributed linear elastic dislocations was recovered, as the Γ-limit of the stored energy, as the lattice parameter and Burgers vectors became vanishingly small.

Discrete Crystal Elasticity and Discrete Dislocations in Crystals. M.P.Ariza, M.Ortiz: Archive for Rational Mechanics and Analysis, 2005, 178[2], 149-226