Discrete Dislocation Dynamics in Crystals

We present a study of 3D dislocation dynamics in BCC crystals based on discrete crystal elasticity. Ideas are borrowed from discrete differential calculus and algebraic geometry to construct a mechanics of discrete lattices. The notion of lattice complexes provides a convenient means of manipulating forms and fields defined over the crystal. Atomic interactions are accounted for via linearized embedded atom potentials thus allowing for the application of efficient fast Fourier transforms. Dislocations are treated within the theory as energy minimizing structures that lead to locally lattice-invariant but globally incompatible eigendeformations. The discrete nature of the theory automatically eliminates the need for core cutoffs. The quantization of slip to integer multiples of the Burgers vector along each slip system leads to a large integer optimization problem. We suggest a new method for solving this NP-hard optimization problem and the simulation of large 3D systems.