A discrete mechanics approach to modeling the dynamics of dislocations in body-centered cubic single crystals was presented. Ideas were incorporated from discrete differential calculus and algebraic topology and adapted to crystal lattices. In particular, the extension of a crystal lattice to a CW complex permitted the convenient manipulation of forms and fields defined over the crystal. Dislocations were treated within the theory as energy-minimizing structures that led to locally lattice-invariant but globally incompatible eigen-deformations. The discrete nature of the theory eliminated the need for regularization of the core singularity and inherently permitted dislocation reactions and complicated topological transitions. The quantization of slip to integer multiples of the Burgers’ vector led to a large integer optimization problem. A novel approach to solving this NP-hard problem based upon considerations of metastability was proposed. A numerical example that used the method to study the emanation of dislocation loops from a point source of dilatation in a large body-centered cubic crystal was presented. The structure and energetics of body-centered cubic screw dislocation cores, as obtained via the present formulation, were also considered and shown to be in good agreement with available atomistic studies. The method thus provided a realistic avenue for mesoscale simulations of dislocation based crystal plasticity with fully atomistic resolution.

A Discrete Mechanics Approach to Dislocation Dynamics in BCC Crystals. A.Ramasubramaniam, M.P.Ariza, M.Ortiz: Journal of the Mechanics and Physics of Solids, 2007, 55[3], 615-47