Size-dependent plasticity: Discrete dislocations and gradient continuum theory

Classical crystal plasticity is size-independent. Plastic deformation in small scales is size-dependent, and mainly governed by dislocation motion and its interactions. In small volumes the motion of dislocations is confined, resulting in dislocation pile-ups known as geometrically necessary dislocations (GNDs). The size-effects are the result of high energy configurations—stacked pile-ups of dislocations.;We present a systematic study to understand the kinematic and thermodynamic effects of representing discrete dislocations as continuous distributions in their slip planes. We compute the error in microstructural energy in representing discrete dislocations as continuous entities. Using the continuum formulation, we solve for slip distributions and compute the orientation dependence of interface energy. Then, we solve single and double slip problems and compare our results to existing discrete dislocation simulations.;In general, three kinds of representations of GNDs are used: discrete, semidiscrete, and, continuous representation. The discrete representations are closest to reality. Therefore, the corresponding solutions are considered exact. In the semi-discrete representation, the discrete dislocations are smeared out into continuous planar distributions within discrete slip planes. The solutions to problems formulated using different descriptions are different. We consider the errors in: dislocation distributions (number of dislocations), and, microstructural energies; when the discrete description is replaced by the semi-discrete one. Asymptotic expressions are derived for: number of dislocations, maximum slip, and, microstructural energy density. Then, we consider systems without boundary relaxation and compute the coarsening error in microstructural energy and express them in terms of continuum fields.;Two characteristic lengths emerge from the analysis: the ratio of pile-up length to slip plane spacing, and, the ratio of slip plane spacing to the Burgers vector. For large enough value of the former parameter and large enough number of dislocations, both the discrete and semi-discrete solutions are well-approximated by asymptotic solutions. The coarsening error in microstructural energy is localized and the orientation dependence of interface energy was determined from comparing continuum solution to numerical results. The elastic-plastic stiffness for symmetric double slip was lower than the single slip and overall stress-strain response, between the continuum theory and simulations was a good match, for both cases.