Size-dependent plasticity: Discrete dislocations and gradient continuum theory
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Classical crystal plasticity is size-independent. Plastic deformation in smallscales 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-upsknown as geometrically necessary dislocations (GNDs). The size-effects are the result ofhigh energy configurations - stacked pile-ups of dislocations.We present a systematic study to understand the kinematic and thermodynamiceffects of representing discrete dislocations as continuous distributions in their slipplanes. We compute the error in microstructural energy in representing discretedislocations as continuous entities. Using the continuum formulation, we solve for slipdistributions and compute the orientation dependence of interface energy. Then, we solvesingle and double slip problems and compare our results to existing discrete dislocationsimulations.In general, three kinds of representations of GNDs are used: discrete, semidiscrete,and, continuous representation. The discrete representations are closest toreality. Therefore, the corresponding solutions are considered exact. In the semi-discreterepresentation, the discrete dislocations are smeared out into continuous planardistributions within discrete slip planes. The solutions to problems formulated usingdifferent descriptions are different. We consider the errors in: dislocation distributions(number of dislocations), and, microstructural energies; when the discrete description isreplaced by the semi-discrete one. Asymptotic expressions are derived for: number ofdislocations, maximum slip, and, microstructural energy density. Then, we considersystems without boundary relaxation and compute the coarsening error in microstructuralenergy and express them in terms of continuum fields.Two characteristic lengths emerge from the analysis: the ratio of pile-up length toslip plane spacing , and, the ratio of slip plane spacing to the Burgers vector. Forlarge enough value of the former parameter and large enough number of dislocations, both the discrete andsemi-discrete solutions are well-approximated by asymptotic solutions. The coarseningerror in microstructural energy is localized and the orientation dependence of interfaceenergy 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.