Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals

Abstract

We present a continuum dislocation dynamics model that predicts the formation of dislocation cell structure in single crystals at low strains. The model features a set of kinetic equations of the curl type that govern the space and time evolution of the dislocation density in the crystal. These kinetic equations are coupled to stress equilibrium and deformation kinematics using the eigenstrain approach. A custom finite element method has been developed to solve the coupled system of equations of dislocation kinetics and crystal mechanics. The results show that, in general, dislocations self-organize in patterns under their mutual interactions. However, the famous dislocation cell structure has been found to form only when cross slip is implemented in the model. Cross slip is also found to lower the yield point, increase the hardening rate, and sustain an increase in the dislocation density over the hardening regime. Analysis of the cell structure evolution reveals that the average cell size decreases with the applied stress, which is consistent with the similitude principle.

The discrete-continuum connection in dislocation dynamics: I. Time coarse graining of cross slip

“…The continuum dislocation dynamics model summarized here predicts the dislocation cell structure as well as the famous similitude law for the evolution of the average cell size with stress [1].…”

“…The cross slip rate values used to test the effect of cross slip were so high that the yield point and hardening rate were overestimated and, over the strain range used in the simulations, the average cell size of the emerging cell structure was found to be smaller than the experimental values. It was argued in [1] that a properly calibrated cross slip model should yield better results as far as stress-strain behaviour and cell size evolution are concerned. Such a calibration is the subject of the current work.…”

“…Here M is the mobility and v is a unit vector along the velocity direction in the slip plane, which is normal to l ρ ; see [1] for details.…”

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AbstractA recent continuum dislocation dynamics formalism [1] has been enriched by incorporating an improved cross slip model. 3D discrete dislocation dynamics simulations were used to collect cross slip rate data in the form of time series that were analysed to estimate the correlation time for cross slip, which was subsequently used as a time scale for local window averaging of the collected cross slip rate data.

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Dislocation patterning in a two-dimensional continuum theory of dislocations

“…[26] the authors also discussed a possible extension of the 2D continuum theory of Groma et al [21] leading to the instability of the homogeneous dislocation density in a deforming crystal (details are discussed below). Some remarkable steps towards modeling pattern formation have been made by Kratochvil et al [38] and recently in a 3D mean field theory by Xia and El-Azab [39]. In order to obtain patterns, however, in both models specific microscopic dislocation mechanisms (sweeping narrow dipoles by moving curved dislocations, or cross slip, respectively) had to be invoked.…”