Abstract
We present a self-contained review of the discrete dislocation dynamics (DDD) method for the numerical investigation of plasticity in crystals, focusing on recent development and implementation progress. The review covers the theoretical foundations of DDD within the framework of incompatible elasticity, its numerical implementation via the nodal method, the extension of the method to finite domains and several implementation details. Applications of the method to current topics in micro-plasticity are presented, including the size effects in nano-indentation, the evolution of the dislocation microstructure in persistent slip bands, and the phenomenon of dislocation avalanches in micro-pillar compression.
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Notes
The more general case involving thermal and vacancy diffusion causing dislocation climb is treated in Ref. 24.
At higher dislocation velocity, the quadratic form of π cannot be used and non-linear dislocation mobility laws are obtained. Moreover, the quadratic form of π does not capture lattice-controlled mobility typical of bcc metals at low temperature.
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Acknowledgements
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Number DE-FG02-03ER54708, and the US Air Force Office of Scientific Research (AFOSR), under award number FA9550-11-1-0282.
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Appendix A: A Elastic Fields of Dislocation Loops: Vector Notation
Appendix A: A Elastic Fields of Dislocation Loops: Vector Notation
The differential operators U ikl and S ijkl appearing in Eqs. 5 and 6 are, respectively
where Δ is the Laplace operator, ν is the Poisson ratio, and ji the shear modulus of the material. Using Eq. 21, the displacement field generated by a dislocation becomes
where Ω(x) is the solid angle subtended by the loop, R = x − x’, \( R = \sqrt {\varvec{R}\cdot\varvec{R}} \), and \( \widehat{\varvec{R}} = \varvec{R}/R \). With the same notation, using Eq. 22 the stress field caused by the loop becomes:
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Po, G., Mohamed, M.S., Crosby, T. et al. Recent Progress in Discrete Dislocation Dynamics and Its Applications to Micro Plasticity. JOM 66, 2108–2120 (2014). https://doi.org/10.1007/s11837-014-1153-2
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DOI: https://doi.org/10.1007/s11837-014-1153-2