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An iterative scheme for the generalized Peierls-Nabarro model based on the inverse Hilbert transform

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Abstract

A new semi-analytical iterative scheme is proposed in this work for solving the generalized Peierls-Nabarro model. The numerical method developed here exploits certain basic properties of the Hilbert transform to achieve the desired reduction of the non-local and non-linear equations characterizing the generalized Peierls-Nabarro model to a local fixed point iteration scheme. The method is validated with simple examples involving the 1D Peierls-Nabarro model corresponding to a sinusoidal stacking fault energy, and with calculations of the core structure of both edge and screw dislocations on the close-packed $$\{111\}$$ planes in Aluminium. An approximate technique to incorporate external stresses within the framework of the proposed iterative scheme is also discussed with applications to the equilibration of a dislocation dipole. Finally, the advantages, limitations and avenues for future extension of the proposed method are discussed.

Most cited references22

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The size of a dislocation

(1940)
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Dislocations in a simple cubic lattice

(1947)
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Intrinsic stacking faults in body-centred cubic crystals

(1968)
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Author and article information

Journal
02 July 2019
Article
1907.01165