Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space

Yuan Gao; Jian-Guo Liu; Tao Luo; Yang Xiang

doi:10.3934/dcdsb.2020224

Article Contents

2021, Volume 26, Issue 6: 3177-3207. Doi: 10.3934/dcdsb.2020224

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Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space

1.
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
2.
Department of Mathematics, Duke University, Durham NC 27708, USA
3.
Department of Mathematics and Department of Physics, Duke University, Durham NC 27708, USA
4.
Department of Mathematics, Purdue University, West Lafayette IN 47907, USA

Received: May 2020

Early access: July 2020

Published: June 2021

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Abstract

In this paper, we revisit the mathematical validation of the Peierls–Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation in Hilbert space. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy, which is always infinite, across the slip plane. We revisit the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. The shear displacement jump is determined only by the reduced problem on the slip plane while the displacement fields in the two half spaces are determined by linear elasticity. We establish the existence and sharp regularities of classical solutions in Hilbert space. For both the reduced problem and the full PN model, we prove that a static solution is a global minimizer in a perturbed sense. We also show that there is a unique classical, global in time solution of the dynamic PN model.

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Mathematics Subject Classification: 35R11, 35Q74, 35S15, 35J50.

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Figure 1. Schematic illustration of the PN model for an edge dislocation. The dislocation locates along the $ z $ axis with $ +z $ direction, and its slip plane is the $ y = 0 $ plane. $ \mathbf b $ is the Burgers vector and $ d $ is the interplanar distance in the direction normal to the slip plane. The black dots and red circles show the locations of atoms of the two atomic planes $ y = 0^+ $ and $ y = 0^- $ in the lattice with the dislocation and in the reference states before elastic deformation, respectively, based on a simple cubic lattice. The Burgers vector enclosed by a loop $ L $ enclosing the dislocation is $ \mathbf b_L = \oint_L \,\mathrm{d} \mathbf u $

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