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Existence and stability of a screw dislocation under anti-plane deformation
Analysis of PDEs · Submitted 9 Apr 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model.
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We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. | method 0.54 | 0.94 | 0.04 | asserted 0.98 |
| s1 | Invariance of the energy functional under lattice symmetries renders the problem non-coercive. | limitation 0.56 | 0.53 | 0.03 | asserted 1.00 |
| s2 | Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model. | result 0.91 | 0.96 | 0.93 | asserted 1.00 |
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The paper
Original abstract
We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model.
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How the literature uses it
2 arXiv papers cite this in 12 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 2 papers
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Following the theory developed in [ref] for point defects and straight dislocations, we formulate the static crack model as a minimisation problem [equation] with the energy difference functional given by [equation] where $V ^ →$ is an interatomic potential, $û →$ is the far-field predictor and $u$ a core correction, thus giving us the actual displacement as $û + u$.
Analysis of an atomistic model for anti-plane fracture · Atomistic model for anti-plane fracture · 0.86
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It is clear that the energy difference in [ref] is well-defined if $y-ỹ ∈_c(_0)$ and a straightforward adjustment of results in [ref] ensures that, under certain conditions to be subsequently discussed, $$ can be extended by continuity to displacements such as $y -ŷ ∈(_0)$ defined in [ref].
Atomistic modelling of near-crack-tip plasticity · Energy difference · 0.86
Uses its method or tool 1 paper
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With the underlying domain of infinite size, the atomistic energy difference defined in [ref] can be used to model a crack opening provided that the predictor-corrector approach championed in [ref] , [ref] and [ref] is employed.
Atomistic modelling of near-crack-tip plasticity · Far-field prediction · 0.95
Compares with it 1 paper
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This is in contrast with dislocation problems, where, except for specific cases with stringent assumptions as e.g. in [ref] , we simply assume that a solution exists.
Analysis of an atomistic model for anti-plane fracture · Conclusion and discussion · 0.97
Cites it as background 1 paper
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On a mathematical level, while there is a wealth of recent work about atomistic modelling of materials and in particular atomistic approaches to defects in crystals (c.f., among many other, [ref] ), many of the mathematical techniques employed rely at least in part on exploiting crystal symmetries.
Atomistic modelling of near-crack-tip plasticity · Introduction · 0.90
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