Dynamics of dislocations described as evolving curves interacting with obstacles

Pauš, Petr; Beneš, Michal; Kolář, Miroslav; Kratochvı́l, Jan

doi:10.1088/0965-0393/24/3/035003

Cited by 10 publications

(14 citation statements)

References 57 publications

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“…Various nontrivial choices of

α^{k}

have been proposed in the literature. We refer the reader to papers by Mikula and Ševčovič, Beneš et al, and references therein. A general framework yielding the so‐called curvature adjusted tangential velocity has been proposed by Ševčovič and Yazaki in a previous article .…”

Section: Lagrangian Description Of a Motion Of Interacting Curvesmentioning

confidence: 99%

Curvature driven flow of a family of interacting curves with applications

2020

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In this paper, we investigate a system of geometric evolution equations describing a curvature‐driven motion of a family of planar curves with mutual interactions that can have local as well as nonlocal character, and the entire curve may influence evolution of other curves. We propose a direct Lagrangian approach for solving such a geometric flow of interacting curves. We prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical solution to the governing system has been constructed by means of the method of flowing finite volumes. We also discuss various applications of the motion of interacting curves arising in nonlocal geometric flows of curves as well as an interesting physical problem of motion of two interacting dislocation loops in the material science.

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“…Various nontrivial choices of

α^{k}

Section: Lagrangian Description Of a Motion Of Interacting Curvesmentioning

confidence: 99%

Curvature driven flow of a family of interacting curves with applications

2020

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“…In our understanding of the DDD modeling, dislocations are represented by curves moving by the (mean) curvature (see [5] or [6]) which schematically reads as normal velocity = curvature + force.…”

Section: Modelmentioning

confidence: 99%

Modeling of Double Cross-Slip by Means of Geodesic Curvature Driven Flow

et al. 2018

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In this paper we develop a model for double cross-slip in fcc crystals. The double cross-slip mechanism is demonstrated by a simulation of an overcoming of a particle exerting spherically symmetric repulsive stress field. The cross-slip is treated as a deterministic, stress-controlled process. For the identification of the cross-slip, we use a criterion based on evaluation of stresses exerted on a tip of a screw part of a dislocation resolved in the primary plane and in the cross-slip plane. The motion of a dislocation is described by the geodesic curvature driven flow on surfaces, and treated by means of the parametric approach. The results of numerical simulations are validated by analytical calculations.

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“…To construct a stable numerical computational scheme, several nontrivial choices of α have been proposed in the literature. We refer the reader to papers by Mikula andŠevčovič [22,23], Beneš et al [5,16,24] and references therein. A general framework yielding the so-called curvature adjusted tangential velocity has been proposed byŠevčovič and Yazaki in [27].…”

Section: Planar Motion Lawmentioning

confidence: 99%

“…We refer the reader to papers [10,4,5] by Deckelnick and Beneš et al where the basics of this approach and its applications are elaborated. In [24] Pauš et al compared the direct approach and other interface-capturing methods, such as the level-set method or the phase-field method. In the paper [24] by Pauš et al, the approximate algorithmic approach for handling topological changes (like self-intersecting) was proposed and analyzed.…”

mentioning

confidence: 99%

“…In [24] Pauš et al compared the direct approach and other interface-capturing methods, such as the level-set method or the phase-field method. In the paper [24] by Pauš et al, the approximate algorithmic approach for handling topological changes (like self-intersecting) was proposed and analyzed. Concerning some of the drawbacks of the direct method, the problem of tangential redistribution was analyzed in [27] byŠevčovič and Yazaki.…”

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confidence: 99%

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Area preserving geodesic curvature driven flow of closed curves on a surface

Kolář¹,

Beneš²,

Ševčovič³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.2010 Mathematics Subject Classification. Primary: 35K57, 35K65, 65N40, 65M08; Secondary: 53C80.

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Dynamics of dislocations described as evolving curves interacting with obstacles

Cited by 10 publications

References 57 publications

Curvature driven flow of a family of interacting curves with applications

Curvature driven flow of a family of interacting curves with applications

Modeling of Double Cross-Slip by Means of Geodesic Curvature Driven Flow

Area preserving geodesic curvature driven flow of closed curves on a surface

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