On the Motion of Curved Dislocations in Three Dimensions: Simplified Linearized Elasticity

Fonseca, Irene; Ginster, Janusz; Wojtowytsch, Stephan

doi:10.1137/20m1325654

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…In materials science, the Peierls-Nabarro (PN) model with Poisson ratio ν ∈ [−1, 1/2] plays a fundamental role in describing dislocations or line defects in materials [6,27]. Understanding this model provides insights on designing new materials with robust performance [8,16,21,24]. However, the existence and rigidity problem regarding the vector-field PN model has not been explored.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions

Gao

Liu

2021

Nonlinearity

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We focus on the existence and rigidity problems of the vectorial Peierls–Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg–Landau equation with an anisotropic positive (if Poisson ratio belongs to (−1/2, 1/3)) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from H 1/2 regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In particular, u 1 in (1. 16) is a layer solution (see definition 3) since it is strictly monotonic in x direction and satisfies assumption (1.9). In fact, (1.16) is a good candidate for minimizers of total energy (1.1) in the sense of definition 1.…”

Section: The Vectorial Peierls-nabarro Model and Its Reduced Scalar E...mentioning

confidence: 99%

Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions

Gao

Liu

2021

Nonlinearity

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“…Optimal scaling methods were pioneered by Kohn and Müller [31] as part of their seminal work on branched structures in martensite, and have been since successfully applied to a number of related problems, including shape-memory alloys, micromagnetics, crystal plasticity, and others [31,32,4,5,7]. We also note that, whereas the line tension approximation pervades the better part of physical metallurgy, (cf., e. g., [27,29]), rigorous results showing that line tension indeed describes the energy of sufficiently dilute dislocations, or, equivalently, dislocations of sufficiently small core radius, have only recently become available [26,6,22].…”

Section: Introductionmentioning

confidence: 99%

A Proof of Taylor Scaling for Curvature-Driven Dislocation Motion Through Random Arrays of Obstacles

Courte

Dondl

Ortiz

2022

Arch Rational Mech Anal

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We prove Taylor scaling for dislocation lines characterized by linetension and moving by curvature under the action of an applied shear stress in a plane containing a random array of obstacles. Specifically, we show-in the sense of optimal scaling-that the critical applied shear stress for yielding, or percolation-like unbounded motion of the dislocation, scales in proportion to the square root of the obstacle density. For sufficiently small obstacle densities, Taylor scaling dominates the linear-scaling that results from purely energetic considerations and, therefore, characterizes the dominant rate-limiting mechanism in that regime.

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Renormalized Energy of a Dislocation Loop in a 3D Anisotropic Body

Šilhavý

2023

J Elast

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The paper presents a rigorous analysis of the singularities of elastic fields near a dislocation loop in a body of arbitrary material symmetry that extends over the entire three-space. Explicit asymptotic formulas are given for the stress, strain and the incompatible distortion near the curved dislocation. These formulas are used to analyze the main object of the paper, the renormalized energy. The core-cutoff method is used to introduce that notion: first, a core in the form of a curved tube along the dislocation loop is removed; then, the energy of the complement is determined (= the core-cutoff energy). As in the case of a straight dislocation, the core-cutoff energy has a singularity that is proportional to the logarithm of the core radius. The renormalized energy is the limit, as the radius tends to 0, of the core-cutoff energy minus the singular logarithmic part. The main result of the paper are novel formulas for the coefficient of logarithmic singularity (the ‘prelogarithmic energy factor’) and for the renormalized energy.

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On the Motion of Curved Dislocations in Three Dimensions: Simplified Linearized Elasticity

Cited by 4 publications

References 19 publications

Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions

Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions

A Proof of Taylor Scaling for Curvature-Driven Dislocation Motion Through Random Arrays of Obstacles

Renormalized Energy of a Dislocation Loop in a 3D Anisotropic Body

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