Stochastic Peierls-Nabarro Model for Dislocations in High Entropy Alloys

Jiang, Tianpeng; Xiang, Yang; Zhang, Luchan

doi:10.48550/arxiv.2004.09375

Cited by 1 publication

(1 citation statement)

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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 [3,20]. Understanding this model provides insights on designing new materials with robust performance [17,15,4,11]. However, the existence and rigidity problem regarding the vector-field PN model is still lack of exploration.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Gao,

Liu,

Liu

2020

Preprint

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We focus on 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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