Non-local Interfacial Energies in Grain Boundaries via Dirichlet-to-Neumann Map and Energetic scaling
Abstract
This paper deals with the variational analysis of edge dislocations arising in small-angle grain boundaries (Read--Shockley) and semi-coherent interfaces (van der Merwe). Using the Nabarro--Peierls approach to model edge dislocations, we reduce the two-dimensional isotropic Navier--Lamé system governing the linearized elastic energy to a one-dimensional non-local energy on the slip interface through a Dirichlet-to-Neumann map. We prove that the resulting boundary energy coincides, up to a multiplicative constant, with the fractional H1/2-Gagliardo seminorm of the displacement jump along the interface. Within this framework, we establish the existence of a unique global minimizer by combining our dimensional-reduction result with the characterization of De Luca, Goldman, and Ponsiglione. Indeed, we show that the optimal defect configuration is uniquely attained, up to translations, by an equispaced periodic array. Finally, we provide a physical rescaling that recovers the classical logarithmic energy regimes of both the Read--Shockley and van der Merwe models.
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BibTeX
@misc{corica2026nonlocal,
title = {{Non-local Interfacial Energies in Grain Boundaries via Dirichlet-to-Neumann Map and Energetic scaling}},
author = {Valerio Corica and Francesco Santilli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35133},
eprint = {2610.35133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35133},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-local Interfacial Energies in Grain Boundaries via Dirichlet-to-Neumann Map and Energetic scaling
AU - Valerio Corica
AU - Francesco Santilli
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35133
UR - https://arxiv.org/abs/2610.35133
AB - This paper deals with the variational analysis of edge dislocations arising in small-angle grain boundaries (Read--Shockley) and semi-coherent interfaces (van der Merwe). Using the Nabarro--Peierls approach to model edge dislocations, we reduce the two-dimensional isotropic Navier--Lamé system governing the linearized elastic energy to a one-dimensional non-local energy on the slip interface through a Dirichlet-to-Neumann map. We prove that the resulting boundary energy coincides, up to a multiplicative constant, with the fractional H^1/2-Gagliardo seminorm of the displacement jump along the interface. Within this framework, we establish the existence of a unique global minimizer by combining our dimensional-reduction result with the characterization of De Luca, Goldman, and Ponsiglione. Indeed, we show that the optimal defect configuration is uniquely attained, up to translations, by an equispaced periodic array. Finally, we provide a physical rescaling that recovers the classical logarithmic energy regimes of both the Read--Shockley and van der Merwe models.
ER -