In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof relies on energy relaxation methods. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.
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Keywords: Disclinations, discrete to continuum limits, non-linear elasticity, relaxation, Gamma-convergence
@article{COCV_2021__27_1_A25_0,
author = {Cesana, Pierluigi and Van Meurs, Patrick},
title = {Discrete-to-continuum limits of planar disclinations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021025},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021025/}
}
TY - JOUR AU - Cesana, Pierluigi AU - Van Meurs, Patrick TI - Discrete-to-continuum limits of planar disclinations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021025/ DO - 10.1051/cocv/2021025 LA - en ID - COCV_2021__27_1_A25_0 ER -
%0 Journal Article %A Cesana, Pierluigi %A Van Meurs, Patrick %T Discrete-to-continuum limits of planar disclinations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021025/ %R 10.1051/cocv/2021025 %G en %F COCV_2021__27_1_A25_0
Cesana, Pierluigi; Van Meurs, Patrick. Discrete-to-continuum limits of planar disclinations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 23. doi: 10.1051/cocv/2021025
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