The recent fabrication of weakly interacting incommensurate two-dimensional layer stacks (A. Geim and I. Grigorieva, Nature 499 (2013) 419–425) requires an extension of the classical notion of the Cauchy–Born strain energy density since these atomistic systems are typically not periodic. In this paper, we rigorously formulate and analyze a Cauchy–Born strain energy density for weakly interacting incommensurate one-dimensional lattices (chains) as a large body limit and we give error estimates for its approximation by finite samples as well as the popular supercell method.
Accepté le :
DOI : 10.1051/m2an/2017057
Mots clés : Two-dimensional materials, heterostructures, incommensurability, Cauchy–Born
@article{M2AN_2018__52_2_729_0,
author = {Cazeaux, Paul and Luskin, Mitchell},
title = {Cauchy{\textendash}Born strain energy density for coupled incommensurate elastic chains},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {729--749},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2017057},
zbl = {1416.74011},
mrnumber = {3834441},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017057/}
}
TY - JOUR AU - Cazeaux, Paul AU - Luskin, Mitchell TI - Cauchy–Born strain energy density for coupled incommensurate elastic chains JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 729 EP - 749 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017057/ DO - 10.1051/m2an/2017057 LA - en ID - M2AN_2018__52_2_729_0 ER -
%0 Journal Article %A Cazeaux, Paul %A Luskin, Mitchell %T Cauchy–Born strain energy density for coupled incommensurate elastic chains %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 729-749 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017057/ %R 10.1051/m2an/2017057 %G en %F M2AN_2018__52_2_729_0
Cazeaux, Paul; Luskin, Mitchell. Cauchy–Born strain energy density for coupled incommensurate elastic chains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 729-749. doi : 10.1051/m2an/2017057. http://archive.numdam.org/articles/10.1051/m2an/2017057/
[1] and , An atomistic-based finite deformation membrane for single layer crystalline films. J. Mech. Phys. Solids 50 (2002) 1941–1977. | DOI | MR | Zbl
[2] and , Continuum mechanics modeling and simulation of carbon nanotubes. Meccanica 40 (2005) 455–469. | DOI | MR | Zbl
[3] ,Dynamik der Kristallgitter, Vol. 4. Teubner, Berlin/Leipzig (1915). | JFM
[4] , and , Generalized Kubo formulas for the transport properties of incommensurate 2D atomic heterostructures. J. Math. Phys. 58 (2017) 063502. | DOI | MR | Zbl
[5] and , Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55 (1985) 2471–2474. | DOI
[6] , , , and , The electronic properties of graphene. Rev. Mod. Phys. 81 (2009) 109–162. | DOI
[7] , and , Analysis of rippling in incommensurate one-dimensional coupled chains. Multiscale Model. Simul. 15 (2017) 56–73. | DOI | MR | Zbl
[8] , , and , Sufficient conditions for the validity of the Cauchy–Born rule close to SO(n). J. Eur. Math. Soc. 8 (2006) 515–530. | DOI | MR | Zbl
[9] , An Introduction to Г-Convergence. Springer, Birkhäuser, Boston (1993). | DOI | MR | Zbl
[10] , There is no pointwise consistent quasicontinuum energy. IMA J. Numer. Anal. 34 (2014) 1541–1553. | DOI | MR | Zbl
[11] , On the Cauchy–Born rule. Math. Mech. Solids 13 (2008) 199–220. | DOI | MR | Zbl
[12] and , Validity and failure of the Cauchy–Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12 (2002) 445–478. | DOI | MR | Zbl
[13] and , Van der Waals heterostructures. Nature 499 (2013) 419–425. | DOI
[14] , Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Vol. 49 of Publications Mathématiques de l’Institut des Hautes Études Scientifiques (1979) 5–233. | DOI | Numdam | MR | Zbl
[15] , in Sur les courbes invariantes par les difféomorphismes de l’anneau, With an appendix by Albert Fathi. Vol. 1. Vol. 103 of Astérisque. Société Mathématique de France, Paris (1983). | Numdam | MR | Zbl
[16] , The discrepancy of random sequences {kx}. Acta Arith. 10 (1964) 183–213. | DOI | MR | Zbl
[17] and , Registry-dependent interlayer potential for graphitic systems. Phys. Rev. B 71 (2005) 235415. | DOI
[18] , Averaging differential operators with almost periodic, rapidly oscillating coefficients. Math. USSR-Sbornik 35 (1979) 481–498. | DOI | MR | Zbl
[19] and , Uniform Distribution of Sequences. Wiley-Interscience, New York (1974). | MR | Zbl
[20] ,, and , Analysis of blended atomistic/continuum hybrid methods. Preprint arXiv: (2014). | arXiv | MR
[21] and , Atomistic-to-continuum coupling. Acta Numer. 22 (2013) 397–508. | DOI | MR | Zbl
[22] and , Continuum model for inextensible incommensurate 1-D bilayer with bending and disregistry manuscript (2015).
[23] , and , Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529–1563. | DOI
[24] and , Justification of the Cauchy–Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207 (2013) 1025–1073. | DOI | MR | Zbl
[25] and , Boundary value problems with rapidly oscillating random coefficients, in Rigorous Results in Statistical Mechanics and Quantum Field Theory. Proceedings of Conference on Random Fields, Esztergom, Hungary, 1979, edited by , and . Seria Colloquia Mathematica Societatis Janos Bolyai, 27. North Holland (1981) 835–873. | MR | Zbl
[26] , and , A surface Cauchy-Born model for nanoscale materials. Int. J. Numer. Meth. Eng. 68 (2006) 1072–1095. | DOI | MR | Zbl
[27] , , Cauchy–Born and homogenized stability criteria for a free-standing monolayer graphene at the continuum level. Eur. J. Mech. A/Solids 55 (2016) 134–148. | DOI | MR | Zbl
[28] and , Bilayers of transition metal dichalcogenides: different stackings and heterostructures. J. Mater. Res. 29 (2014) 373–382. | DOI
[29] , , , , , , et al., Perturbation theory for weakly coupled two-dimensional layers. J. Mater. Res. 31 (2016) 959–966. | DOI
[30] , The Cauchy–Born hypothesis, nonlinear elasticity and mechanical twinning in crystals. Acta Crystallogr. Sect. A 52 (1996) 839–849. | DOI
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