Homogenization of hexagonal lattices

Hervé Le Dret; Annie Raoult; Hervé Le Dret; Annie Raoult

doi:10.3934/nhm.2013.8.541

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 541-572. doi: 10.3934/nhm.2013.8.541

Previous Article Next Article

Homogenization of hexagonal lattices

Hervé Le Dret ^{1
,},
Annie Raoult ^{2
,}

1.
UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005
2.
Laboratoire MAP5, UMR CNRS 8145, Université Paris Descartes, Paris

Received: 01 January 2012 Revised: 01 December 2012
74Q05, 74Q15, 74K35, 49J45.

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using $\Gamma$-convergence.
- Graphene sheet,
- atomistic-to-continuum,
- homogenization,
- $\Gamma$-convergence,
- Cauchy-Born rule
Citation: Hervé Le Dret, Annie Raoult. Homogenization of hexagonal lattices[J]. Networks and Heterogeneous Media, 2013, 8(2): 541-572. doi: 10.3934/nhm.2013.8.541

Related Papers:

Abstract

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using $\Gamma$-convergence.

References

[1]	R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. Partial Diff. Eq., 33 (2008), 267-297. doi: 10.1007/s00526-008-0159-4
[2]	R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36 (2004), 1-37. doi: 10.1137/S0036141003426471
[3]	R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Rational Mech. Anal., 200 (2011), 881-943. doi: 10.1007/s00205-010-0378-7
[4]	S. Bae, et al., Roll-to-roll production of 30-inch graphene films for transparent electrodes, Nature Nanotechnology, 5 (2010), 574-578. doi: 10.1038/nnano.2010.132
[5]	M. Barchiesi and A. Gloria, New counterexamples to the cell formula in nonconvex homogenization, Arch. Rational Mech. Anal., 195 (2010), 991-1024. doi: 10.1007/s00205-009-0226-9
[6]	X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Rational Mech. Anal., 164 (2002), 341-381. doi: 10.1007/s00205-002-0218-5
[7]	A. Braides and M. S. Gelli, From discrete systems to continuous variational problems: An introduction, in "Topics on Concentration Phenomena and Problems with Multiple Scales" (eds. A. Braides and V. Chiado' Piat), Lecture Notes Unione Mat. Ital., 2, Springer, Berlin, (2006), 3-77. doi: 10.1007/978-3-540-36546-4_1
[8]	A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), 41-66. doi: 10.1177/1081286502007001229
[9]	D. Caillerie, A. Mourad and A. Raoult, Discrete homogenization in graphene sheet modeling, J. Elast., 84 (2006), 33-68. doi: 10.1007/s10659-006-9053-5
[10]	S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$, J. Eur. Math. Soc., 8 (2006), 515-539. doi: 10.4171/JEMS/65
[11]	B. Dacorogna, "Direct Methods in the Calculus of Variations," Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8
[13]	W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Rational Mech. Anal., 183 (2007), 241-297. doi: 10.1007/s00205-006-0031-7
[14]	J. L. Ericksen, On the Cauchy-Born rule, Math. Mech. Solids, 13 (2008), 199-220. doi: 10.1177/1081286507086898
[15]	G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), 445-478. doi: 10.1007/s00332-002-0495-z
[16]	A. K. Geim and A. H. MacDonald, Graphene: Exploring carbon flatland, Physics Today, 60 (2007), 35-41. doi: 10.1063/1.2774096
[17]	H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578.
[18]	H. Le Dret and A. Raoult, Homogenization of hexagonal lattices, C. R. Acad. Sci. Paris, 349 (2011), 111-114. doi: 10.1016/j.crma.2010.12.012
[19]	P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152. doi: 10.1007/BF02417888
[20]	N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three point interactions, Math. Mod. Meth. Appl. Sci., 22 (2012), 21 pp. doi: 10.1142/S0218202512500327
[21]	S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506
[22]	G. Odegard, Equivalent-continuum modeling of nanostructured materials, ChemInform, 38 (2007). doi: 10.1002/chin.200723218
[23]	A. Raoult, D. Caillerie and A. Mourad, Elastic lattices: Equilibrium, invariant laws and homogenization, Ann. Univ. Ferrara Sez. VII Sci. Mat., 54 (2008), 297-318. doi: 10.1007/s11565-008-0054-0
[24]	B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190 (2008), 1-55. doi: 10.1007/s00205-008-0138-0

Reader Comments

Your name:*

Email:*
© 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)