Homogenization of hexagonal lattices

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2013, 8(2): 541-572. doi: 10.3934/nhm.2013.8.541

Homogenization of hexagonal lattices

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

Received January 2012 Revised December 2012 Published May 2013

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

Keywords: Cauchy-Born rule., homogenization, $\Gamma$-convergence, Graphene sheet, atomistic-to-continuum.

Mathematics Subject Classification: 74Q05, 74Q15, 74K35, 49J4.

Citation: Hervé Le Dret, Annie Raoult. Homogenization of hexagonal lattices. Networks & Heterogeneous Media, 2013, 8 (2) : 541-572. doi: 10.3934/nhm.2013.8.541

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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-002-0218-5.
[7]	A. Braides and M. S. Gelli, From discrete systems to continuous variational problems: An introduction,, in, 2 (2006), 3. 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. doi: 10.1177/1081286502007001229.
[9]	D. Caillerie, A. Mourad and A. Raoult, Discrete homogenization in graphene sheet modeling,, J. Elast., 84 (2006), 33. 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. doi: 10.4171/JEMS/65.
[11]	B. Dacorogna, "Direct Methods in the Calculus of Variations," Second edition,, Applied Mathematical Sciences, 78 (2008).
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (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. doi: 10.1007/s00205-006-0031-7.
[14]	J. L. Ericksen, On the Cauchy-Born rule,, Math. Mech. Solids, 13 (2008), 199. 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. doi: 10.1007/s00332-002-0495-z.
[16]	A. K. Geim and A. H. MacDonald, Graphene: Exploring carbon flatland,, Physics Today, 60 (2007), 35. 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.
[18]	H. Le Dret and A. Raoult, Homogenization of hexagonal lattices,, C. R. Acad. Sci. Paris, 349 (2011), 111. 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. 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). doi: 10.1142/S0218202512500327.
[21]	S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. 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. 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. doi: 10.1007/s00205-008-0138-0.

show all references

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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-002-0218-5.
[7]	A. Braides and M. S. Gelli, From discrete systems to continuous variational problems: An introduction,, in, 2 (2006), 3. 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. doi: 10.1177/1081286502007001229.
[9]	D. Caillerie, A. Mourad and A. Raoult, Discrete homogenization in graphene sheet modeling,, J. Elast., 84 (2006), 33. 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. doi: 10.4171/JEMS/65.
[11]	B. Dacorogna, "Direct Methods in the Calculus of Variations," Second edition,, Applied Mathematical Sciences, 78 (2008).
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (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. doi: 10.1007/s00205-006-0031-7.
[14]	J. L. Ericksen, On the Cauchy-Born rule,, Math. Mech. Solids, 13 (2008), 199. 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. doi: 10.1007/s00332-002-0495-z.
[16]	A. K. Geim and A. H. MacDonald, Graphene: Exploring carbon flatland,, Physics Today, 60 (2007), 35. 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.
[18]	H. Le Dret and A. Raoult, Homogenization of hexagonal lattices,, C. R. Acad. Sci. Paris, 349 (2011), 111. 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. 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). doi: 10.1142/S0218202512500327.
[21]	S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. 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. 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. doi: 10.1007/s00205-008-0138-0.

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