June  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

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using $\Gamma$-convergence.
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), 267-297. doi: 10.1007/s00526-008-0159-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. L. Ericksen, On the Cauchy-Born rule, Math. Mech. Solids, 13 (2008), 199-220. doi: 10.1177/1081286507086898.  Google Scholar

[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.  Google Scholar

[16]

A. K. Geim and A. H. MacDonald, Graphene: Exploring carbon flatland, Physics Today, 60 (2007), 35-41. doi: 10.1063/1.2774096.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152. doi: 10.1007/BF02417888.  Google Scholar

[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.  Google Scholar

[21]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.  Google Scholar

[22]

G. Odegard, Equivalent-continuum modeling of nanostructured materials, ChemInform, 38 (2007). doi: 10.1002/chin.200723218.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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), 267-297. doi: 10.1007/s00526-008-0159-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. L. Ericksen, On the Cauchy-Born rule, Math. Mech. Solids, 13 (2008), 199-220. doi: 10.1177/1081286507086898.  Google Scholar

[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.  Google Scholar

[16]

A. K. Geim and A. H. MacDonald, Graphene: Exploring carbon flatland, Physics Today, 60 (2007), 35-41. doi: 10.1063/1.2774096.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152. doi: 10.1007/BF02417888.  Google Scholar

[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.  Google Scholar

[21]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.  Google Scholar

[22]

G. Odegard, Equivalent-continuum modeling of nanostructured materials, ChemInform, 38 (2007). doi: 10.1002/chin.200723218.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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