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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 |
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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