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On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth
1. | Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany, Germany |
References:
[1] |
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. |
[2] |
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. |
[3] |
R. Alicandro, M. Focardi and M. S. Gelli, Finite-difference approximation of energies in fracture mechanics, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 671-709. |
[4] |
X. Blanc, C. L. 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. |
[5] |
X. Blanc, C. L. and P.-L. Lions, Atomistic to continuum limits for computational materials science, Math. Model. Numer. Anal., 41 (2007), 391-426.
doi: 10.1051/m2an:2007018. |
[6] |
A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321. |
[7] |
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 1998. |
[8] |
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. (JEMS), 8 (2006), 515-539.
doi: 10.4171/JEMS/65. |
[9] |
B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008. |
[10] |
G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[11] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces, Springer, New York, 2007. |
[12] |
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. |
[13] |
S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192. |
[14] |
N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions, Math. Models Methods Appl. Sci., 22 (2012), 21pp.
doi: 10.1142/S0218202512500327. |
[15] |
S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212.
doi: 10.1007/BF00284506. |
[16] |
B. Schmidt, On the derivation of linear elasticity from atomistic models, Networks and Heterogeneous Media, 4 (2009), 789-812.
doi: 10.3934/nhm.2009.4.789. |
[17] |
B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Rational Mech. Anal., 190 (2008), 1-55.
doi: 10.1007/s00205-008-0138-0. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
R. Alicandro, M. Focardi and M. S. Gelli, Finite-difference approximation of energies in fracture mechanics, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 671-709. |
[4] |
X. Blanc, C. L. 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. |
[5] |
X. Blanc, C. L. and P.-L. Lions, Atomistic to continuum limits for computational materials science, Math. Model. Numer. Anal., 41 (2007), 391-426.
doi: 10.1051/m2an:2007018. |
[6] |
A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321. |
[7] |
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 1998. |
[8] |
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. (JEMS), 8 (2006), 515-539.
doi: 10.4171/JEMS/65. |
[9] |
B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008. |
[10] |
G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[11] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces, Springer, New York, 2007. |
[12] |
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. |
[13] |
S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192. |
[14] |
N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions, Math. Models Methods Appl. Sci., 22 (2012), 21pp.
doi: 10.1142/S0218202512500327. |
[15] |
S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212.
doi: 10.1007/BF00284506. |
[16] |
B. Schmidt, On the derivation of linear elasticity from atomistic models, Networks and Heterogeneous Media, 4 (2009), 789-812.
doi: 10.3934/nhm.2009.4.789. |
[17] |
B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Rational Mech. Anal., 190 (2008), 1-55.
doi: 10.1007/s00205-008-0138-0. |
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