Citation: |
[1] |
M. Arroyo and T. Belytschko, Nonlinear mechanical response and rippling of thick multiwalled Carbon nanotubes, Phys. Rev. Lett., 91 (2003), 215505. |
[2] |
M. Arroyo and T. Belytschko, Finite element methods for the nonlinear mechanics of crystalline sheets and nanotubes, Int. J. Numer. Methods Eng., 59 (2004), 419-456.doi: 10.1002/nme.944. |
[3] |
X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), 341-381.doi: 10.1007/s00205-002-0218-5. |
[4] |
M. Born, Thermodynamics of crystals and melting, J. Chem. Phys., 7 (1939), 591-603. |
[5] |
A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case, Arch. Ration. Mech. Anal., 146 (1999), 23-58.doi: 10.1007/s002050050135. |
[6] |
E. Cancès and M. Lewin, The dielectric permittivity of crystals in the reduced Hartree-Fock approximation, Arch. Ration. Mech. Anal., 197 (2010), 139-177.doi: 10.1007/s00205-009-0275-0. |
[7] |
E. Cancès and G. Stoltz, A mathematical formulation of the random phase approximation for crystals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 887-925.doi: 10.1016/j.anihpc.2012.05.004. |
[8] |
A.-L. Cauchy, Sur l'equilibre et le mouvement d'un système de points materiels sollicités par forces d'attraction ou de répulsion mutuelle, Ex. de Math., 3 (1828), 277-287. |
[9] |
A.-L. Cauchy, De la pression on tension dans un système de points matériels, Ex. de Math., 3 (1828), 253-277. |
[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), 515-530.doi: 10.4171/JEMS/65. |
[11] |
W. E and J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born rule, Arch. Ration. Mech. Anal., 199 (2011), 407-433.doi: 10.1007/s00205-010-0339-1. |
[12] |
W. E and J. Lu, Stability and the continuum limit of the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model, J. Math. Phys., 53 (2012), 115615. |
[13] |
W. E and J. Lu, The Kohn-Sham equation for deformed crystals, Mem. Amer. Math. Soc., 221 (2013), 1040.doi: 10.1090/S0065-9266-2012-00659-9. |
[14] |
W. E, J. Lu and X. Yang, Effective Maxwell equations from time-dependent density functional theory, Acta Math. Sin. (Eng. Ser.), 27 (2011), 339-368.doi: 10.1007/s10114-011-0555-0. |
[15] |
W. E and P. B. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), 241-297.doi: 10.1007/s00205-006-0031-7. |
[16] |
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. |
[17] |
P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. (2), 136 (1964), B864-B871.doi: 10.1103/PhysRev.136.B864. |
[18] |
W. Kohn and L. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), A1133-A1138.doi: 10.1103/PhysRev.140.A1133. |
[19] |
F. A. Lindemann, Uber die Berechnung Molecular Eigenfrequnzen, Physik. Z., 11 (1910), 609-612. |
[20] |
M. P. Marder, Condensed Matter Physics, Second edition, Wiley, 2010. |
[21] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Second edition, Academic Press, Inc., New York, 1980. |
[22] |
E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A, 73 (1996), 1529-1563. |
[23] |
J. Z. Yang and W. E, Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models, Phys. Rev. B, 74 (2006), 184110. |