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Double dispersion equation for nonlinear waves in a graphene-type hexagonal lattice. (English) Zbl 1524.74290

Summary: It is shown that plane longitudinal nonlinear strain waves in a 2D graphene-type hexagonal lattice are described by a nonlinear double dispersion equation previously developed for the description of waves in an elastic rod. A procedure is developed to derive the governing equation as a continuum limit of the original lattice model. The lattice is described by an interaction of two sub-lattices and both translational and angular interactions between the lattice masses are taken into account.

MSC:

74J30 Nonlinear waves in solid mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Samsonov, A. M., Strain Soltons in Solids and How to Construct Them (2001), Chapman & Hall/CRC · Zbl 0983.74004
[2] Dreiden, G. V., Experiments in the propagation of longitudinal strain solitons in a nonlinearly elastic rod, Tech. Phys. Lett., 21, 415-417 (1995)
[3] Samsonov, A. M., On existence of longitudinal strain solitons in an infinite nonlinearly elastic rod, Sov. Phys.-Dokl., 33, 298-300 (1988) · Zbl 0671.73026
[4] Porubov, A. V.; Samsonov, A. M., Refinement of the model for the propagation of longitudinal strain waves in a rod with nonlinear elasticity, Tech. Phys. Lett., 19, 365-366 (1993)
[5] Erofeyev, V. I.; Klyueva, N. V., Solitons and nonlinear periodic strain waves in rods, plates, and shells (A review), Acoust. Phys., 48, 6, 643-655 (2002)
[6] Khusnutdinova, K.; Samsonov, A. M.; Zakharov, A. S., Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures, Phys. Rev. E, 79, Article 056606 pp. (2009)
[7] Born, M.; Huang, K., Dynamic Theory of Crystal Lattices (1954), Clarendon Press: Clarendon Press Oxford · Zbl 0057.44601
[8] Manevich, A. I.; Manevitch, L. I., The Mechanics of Nonlinear Systems with Internal Resonances (2005), Imperial College Press: Imperial College Press London · Zbl 1082.70001
[9] Ostoja-Starzewski, M., Lattice models in micromechanics, Appl. Mech. Rev., 55, 1, 35-60 (2002) · Zbl 1110.74611
[10] Askar, A., Lattice Dynamical Foundations of Continuum Theories (1985), World Scientific: World Scientific Singapore
[11] Maugin, G. A., Nonlinear Waves in Elastic Crystals (1999), Oxford University Press: Oxford University Press UK · Zbl 0943.74002
[12] Andrianov, I. V.; Awrejcewicz, J.; Weichert, D., Improved continuous models for discrete media, Math. Probl. Eng. (Open Access), Article 986242 pp. (2010) · Zbl 1191.76102
[13] Kosevich, A. M.; Savotchenko, S. E., Peculiarities of dynamics of one-dimensional discrete systems with interaction extending beyond nearest neighbors, and the role of higher dispersion in soliton dynamics, Low Temp. Phys., 25, 550-557 (1999)
[14] Kirkwood, J. G., The skeletal modes of vibration of long chain molecules, Chem. Eng. J., 7, 506-509 (1939)
[15] Belomestnykh, V. N.; Soboleva, E. G., Lateral strain ratios for cubic ionic crystals, Lett. Mater., 1, 84-87 (2011)
[16] Erofeev, V. I.; Pavlov, I. S., Parametric identification of crystals having a cubic lattice with negative Poissons ratios, J. Appl. Mech. Tech. Phys., 56, 1015-1022 (2015) · Zbl 1381.74043
[17] Askes, H.; Metrikine, A., Higher-order continua derived from discrete media: Continualisation aspects and boundary conditions, Int. J. Solids Struct., 42, 187-202 (2005) · Zbl 1111.74002
[18] Metrikine, A. V.; Askes, H., An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice, Phil. Mag., 86, 3259-3286 (2006)
[19] Vasiliev, A. A.; Dmitriev, S. V.; Miroshnichenko, A. E., Mutlti-field continuum theory for medium with microscopic rotations, Intern. J. Solids Struct., 42, 6245-6260 (2005) · Zbl 1119.74340
[20] Brenner, D. W., Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films, Phys. Rev. B, 42, 15, 9458-9471 (1990)
[21] Pavlov, I. S.; Potapov, A. I.; Maugin, G. A., A 2D granular medium with rotating particles, Intern. J. Solids Struct., 43, 6194-6207 (2006) · Zbl 1120.74410
[22] Vasiliev, A., Elastic properties of a two-dimensional model of crystals containing particles with rotational degrees of freedom, Phys. Rev. E, 65, Article 094101 pp. (2002)
[23] Berinskii, I.; Krivtsov, A., On using many-particle interatomic potentials to compute elastic properties of graphene and diamond, Mech. Solids, 45, 6, 815-834 (2010)
[24] Zhang, P., The elastic modulus of single-wall carbon nanotubes: A continuum analysis incorporating interatomic potentials, Intern. J. Solids Struct., 39, 3893-3906 (2002) · Zbl 1049.74753
[25] Huang, Y.; Wu, J.; Hwang, K. C., Thickness of graphene and single-wall carbon nanotubes, Phys. Rev. B, 74, Article 245413 pp. (2006)
[26] Falkovsky, L. A., Symmetry constraints on phonon dispersion in graphene, Phys. Lett. A, 372, 31, 5189-5192 (2008) · Zbl 1221.82171
[27] Tovstik, P. E.; Tovstik, T. P., Static and dynamic analysis of two-dimensional graphite lattices, Mech. Solids, 47, 517-524 (2012)
[28] Ansari, R.; Sahmani, S.; Arash, B., Nonlocal plate model for free vibrations of single-layered graphene sheets, Phys. Lett. A, 375, 53-62 (2010)
[29] Jiang, Jin-Wu; Chang, T.; Guo, X.; Park, H. S., Intrinsic negative Poisson’s ratio for single-layer graphene, Nano Lett., 16, 8, 5286-5290 (2016)
[30] Qin, Zh; Qin, G.; Hu, M., Origin of anisotropic negative Poisson’s ratio in graphene, Nanoscale, 10, 10365-10370 (2018)
[31] Pedrielli, A., Designing graphene based nanofoams with nonlinear auxetic and anisotropic mechanical properties under tension or compression, Carbon, 111, 796-806 (2017)
[32] Shodja, H. M.; Delfani, M. R., A novel nonlinear constitutive relation for graphene and its consequence for developing closed-form expressions for young’s modulus and critical buckling strain of single-walled carbon nanotubes, Acta Mech., 222, 91-101 (2011) · Zbl 1271.74016
[33] Davydov, S. Yu, Third-order elastic moduli of single-layer graphene, Phys. Solid State, 53, 3, 665-668 (2011)
[34] Wei, X., Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description, Phys. Rev. B, 80, Article 205407 pp. (2009)
[35] Wang, R.; Wang, S.; Wu, X.; Liang, X., First-principles calculations on third-order elastic constants and internal relaxation for monolayer graphene, Physica B, 405, 3501-3506 (2010)
[36] Savin, A. V.; Savina, O. I., Nonlinear dynamics of carbon molecular lattices: Soliton plane waves in graphite layers and supersonic acoustic solitons in nanotubes, Phys. Solid State, 46, 2, 383-391 (2004)
[37] Barani, E., Transverse discrete breathers in unstrained graphene, Eur. Phys. J. B, 90, 38 (2017)
[38] Astakhova, T. Yu; Gurin, O. D.; Menon, M.; Vinogradov, G. A., Longitudinal solitons in carbon nanotubes, Phys. Rev. B, 64, Article 035418 pp. (2001)
[39] Porubov, A.; Berinskii, I., Non-linear plane waves in materials having hexagonal internal structure, Int. J. Non-Linear Mech., 67, 27-33 (2014)
[40] Porubov, A. V.; Krivtsov, A. M.; Osokina, A. E., Two-dimensional waves in extended square lattice, Int. J. Non-Linear Mech., 99, 281-287 (2018)
[41] Porubov, A. V.; Pastrone, F., Nonlinear bell-shaped and kink-shaped strain waves in microstructured solids, Intern. J. Non-Linear Mech., 39, 8, 1289-1299 (2004) · Zbl 1348.74273
[42] Ahmadpoor, F.; Wang, P.; Huang, R.; Sharma, P., Thermal fluctuations and effective bending stiffness of elastic thin sheets and graphene: A nonlinear analysis, J. Mech. Phys. Solids, 107, 294-319 (2017)
[43] Genoese, A.; Genoese, A.; Rizzi, N. L.; Salerno, G., On the derivation of the elastic properties of lattice nanostructures: The case of graphene sheets, Composites B, 115, 316-329 (2017)
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