Deformation state of a graphene sheet within the framework of the continuum moment-membrane theory of elasticity
DOI:
https://doi.org/10.7242/1999-6691/2024.17.1.4Keywords:
continuum moment-membrane theory of elasticity, plane stress state, transverse bending, nanostructures with graphene, statics and natural vibrations, finite element methodAbstract
The paper proposes an approach to finding the stress-strain state (SSS) of structures containing graphene, a novel nanomaterial that has currently found а wide range of practical applications in nanoelectromechanical systems. Graphene is a 2D basic building block for other carbon structures such as membranes, sheets, nanotubes, etc. To describe the SSS of a graphene sheet, the phenomenological continuum moment-membrane theory of plates is used, from which, due to the fact that graphene is an ultrathin material, the concept of thickness is excluded. The physical elasticity relationships of a graphene sheet are expressed through its rigidity characteristics, determined using the harmonic potential of interatomic interactions in carbon. A differential formulation and the corresponding variational formulation are given for the problem of static deformation and determination of the natural frequencies and modes of vibration of a graphene sheet. The variational formulation is based on the Lagrange principle and is implemented numerically using the finite element method. Finite element relations are constructed taking into account moment effects of the behavior of a graphene sheet. For approximation, a 4-node rectangular finite element is used. Numerical solutions to several problems of static deformation of a graphene sheet under conditions of a plane stress state and transverse bending are presented, and the analysis of its natural vibrations is also performed. Good convergence of numerical simulation results in all considered problems is demonstrated. The obtained numerical solutions are essential in designing and calculating resonators in which ultrathin nanostructures are used. The establishment of the fact that a graphene sheet has a high intrinsic frequency falling in the GHz region (for example, quartz resonators are characterized by megahertz frequencies) opens up new prospects for using graphene itself as an ultrasensitive nanomechanical resonator for detecting small masses and ultrasmall displacements.Downloads
References
Geim A.K., Novoselov K.S. The rise of graphene. Nature Materials. 2007. No. 3. P. 183–191. DOI: 10.1038/nmat1849.
Baimova Y.A., Mulyukov R.R. Grafen, nanotrubki i drugiye uglerodnyye nanostruktury. Moscow: RAN, 2018. 212 p.
Kang J.W., Kim H.-W., Kim K.-S., Lee J.H. Molecular dynamics modeling and simulation of a graphene-based nanoelectromechanical resonator. Current Applied Physics. 2013. Vol. 13, no. 4. P. 789–794. DOI: 10 . 1016 / j . cap . 2012.12.007.
Wang J., Li T.T. Molecular dynamics simulation of the resonant frequency of graphene nanoribbons. Ferroelectrics. 2019. Vol. 549, no. 1. P. 87–95. DOI: 10.1080/00150193.2019.1592547.
Popov A.M. Vychislitelnyye nanotekhnologii. Moscow: KnoRus, 2017. 312 p.
Korobeynikov S.N., Alyokhin V.V., Babichev A.V. Simulation of mechanical parameters of graphene using the DREIDING force field. Acta Mechanica. 2018. Vol. 229, no. 6. P. 2343–2378. DOI: 10.1007/s00707-018-2115-5.
Korobeynikov S.N., Alyokhin V.V., Babichev A.V. On the molecular mechanics of single layer graphene sheets. International Journal of Engineering Science. 2018. Vol. 133. P. 109–131. DOI: 10.1016/j.ijengsci.2018.09.001.
Korobeynikov S.N., Alyokhin V.V., Babichev A.V. Advanced nonlinear buckling analysis of a compressed single layer graphene sheet using the molecular mechanics method. International Journal of Mechanical Sciences. 2021. Vol. 209. 106703 DOI: 10.1016/j.ijmecsci.2021.106703.
Annin B.D., Baimova Y.A., Mulyukov R.R. Mechanical Properties, Stability, and Correction of Graphene Sheets and Carbon Nanotubes (Review). Journal of Applied Mechanics and Technical Physics. 2020. Vol. 61, no. 5. P. 834–846. DOI: 10.1134/ S0021894420050193.
Kvashnin A.G., Sorokin P.B., Kvashnin D.G. Theoretic Investigation of Mechanical Properties of Graphene Membranes by Means of Molecular Mechanics. Journal of Siberian Federal University. Mathematics & Physics. 2009. Vol. 2, no. 4. P. 426–431.
Odegard G.M., Gates T.S., Nicholson L.M., Wise K.E. Equivalent-Continuum Modeling of Nano-structured Materials: Technical Memorandum / NASA Langley Research Center. 2001. NASA/TM-2001-210863–2001.
Li C., Chou T.-W. A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures. 2003. Vol. 40. P. 2487–2499. DOI: 10.1016/S0020-7683(03)00056-8.
Goldshtein R.V., Chentsov A.V. Discrete-continuous model of a nanotube. Mechanics of Solids. 2005. Vol. 40, no. 4. P. 45–59.
Wan H., Delale F. A structural mechanics approach for predicting the mechanical properties of carbon nanotubes. Meccanica. 2009. Vol. 45. P. 43–51. DOI: 10.1007/s11012-009-9222-2.
Berinskii I.E., Krivtsov A.M., Kudarova A.M. Bending stiffness of a graphene sheet. Physical Mesomechanics. 2014. Vol. 17, no. 4. P. 356–364. DOI: 10.1134/S1029959914040134.
Ivanova E.A., Morozov N.F., Semenov B.N., Firsova A.D. Determination of elastic moduli of nanostructures: theoretical estimates and experimental techniques. Mechanics of Solids. 2005. Vol. 40, no. 4. P. 60–68.
Ivanova E.A., Krivtsov A.M., Morozov N.F., Firsova A.D. Inclusion of the moment interaction in the calculation of the flexural rigidity of nanostructures. Doklady Physics. 2003. Vol. 48, no. 8. P. 455–458. DOI: 10.1134/1.1606763.
Ivanova E.A., Krivtsov A.M., Morozov N.F. Derivation of macroscopic relations of the elasticity of complex crystal lattices taking into account the moment interactions at the microlevel. Applied Mathematics and Mechanics. 2007. Vol. 71, no. 4. P. 543–561. DOI: 10.1016/j.jappmathmech.2007.09.009.
Kuzkin V.A., Krivtsov A.M. Description for mechanical properties of graphene using particles with rotational degrees of freedom. Doklady Physics. 2011. Vol. 56, no. 10. P. 527–530. DOI: 10.1134/S102833581110003X.
Sovremennyye problemy mekhaniki. Mekhanicheskiye svoystva kovalentnykh kristallov / ed. by A.M. Krivtsov, O.S. Loboda. St. Petersburg: Publishing house of Politekhn. University, 2014. 160 p.
Sargsyan S.H. Beam and Continuous-Moment Models of Deformation of Two-Dimensional Nanomaterials. Physical Mesomechanics. 2022. Vol. 25, no. 4. P. 373–384. DOI: 10.1134/S1029959922040117.
Sargsyan S.H. A Moment-Elasticity Thin Shell Model for Shear-Plus-Rotation Deformation. Physical Mesomechanics. 2021. Vol. 24, no. 2. P. 140–145. DOI: 10.1134/S102995992102003X.
Sargsyan S.H. Variation Principles of Moment-Membrane Theory of Shells. Moscow University Mechanics Bulletin. 2022. Vol. 77, no. 1. P. 1–11. DOI: 10.3103/S0027133022010046.
Sachio N., Benedict R., Lakes R. Finite element method for orthotropic micropolar elasticity. International Journal of Engineering Science. 1984. Vol. 22, no. 3. P. 319–330. DOI: 10.1016/0020-7225(84)90013-2.
Nakamura S., Lakes R.S. Finite element analysis of stress concentration around a blunt crack in a cosserat elastic solid. Computer Methods in Applied Mechanics and Engineering. 1988. Vol. 66, no. 3. P. 257–266. DOI: 10.1016/0045-7825(88)90001-1.
Korepanov V.V., Matveenko V.P., Shardakov I.N. Numerical study of two-dimensional problems of nonsymmetric elasticity. Mechanics of Solids. 2008. Vol. 43, no. 2. P. 218–224. DOI: 10.3103/S0025654408020064.
Sargsyan S.H. Boundary-value problems of the asymmetric theory of elasticity for thin plates. Journal of Applied Mathematics and Mechanics. 2008. Vol. 72, no. 1. P. 77–86. DOI: 10.1016/j.jappmathmech.2008.03.018.
Sargsyan S.H. The theory of micropolar thin elastic shells. Journal of Applied Mathematics and Mechanics. 2012. Vol. 76, no. 2.
P. 235–249. DOI: 10.1016/j.jappmathmech.2012.05.015.
Palmov V.A. The plane problem in the theory of nonsymmetrical elasticity. Journal of Applied Mathematics and Mechanics. 1964. Vol. 28, no. 6. P. 1341–1345. DOI: 10.1016/0021-8928(64)90046-2.
Bulygin A.N., Kuvshinskii E.V. Plane strain in the asymmetric theory of elasticity. Journal of Applied Mathematics and Mechanics. 1967. Vol. 31, no. 3. P. 569–573. DOI: 10.1016/0021-8928(67)90047-0.
Pelekh B.L. Kontsentratsiya napryazheniy okolo otverstiy pri izgibe transversalno-izotropnykh plastin. Kyiv: Naukova dumka, 1977. 183 p.
Norrie D.H., de Vries J. An Introduction to Finite Element Analisys. Academic Press, 1978. 301 p.
Gallagher R.H. Finite Element Analisys. Fundamentals. Englewood Cliffs, N.J., Prentice-Hall, 1974. 428 p.
Belkin A.E., Gavryushin S.S. Raschet plastin metodom konechnykh elementov. Moscow: Publishing house of MSTU im. N.E. Bauman, 2008. 232 p.
Nowacki W. Theory of Asymmetric Elasticity. Oxford, New York: Pergamon Press, 1986. 383 p.
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