Stability of the plane stressed state of the graphene sheet based on the moment-membrane theory of elastic plates

Мұқаба

Дәйексөз келтіру

Толық мәтін

Ашық рұқсат Ашық рұқсат
Рұқсат жабық Рұқсат берілді
Рұқсат жабық Тек жазылушылар үшін

Аннотация

Two-dimensional nanomaterials (graphene, carbon nanotube) are high-strength and ultra-light materials that have several promising areas of application. From theoretical and applied perspectives, it is relevant to study various problems of their statics, stability, vibrations, and calculations of the required mechanical characteristics based on the corresponding continuum theory of the deformation behavior of two-dimensional nanomaterials.

In this work, based on the moment-membrane theory of elastic plates, which is interpreted as the continuum theory of the deformation behavior of graphene, stability problems of a freely supported graphene sheet (rectangular plate) are studied. The sheet is uniformly compressed in one direction, compressed in two directions, and subjected to shear stresses in its plane. The stability problem of uniformly compressed graphene sheets, freely supported on two opposite sides and having different boundary conditions on the other two sides, is also considered.

When solving stability problems of the graphene sheet (rectangular plate), the Euler method is applied, considering a form of equilibrium that is slightly deviated from the initial (moment-free) position (buckled plate). Differential equilibrium equations and boundary conditions are formulated for this shape. The critical load value is determined from the solution of these boundary problems, i.e., the load value at which the initial flat form of the plate becomes unstable. All solutions are accompanied by numerical results: tables or diagrams providing the critical load values for each particular case.

Толық мәтін

Рұқсат жабық

Авторлар туралы

А. Sargsyan

State University of Shirak after M. Nalbandyan

Хат алмасуға жауапты Автор.
Email: armenuhis@gmail.com
Армения, Gyumri

S. Sargsyan

State University of Shirak after M. Nalbandyan

Email: s_sargsyan@yahoo.com
Армения, Gyumri

Әдебиет тізімі

  1. Allen M.P., Tildesley D.I. Computer simulation of liquids. Oxford Science Publications, 2000. 385 p.
  2. Kang J.W. et al. Molecular dynamics modeling and simulation of a graphene-based nanoelectromechanical resonator // Curr. Appl. Phys. 2013. V. 13. № 4. P 789–794. https://doi.org/10.1016/j.cap.2012.12.007
  3. Wang J., Li T.T. Molecular dynamics simulation of the resonant frequency of graphene nanoribbons // Ferroelectrics. 2019. V. 549. № 1. P. 87–95. https://doi.org/10.1080/00150193.2019.1592547
  4. Ivanova E.A., Morozov N.F., Semenov B.N., Firsova A.D. On the determination of elastic moduli of nanostructures, theoretical calculation and experimental technique // Izv. RAS. MTT. 2005. № 4. P. 75–84 [in Russian].
  5. Ivanova E.A., Krivtsov A.M., Morozov N.F. Obtaining macroscopic relations of elasticity of complex crystal lattices taking into account moment interactions at the micro level // PMM. 2007. V. 71. № 4. P. 595–615 [in Russian].
  6. Ivanova E.A., Krivtsov A.M., Morozov N.F., Firsova A.D. Description of the crystal packing of particles taking into account moment interactions // Izv. RAS. MTT. 2003. № 4. P. 110–127 [in Russian].
  7. Berinsky I.E., Ivanova E.A., Krivtsov A.M., Morozov N.F. Application of moment interaction to the construction of a stable model of the graphite crystal lattice // Izv. RAS. MTT. 2007. № 5. P. 6–16 [in Russian].
  8. Kuzkin V.A., Krivtsov A.M. Description of mechanical properties of graphene using particles with rotational properties of degrees of freedom // Reports of the Russian Academy of Sciences. 2011. V. 440. № 4. P. 476–479 [in Russian].
  9. Berinsky I.E. et al. Modern problems of mechanics. Mechanical properties of covalent crystals: textbook. Manual / Ed. by A.M. Krivtsov, O.S. Loboda. St. Petersburg: Publishing house of the Polytechnic University, 2014. 160 p. [in Russian].
  10. Ivanova E.A., Krivtsov A.M., Morozov N.F., Firsova A.D. Theoretical Mechanics. Determination of Equivalent Elastic Characteristics of Discrete Systems. St. Petersburg: Publishing House of St. Petersburg State Polytechnical University. 2004. 32 p. [in Russian].
  11. Savinsky S.S., Petrovsky V.A. Discrete and continuous models for calculating the phonon spectra of carbon nanotubes. // Solid State Physics. 2002. V. 44. Iss. 9. P. 1721–1726 [in Russian].
  12. Savin A.V., Kivshar Y.S., Hu B. Suppression of thermal conductivity in graphene nanoribbons with rough edges. // Phys. Rev. B. 2010. V. 82. 195422. https://doi.org/10.1103/PhysRevB.82.195422
  13. Abdullina D.U., Korznikova E.A., Dubinko V.I., Laptev D.V., Kudreyko A.A., Soboleva E.G. et al. Mechanical response of carbon nanotube bundle to lateral compression // Computation. 2020. V. 8. № 2. P. 27. https://doi.org/10.3390/computation8020027
  14. Evazzade I., Lobzenko I.P., Korznikova E.A., Ovid’ko I.A., Roknabadi M.R., Dmitriev S.V. Energy transfer in strained graphene assisted by discrete breathers excited by external ac driving // Phys. Rev. B. 95. 2017. P. 035423. https://doi.org/10.1103/PhysRevB.95.035423
  15. Dmitriev S.V., Sunagatova I.R., Ilgamov M.A., Pavlov I.S. Natural frequencies of bending vibrations of carbon nanotubes. // J. Tech. Phys. 2021. V. 91. Iss. 11. P. 1732–1737. https://doi.org/10.21883/JTF.2021.11.51536.127-21 [in Russian].
  16. Krivtsov A.M. Deformation and destruction of solids with microstructure. M.: Fizmatlit, 2007. 304 p. [in Russian].
  17. Kormilitsin O.P., Shukeilo Yu.A. Mechanics of materials and structures of nano- and microengineering. M.: Publishing Center “Academy”, 2008. 224 p. [in Russian].
  18. Odegard G.M., Gates T.S., Nicholson L.M., Wise K.E. Equivalent-continuum modeling of nano-structured materials // NASA Langley Research Center: Technical Memorandum NASA/TM. 2001. P. 1869–1880.
  19. Goldstein R.V., Chentsov A.V. Discrete-continuous model of a nanotube // Izv. RAS. MTT. 2005. № 4. P. 57–74 [in Russian].
  20. Goldstein R.V., Chentsov A.V. Discrete-continuous model of nanotube deformation. Moscow: IPM RAS, 2003. Preprint № 739. 67 p. [in Russian].
  21. Lisovenko D.S., Gorodtsov V.A. From graphite (rods, plates, shells) to carbon nanotubes. Elastic properties. Moscow: IPM RAS, 2004. Preprint № 747. 67 p. [in Russian].
  22. Goldstein R.V., Gorodtsov V.A., Lisovenko D.S. Mesomechanics of multilayer carbon nanotubes and nanowhiskers // Physical mesomechanics. 2008. V. 11. № 6. P. 25–42 [in Russian].
  23. Li C.A., Chou T.W. A structural mechanics approach for the analysis of carbon nanotubes // Int. J. Solids Struct. 2003. V. 40. № 10. P. 2487–2499. https://doi.org/10.1016/S0020-7683(03)00056-8
  24. Wan H., Delale F. A structural mechanics approach for predicting the mechanical properties of carbon nanotubes // Mechanica. 2010. V. 45. P. 43–51. https://doi.org/10.1007/s11012-009-9222-2
  25. Berinsky I.E., Krivtsov A.M., Kudarova A.M. Determination of the bending rigidity of a graphene sheet // Phys. mesomech. 2014. V. 17. № 1. P. 57–65 [in Russian].
  26. Ustinov K.B., Chentsov A.V. On the deformation of carbon nanoplates: discrete and continuous modeling. M.: IPM. RAS, 2007. Preprint № 824. 31 p. [in Russian].
  27. Annin B.D., Korobeynikov S.N., Babichev A.V. Computer modeling of nanotube buckling under torsion // Sib. J. Ind. Mat. 2008. V. 11. № 1. P. 3–22 [in Russian].
  28. Annin B. D., Baimova Yu. A., Mulyukov R. R. Mechanical properties, stability, warping of graphene sheets in carbon nanotubes (review) // Applied Mechanics and Technical Physics. 2020. Vol. 61. No. 5. P. 175–189. https://doi.org/10.15372/PMTF20200519 [in Russian].
  29. Korobeynikov S. N., Alyokhin V. V., Babichev A. V. Simulation of mechanical parameters of graphene using the DREIDING force field // Acta Mechanica. 2018. V. 229. № 6. P. 2343–2378. https://doi.org/10.1007/s00707-018-2115-5
  30. Korobeynikov S.N., Alyokhin V.V., Babichev A.V. On the molecular mechanics of single layer graphene sheets // Int. J. Eng. Sci. 2018. V. 133. P. 109–131. https://doi.org/10.1016/j.ijengsci.2018.09.001
  31. 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 // Int. J. Mech. Sci. 2021. V. 209. P. 106703. https://doi.org/10.1016/j.ijmecsci.2021.106703
  32. Krivtsov A.M., Morozov N.F. Anomalies of mechanical characteristics of nanosized objects // Reports of the Russian Academy of Sciences. 2001. V. 381. № 3. P. 825–827 [in Russian].
  33. Sargsyan S.O. Rod and continuous-moment models of deformations of two-dimensional nanomaterials // Physical Mesomechanics. 2022. V. 25. № 2. P. 109–121 [in Russian].
  34. Sargsyan S.O. Thin shell model in the moment theory of elasticity with the deformation concept of “shear plus rotation” // Physical Mesomechanics. 2020. V. 23. № 4. P. 13–19. https://doi.org/10.24411/1683-805X-2020-14002 [in Russian].
  35. Sargsyan S.O. Variational principles of the moment-membrane theory of shells // Bulletin of Moscow University. Series 1: Mathematics. Mechanics. 2022. № 1. P. 38–47 [in Russian].
  36. Sargsyan S.H. Moment-membrane theory of elastic cylindrical shells as a continual model of deformation of a single-layer carbon nanotube // Materials Physics and Mechanics. 2024. V. 52. № 1. P. 26–38. https://doi.org/10.18149/MPM.5212024_3
  37. Timoshenko S.P. Stability of elastic systems. M.: Gostekhizdat, 1955. 569 p. [in Russian].
  38. Volmir A.S. Stability of deformable systems. M.: Nauka, 1967. 984 p. [in Russian].

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Жүктеу
2. Fig. 1. Rectangular plate subjected to uniformly distributed compressive forces T11= -P.

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3. Fig. 2. Dependence of coefficient k on the ratio s = a/b for different values of m.

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4. Fig. 3. Rectangular plate subjected to uniformly distributed compressive forces T11 = P1 and T22 = P2.

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5. Fig. 4. Dependence of minimum coefficient k on the ratio of the sides of the plate s.

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6. Fig. 5. Rectangular plate subjected to uniformly distributed tangential forces S12 = S21 = P.

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