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Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions

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Abstract

In the present study, the free vibration, mechanical buckling and thermal buckling analyses of multi-layered graphene sheets (MLGSs) are investigated. Eringen’s nonlocal elasticity equations are incorporated in new two-variable plate theories accounting for small-scale effects. The MLGSs are taken to be perfectly bonded to the surrounding medium. The governing differential equations of this model are solved analytically under various boundary conditions and taking into account the effect of van der Waals forces between adjacent layers. New functions for the displacements are proposed here to satisfy the different boundary conditions. Comparison of the results with those being in the open literature is made. This comparison illustrates that the present scheme yields very accurate results. Furthermore, the influences of nonlocal coefficient, moduli of the surrounding elastic medium and aspect ratio on the frequencies and buckling of the embedded MLGSs are examined.

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References

  1. Geim A.K., Novoselov K.S.: The rise of graphene. Nat. Mater. 6, 183–191 (2007)

    Google Scholar 

  2. Thostenson E.T., Ren Z.F., Chou T.W.: Advances in the science and technology of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 61, 1899–1912 (2001)

    Article  Google Scholar 

  3. Stankovich S., Dikin D.A., Dommett G.H.B., Kohlhaas K.M., Zimney E.J., Stach E.A. et al.: Graphene-based composite materials. Nature 442, 282–286 (2006)

    Article  Google Scholar 

  4. Behfar K., Naghdabadi R.: Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium. Compos. Sci. Technol. 65, 1159–1164 (2005)

    Article  Google Scholar 

  5. Liew K.M., He X.Q., Kitipornchai S.: Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix. Acta Mater. 54, 4229–4236 (2006)

    Article  Google Scholar 

  6. Pradhan S.C., Phadikar J.K.: Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys. Lett. A 373, 1062–1069 (2009)

    Article  MATH  Google Scholar 

  7. Jomehzadeh E., Saidi A.R.: A study on large amplitude vibration of multilayered graphene sheets. Comput. Mater. Sci. 50, 1043–1051 (2011)

    Article  Google Scholar 

  8. Lin R.M.: Nanoscale vibration characteristics of multi-layered graphene sheets. Mech. Syst. Signal Process. 29, 251–261 (2012)

    Article  Google Scholar 

  9. Lin R.M.: Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium. Comput. Mater. Sci. 53, 44–52 (2012)

    Article  Google Scholar 

  10. Ansari R., Rajabiehfard R., Arash B.: Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput. Mater. Sci. 49, 8318 (2010)

    Article  Google Scholar 

  11. Ansari R., Arash B., Rouhi H.: Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos. Struct. 93, 2419–2429 (2011)

    Article  Google Scholar 

  12. Ansari R., Arash B., Rouhi H.: Nanoscale vibration analysis of embedded multi-layered graphene sheets under various boundary conditions. Comput. Mater. Sci. 53, 44–52 (2012)

    Article  Google Scholar 

  13. Kitipornchai S., He X.Q., Liew K.M.: Continuum model for the vibration of multilayered graphene sheets. Phys. Rev. B 72, 075443 (2005)

    Article  Google Scholar 

  14. He X.Q., Kitipornchai S., Liew K.M.: Resonance analysis of multi-layered graphene sheets used as nanoscale resonators. Nanotechnology 16, 2086–2091 (2005)

    Article  Google Scholar 

  15. Samaei A.T., Abbasion S., Mirsayar M.M.: Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory. Mech. Res. Commun. 38, 481–485 (2011)

    Article  MATH  Google Scholar 

  16. Pradhan S.C., Kumar A.: Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput. Mater. Sci. 50, 239–245 (2010)

    Article  Google Scholar 

  17. Zenkour A.M., Sobhy M.: Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium. Phys. E 53, 251–259 (2013)

    Article  Google Scholar 

  18. Alzahrani E.O., Zenkour A.M., Sobhy M.: Small scale effect on hygro-thermo-mechanical bending of nanoplates embedded in an elastic medium. Compos. Struct. 105, 163–172 (2013)

    Article  Google Scholar 

  19. He X.Q., Kitipornchai S., Liew K.M.: Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J. Mech. Phys. Solids 53, 303–326 (2005)

    Article  MATH  Google Scholar 

  20. Zhang Y.Y., Wang C.M., Tan V.B.C.: Buckling of multiwalled carbon nanotubes using Timoshenko beam theory. J. Eng. Mech. 132, 952 (2006)

    Article  Google Scholar 

  21. Khosrozadeh A., Hajabasi M.A.: Free vibration of embedded double-walled carbon nanotubes considering nonlinear interlayer van der Waals forces. Appl. Math. Model. 36, 997–1007 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang X., Lu G., Lu Y.J.: Buckling of embedded multi-walled carbon nanotubes under combined torsion and axial loading. Int. J. Solids Struct. 44, 336–351 (2007)

    Article  MATH  Google Scholar 

  23. Hsu J.-C., Lee H.-L., Chang W.-J.: Thermal buckling of double-walled carbon nanotubes. J. Appl. Phys. 105, 103512 (2009)

    Article  Google Scholar 

  24. Wang X.Y., Wang X., Chen X.: Thermal buckling of multi-walled carbon nanotubes based on a rigorous van der Waals interaction. J. Therm. Stresses 30, 343–355 (2007)

    Article  Google Scholar 

  25. Li R., Kardomateas G.A.: Thermal buckling of multi-walled carbon nanotubes by nonlocal elasticity. J. Appl. Mech. 74, 399–405 (2007)

    Article  MATH  Google Scholar 

  26. Wang Y.-Z., Cui H.-T., Li F.-M., Kishimoto K.: Thermal buckling of a nanoplate with small-scale effects. Acta Mech. 224, 1299–1307 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sobhy M.: Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium. Phys. E 56, 400–409 (2014)

    Article  Google Scholar 

  28. Shimpi R.P., Ghugal Y.M.: A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams. Compos. Sci. Technol. 61, 1271–1283 (2001)

    Article  Google Scholar 

  29. Reissner E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–191 (1944)

    MATH  MathSciNet  Google Scholar 

  30. Mindlin R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  31. Wang C.M., Lim G.T., Reddy J.N., Lee K.H.: Relationships between bending solutions of Reissner and Mindlin plate theories. Eng. Struct. 23, 838–849 (2001)

    Article  Google Scholar 

  32. Reddy J.N.: A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20, 881– 896 (1984)

    Article  MATH  Google Scholar 

  33. Reddy J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47, 663–684 (2000)

    Article  MATH  Google Scholar 

  34. Touratier M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991)

    Article  MATH  Google Scholar 

  35. Soldatos K.P.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 94, 195–200 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  36. Karama M., Afaq K.S., Mistou S.: Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003)

    Article  MATH  Google Scholar 

  37. Shimpi R.P.: Refined plate theory and its variants. AIAA J. 40, 137–146 (2002)

    Article  Google Scholar 

  38. Shimpi R.P., Patel H.G.: A two variable refined plate theory for orthotropic plate analysis. Int. J. Solids Struct. 43, 6783–6799 (2006)

    Article  MATH  Google Scholar 

  39. Shimpi R.P., Patel H.G.: Free vibrations of plate using two variable refined plate theory. J. Sound Vib. 296, 979–999 (2006)

    Article  Google Scholar 

  40. Benachour A., Tahar H.D., Atmane H.A., Tounsi A., Ahmed M.S.: A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient. Compos. Part B 42, 1386–1394 (2011)

    Article  Google Scholar 

  41. Bourada M., Tounsi A., Houari M.S.A., Bedia E.A.: A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates. J. Sandw. Struct. Mater. 14, 5–33 (2012)

    Article  Google Scholar 

  42. Fekrar A., El Meiche N., Bessaim A., Tounsi A., Adda Bedia E.A.: Buckling analysis of functionally graded hybrid composite plates using a new four variable refined plate theory. Steel Compos. Struct. 13, 91–107 (2012)

    Article  Google Scholar 

  43. Bouiadjra M.B., Houari M.S.A., Tounsi A.: Thermal buckling of functionally graded plates according to a four-variable refined plate theory. J. Therm. Stresses 35, 677–694 (2012)

    Article  Google Scholar 

  44. Kettaf F.Z., Houari M.S.A., Benguediab M., Tounsi A.: Thermal buckling of functionally graded sandwich plates using a new hyperbolic shear displacement model. Steel Compos. Struct 15, 399–423 (2013)

    Article  Google Scholar 

  45. Bouderba B., Houari M.S.A, Tounsi A.: Thermomechanical bending response of FGM thick plates resting on Winkler–Pasternak elastic foundations. Steel Compos. Struct. 14, 85–104 (2013)

    Article  Google Scholar 

  46. Bouiadjra R.B., Adda Bedia E.A., Tounsi A.: Nonlinear thermal buckling behavior of functionally graded plates using an efficient sinusoidal shear deformation theory. Struct. Eng. Mech. 48, 547–567 (2013)

    Article  Google Scholar 

  47. Tounsi A., Houari M.S.A., Benyoucef S., Adda Bedia E.A.: A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp. Sci. Technol. 24, 209–220 (2013)

    Article  Google Scholar 

  48. Narendar S., Gopalakrishnan S.: Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory. Acta Mech. 223, 395–413 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  49. Kim S.E., Thai H.T., Lee J.: A two variable refined plate theory for laminated composite plates. Compos. Struct. 89, 197–205 (2009)

    Article  Google Scholar 

  50. Sobhy M.: Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87 (2013)

    Article  Google Scholar 

  51. Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  52. Zenkour A.M., Sobhy M.: Thermal buckling of various types of FGM sandwich plates. Compos. Struct. 93, 93–102 (2010)

    Article  Google Scholar 

  53. Zenkour A.M., Sobhy M.: Thermal buckling of functionally graded plates resting on elastic foundations using the trigonometric theory. J. Therm. Stresses 34, 1119–1138 (2011

    Article  Google Scholar 

  54. Zenkour A.M., Sobhy M.: Elastic foundation analysis of uniformly loaded functionally graded viscoelastic sandwich plates. J. Mech. 28, 439–452 (2012)

    Article  Google Scholar 

  55. Zenkour A.M., Sobhy M.: Dynamic bending response of thermoelastic functionally graded plates resting on elastic foundations. Aerosp. Sci. Technol. 29, 7–17 (2013)

    Article  Google Scholar 

  56. Brush D.O., Almroth B.O.: Buckling of Bars, Plates and Shells. McGraw Hill, New York (1975)

    MATH  Google Scholar 

  57. Aghababaei R., Reddy J.N.: Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289 (2009)

    Article  Google Scholar 

  58. Pradhan S.C., Murmu T.: Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput. Mater. Sci. 47, 268

    Article  Google Scholar 

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  1. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

    Mohammed Sobhy

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  1. Mohammed Sobhy
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Sobhy, M. Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions. Acta Mech 225, 2521–2538 (2014). https://doi.org/10.1007/s00707-014-1093-5

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  • DOI: https://doi.org/10.1007/s00707-014-1093-5

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