Acta Mechanica

, Volume 223, Issue 2, pp 395–413 | Cite as

Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory



This article presents the buckling analysis of orthotropic nanoplates such as graphene using the two-variable refined plate theory and nonlocal small-scale effects. The two-variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the monolayer graphene are derived from the principle of virtual displacements. The closed-form solution for buckling load of a simply supported rectangular orthotropic nanoplate subjected to in-plane loading has been obtained by using the Navier’s method. Numerical results obtained by the present theory are compared with first-order shear deformation theory for various shear correction factors. It has been proven that the nondimensional buckling load of the orthotropic nanoplate is always smaller than that of the isotropic nanoplate. It is also shown that small-scale effects contribute significantly to the mechanical behavior of orthotropic graphene sheets and cannot be neglected. Further, buckling load decreases with the increase of the nonlocal scale parameter value. The effects of the mode number, compression ratio and aspect ratio on the buckling load of the orthotropic nanoplate are also captured and discussed in detail. The results presented in this work may provide useful guidance for design and development of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Geim A.K., Novoselov K.S.: The rise of graphene. Nat. Mater. 6, 183 (2007)CrossRefGoogle Scholar
  2. 2.
    Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubonos S.V., Grigorieva I.V., Firsov A.A.: Electric field effect in atomically thin carbon films. Science 306, 666 (2004)CrossRefGoogle Scholar
  3. 3.
    Ohta T., Bostwick A., Seyller T., Horn K., Rotenberg E.: Controlling the electronic structure of bilayer graphene. Science 313, 951 (2006)CrossRefGoogle Scholar
  4. 4.
    Oshima C., Nagashima A.: Ultra-thin epitaxial films of graphite and hexagonal boron nitride on solid surfaces. J. Phys. Condens. Matter 9, 1 (1997)CrossRefGoogle Scholar
  5. 5.
    Obraztsov A.N., Obraztsova E.A., Tyurnina A.V., Zolotukhin A.A.: Chemical vapor deposition of thin graphite films of nanometer thickness. Carbon 45, 2017 (2007)CrossRefGoogle Scholar
  6. 6.
    Gomez-Navarro C., Weitz R.T., Bittner A.M., Scolari M., Mews A., Burghard M., Kern K.: Electronic transport properties of individual chemically reduced graphene oxide sheets. Nano Lett. 7, 3499 (2007)CrossRefGoogle Scholar
  7. 7.
    Li X.L., Wang X.R., Zhang L., Lee S.W., Dai H.J.: Chemically derived, ultrasmooth graphene nanoribbon semiconductors. Science 319, 1229 (2008)CrossRefGoogle Scholar
  8. 8.
    Stankovich S. et al.: Graphene-based composite materials. Nature 442, 282 (2006)CrossRefGoogle Scholar
  9. 9.
    Stankovich S. et al.: Synthesis of graphene-based nanosheets via chemical reduction of exfoliated graphite oxide. Carbon 7, 1558–1565 (2007)CrossRefGoogle Scholar
  10. 10.
    Ferrari A.C.: Raman spectroscopy of graphene and graphite: disorder, electronphonon coupling, doping and nonadiabatic effects. Solid State Commun. 143, 47–57 (2007)CrossRefGoogle Scholar
  11. 11.
    Katsnelson M.I., Novoselov K.S.: Graphene: new bridge between condensed matter physics and quantum electrodynamics. Solid State Commun. 143, 3–13 (2007)CrossRefGoogle Scholar
  12. 12.
    Meyer C.J. et al.: The structure of suspended graphene sheets. Nature 446, 60 (2006)CrossRefGoogle Scholar
  13. 13.
    Bunch J., van der Zande A.M., Scott S.V., Ian W.F., David M.T., Jeevak M.P., Harold G.C., Paul L.M.E: Electromechanical resonators from graphene sheets. Science 315, 490–493 (2007)CrossRefGoogle Scholar
  14. 14.
    Ball P.: Roll up for the revolution. Nature (London) 414, 142 (2001)CrossRefGoogle Scholar
  15. 15.
    Baughman R.H., Zakhidov A.A., de Heer W.A.: Carbon nanotubes the route towards applications. Science 297, 787 (2002)CrossRefGoogle Scholar
  16. 16.
    Bodily B.H., Sun C.T.: Structural and equivalent continuum properties of single-walled carbon nanotubes. Int. J. Mat. Prod. Tech. 18, 381 (2003)Google Scholar
  17. 17.
    Li C., Chou T.W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487 (2003)MATHCrossRefGoogle Scholar
  18. 18.
    Li C., Chou T.W.: Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys. Rev. B 68, 073405 (2003)CrossRefGoogle Scholar
  19. 19.
    Sharma P., Ganti S., Bhate N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535 (2003)CrossRefGoogle Scholar
  20. 20.
    Sun C.T., Zhang H.: Size-dependent elastic moduli of platelike nanomaterials. J. Appl. Phys. 93, 1212 (2003)CrossRefGoogle Scholar
  21. 21.
    Sheehan P.E., Lieber C.M.: Nanotribology and nanofabrication of MoO3 structures by atomic force microscopy. Science 272, 1156 (1996)CrossRefGoogle Scholar
  22. 22.
    Yakobson B.I., Smalley R.: Fullerene nanotubes: C1,000,000 and beyond. Am. Sci. 85, 324 (1997)Google Scholar
  23. 23.
    Terrones M., Grobert N., Hsu W., Hu Y., Terrones J., Kroto H., Ealton D.: Bulk glass-forming metallic alloys: science and technology. Mater. Res. Bull. 24, 43 (1999)Google Scholar
  24. 24.
    Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)CrossRefGoogle Scholar
  25. 25.
    Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York, NY (2002)MATHGoogle Scholar
  26. 26.
    Eringen A.C., Edelen D.G.B.: On non-local elasticity. Int. J. Eng. Sci. 10, 233 (1972)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Eringen A.C.: Linear theory of non-local elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425 (1972)MATHCrossRefGoogle Scholar
  28. 28.
    Eringen A.C.: Non-local Polar Field Models. Academic, New York (1996)Google Scholar
  29. 29.
    Lu P., Lee H.P., Lu C., Zhang P.Q.: Dynamic properties of flexural beams using a non-local elasticity model. J. Appl. Phys. 99, 073510 (2006)CrossRefGoogle Scholar
  30. 30.
    Chen Y., Lee J.D., Eskandarian A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41, 2085–2097 (2004)MATHCrossRefGoogle Scholar
  31. 31.
    Peddieson J., Buchanan G.R., McNitt R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305 (2003)CrossRefGoogle Scholar
  32. 32.
    Lazar M., Maugin G., Aifantis E.C.: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43, 1404–1421 (2006)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Zhou S.J., Li Z.Q.: Length scales in the static and dynamic torsion of a circular cylindrical micro-bar. J. Shandong Univ. Technol. 31, 401–407 (2001)Google Scholar
  34. 34.
    Fleck N.A., Hutchinson J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 296–358 (1997)Google Scholar
  35. 35.
    Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)MATHCrossRefGoogle Scholar
  36. 36.
    Ozer T.: On the symmetry group properties of equations of nonlocal elasticity. Mech. Res. Commun. 26, 725–733 (1999)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Murmu T., Pradhan S.C.: Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory. J. Appl. Phys. 105, 064319 (2009)CrossRefGoogle Scholar
  38. 38.
    Murmu T., Pradhan S.C.: Buckling of biaxially compressed orthotropic plates at small scales. Mech. Res. Commun. 36, 933–938 (2009)CrossRefGoogle Scholar
  39. 39.
    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 (2009)CrossRefGoogle Scholar
  40. 40.
    Duan W.H., Wang C.M.: Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology 18, 385704 (2007)CrossRefGoogle Scholar
  41. 41.
    Sakhaee-Pour A.: Elastic buckling of single-layered graphene sheet. Comput. Mater. Sci. 45, 266–270 (2009)CrossRefGoogle Scholar
  42. 42.
    Aydogdu M.: Axial vibration of the nanorods with the nonlocal continuum rod model. Phys. E 41, 861–864 (2009)CrossRefGoogle Scholar
  43. 43.
    Pradhan S.C., Phadikar J.K.: Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys. Lett. A. 37, 1062–1069 (2009)CrossRefGoogle Scholar
  44. 44.
    Wang C.M., Duan W.H.: Free vibration of nanorings/arches based on nonlocal elasticity. J. Appl. Phys. 104, 014303 (2008)CrossRefGoogle Scholar
  45. 45.
    Yang J., Jia X.L., Kitipornchai S.: Pull-in instability of nano-switches using nonlocal elasticity theory. J. Phys. D 41, 035103 (2008)CrossRefGoogle Scholar
  46. 46.
    Reddy J.N., Pang S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103, 023511 (2008)CrossRefGoogle Scholar
  47. 47.
    Murmu T., Pradhan S.C.: Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Phys. E 41, 1232–1239 (2009)CrossRefGoogle Scholar
  48. 48.
    Heireche H., Tounsi A., Benzair A., Maachou M., Adda Bedia E.A.: Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity. Phys. E 40, 2791–2799 (2008)CrossRefGoogle Scholar
  49. 49.
    Wang L.: Wave propagation of fluid-conveying single-walled carbon nanotubes via gradient elasticity theory. Comput. Mater. Sci. 45, 584–588 (2009)CrossRefGoogle Scholar
  50. 50.
    Sudak L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281 (2003)CrossRefGoogle Scholar
  51. 51.
    Zhang Y.Q., Liu G.R., Xie X.Y.: Free transverse vibration of double-walled carbon nanotubes using a theory of nonlocal elasticity. Phys. Rev. B 71, 195404 (2005)CrossRefGoogle Scholar
  52. 52.
    Narendar S., Gopalakrishnan S.: Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes. Comput. Mater. Sci. 47, 526 (2009)CrossRefGoogle Scholar
  53. 53.
    Narendar S., Gopalakrishnan S.: Terahertz wave characteristics of a single-walled carbon nanotube containing a fluid flow using the nonlocal Timoshenko beam model. Phys. E 42, 1706 (2010)CrossRefGoogle Scholar
  54. 54.
    Narendar S., Gopalakrishnan S.: Nonlocal scale effects on ultrasonic wave characteristics of nanorods. Phys. E 42, 1601 (2010)CrossRefGoogle Scholar
  55. 55.
    Narendar S., Gopalakrishnan S.: Theoretical estimation of length dependent in-plane stiffness of single walled carbon nanotubes using the nonlocal elasticity theory. J. Comput. Theor. Nanosci. 7(11), 2349 (2010)CrossRefGoogle Scholar
  56. 56.
    Narendar S., Gopalakrishnan S.: Investigation of the effect of nonlocal scale on ultrasonic wave dispersion characteristics of a monolayer graphene. Comput. Mater. Sci. 49, 734 (2010)CrossRefGoogle Scholar
  57. 57.
    Narendar S., Gopalakrishnan S.: Ultrasonic wave characteristics of nanorods via nonlocal strain gradient models. J. Appl. Phys. 107, 084312 (2010)CrossRefGoogle Scholar
  58. 58.
    Narendar S., Gopalakrishnan S.: Strong nonlocalization induced by small scale parameter on terahertz flexural wave dispersion characteristics of a monolayer graphene. Phys. E 43, 423–430 (2010)CrossRefGoogle Scholar
  59. 59.
    Xu M.: Transverse vibrations of nano-to-micron scale beams. Proc. Royal Soc. A Math. Phys. Eng. Sci. 462, 2977 (2006)MATHCrossRefGoogle Scholar
  60. 60.
    Wang Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 124301 (2005)CrossRefGoogle Scholar
  61. 61.
    Wang Q., Varadan V.K.: Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater. Struct. 16, 178 (2007)CrossRefGoogle Scholar
  62. 62.
    Pradhan S.C.: Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Phys. Lett. A 373, 4182–4188 (2009)CrossRefGoogle Scholar
  63. 63.
    Lu Q., Huang R.: Nonlinear mechanics of single-atomic-layer graphene sheets. Int. J. Appl. Mech. 1(3), 443–467 (2009)CrossRefGoogle Scholar
  64. 64.
    Fasolino A., Los J.H., Katsnelson M.I.: Intrinsic ripples in graphene. Nat. Mater. 6(11), 858–861 (2007)CrossRefGoogle Scholar
  65. 65.
    Meyer J.C., Geim A.K., Katsnelson M.I., Novoselov K.S., Booth T.J., Roth S.: The structure of suspended graphene sheets. Nature 446(7131), 60–63 (2007)CrossRefGoogle Scholar
  66. 66.
    Nelson D.R., Piran T., Weinberg S.: Statistical Mechanics of Membranes and Surfaces. World Scientific Pub, Singapore (2004)MATHGoogle Scholar
  67. 67.
    Arroyo M., Belytschko T.: Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys. Rev. B 69(11), 115415 (2004)CrossRefGoogle Scholar
  68. 68.
    Huang Y., Wu J., Hwang K.C.: Thickness of graphene and single-wall carbon nanotubes. Phys. Rev. B 74(24), 033524 (2006)CrossRefGoogle Scholar
  69. 69.
    Wong E.W., Sheehan P.E., Lieber C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, 1971–1975 (1997)CrossRefGoogle Scholar
  70. 70.
    Sorop T.G., de Jongh L.J.: Size-dependent anisotropic diamagnetic screening in superconducting Sn nanowires. Phys. Rev. B 75, 014510 (2007)CrossRefGoogle Scholar
  71. 71.
    Reddy J.N.: Mechanics of Laminated Composite Plates, Theory and Analysis. Chemical Rubber Company, Boca Raton, FL (1997)MATHGoogle Scholar
  72. 72.
    Hu Y.G., Liew K.M., Wang Q., He X.Q., Yakobson B.I.: Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys. Solids 56(12), 3475 (2008)MATHCrossRefGoogle Scholar
  73. 73.
    Wang L.F., Hu H.Y.: Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B 71, 195412 (2005)CrossRefGoogle Scholar
  74. 74.
    Zhang X., Jiao K., Sharma P., Yakobson B.I.: An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene. J. Mech. Phys. Solids 54, 2304 (2006)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Wang Q., Han Q.K., Wen B.C.: Estimate of material property of carbon nanotubes via nonlocal elasticity. Adv. Theor. Appl. Mech. 1(1), 1–10 (2008)Google Scholar
  76. 76.
    Zhang Y.Y., Wang C.M., Tan V.B.C.: Assessment of Timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics. Adv. Appl. Math. Mech. 1(1), 89–106 (2009)MathSciNetGoogle Scholar
  77. 77.
    Yakobson B.I., Brabec C. J., Bernholc J.: Nanomechanics of carbon tubes: instabilities beyond the linear response. Phys. Rev. Lett. 76, 2511–2514 (1996)CrossRefGoogle Scholar
  78. 78.
    Duan W.H., Wang C.M., Zhang Y.Y.: Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys. 101, 024305 (2007)CrossRefGoogle Scholar
  79. 79.
    Narendar S., Roy Mahapatra D., Gopalakrishnan S.: Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation. Int. J. Eng. Sci 49, 509–522 (2011)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Duan W.H., Wang C.M.: Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology 18, 385704 (2007)CrossRefGoogle Scholar
  81. 81.
    Wang Q., Wang C.M.: The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18, 075702 (2007)CrossRefGoogle Scholar
  82. 82.
    Reddy J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288 (2007)MATHCrossRefGoogle Scholar
  83. 83.
    Reddy C.D., Rajendran S., Liew K.M.: Equilibrium configuration and elastic properties of finite graphene. Nanotechnology 17, 864–870 (2006)CrossRefGoogle Scholar
  84. 84.
    Kim S.E., Thai H.T., Lee J.: Buckling analysis of plates using the two variable refined plate theory. Thin-Walled Struct. 47, 455–462 (2009)CrossRefGoogle Scholar
  85. 85.
    Shimpi R.P., Patel H.G.: A two variable refined plate theory for orthotropic plate analysis. Int. J. Solids Struct. 43, 6783–6799 (2006)MATHCrossRefGoogle Scholar
  86. 86.
    Hernandez E., Goze C., Bernier P., Rubio A.: Elastic properties of C and B x C y N z composite nanotubes. Phys. Rev. B 80, 4502–4505 (1998)CrossRefGoogle Scholar
  87. 87.
    Wang Q.: Effective in-plane stiffness and bending rigidity of armchair and zigzag carbon. Int. J. Solids Struct. 41, 5451–5461 (2004)MATHCrossRefGoogle Scholar
  88. 88.
    Li C., Chou T.W.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003)MATHCrossRefGoogle Scholar
  89. 89.
    Pradhan S.C., Phadikar J.K.: Phadikar nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Defence Research and Development LaboratoryDRDOKanchanbagh, HyderabadIndia
  2. 2.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

Personalised recommendations