Acta Mechanica

, Volume 230, Issue 3, pp 885–907 | Cite as

Comparison of nano-plate bending behaviour by Eringen nonlocal plate, Hencky bar-net and continualised nonlocal plate models

  • Y. P. Zhang
  • N. ChallamelEmail author
  • C. M. Wang
  • H. Zhang
Original Paper


This paper is concerned with the bending behaviour of small-scale simply supported plates as predicted by using the Eringen nonlocal plate model (ENM), the Hencky bar-net model (HBM) and the continualised nonlocal plate model (CNM). HBM comprises rigid beam segments connected by rotational and torsional springs. CNM is a nonlocal model derived by using a continualisation approach that does away with the unknown scale coefficient \(e_{0}\) in ENM. The exact bending solutions for simply supported rectangular nano-plates are derived by using ENM, HBM and CNM. By making the segment length \(\ell \) of HBM equal to the scale length of continualised and Eringen’s nonlocal plate model and noting the phenomenological similarities between ENM, HBM and CNM, the Eringen’s length scale value \(e_0 \) is found to be dependent on the aspect ratio of the simply supported plate and independent of the applied transverse loading. For a very small scale length \(\ell \), \(e_0\) of ENM converges to values ranging from \(1/\sqrt{8}\) to \(1/\sqrt{6}\) for square plate to longish rectangular plate when calibrated by either HBM or CNM.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Peerlings, R.H.J., Geers, M.G.D., De Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38, 7723–7746 (2001). CrossRefzbMATHGoogle Scholar
  3. 3.
    Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007). CrossRefGoogle Scholar
  6. 6.
    Wang, C.M., Xiang, Y., Yang, J., Kitipornchai, S.: Buckling of nano-rings/arches based on nonlocal elasticity. Int. J. Appl. Mech. 04, 1250025 (2012). CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Wang, Q.: Axisymmetric wave propagation of carbon nanotubes with non-local elastic shell model. Int. J. Struct. Stab. Dyn. 06, 285–296 (2006). CrossRefGoogle Scholar
  9. 9.
    Aifantis, E.C.: On the gradient approach—relation to Eringen’s nonlocal theory. Int. J. Eng. Sci. 49, 1367–1377 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Romano, G., Barretta, R., Diaco, M., Marotti de Sciarra, F.: Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017). CrossRefGoogle Scholar
  11. 11.
    Irschik, H., Heuer, R.: Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform. Acta Mech. 229, 867–879 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Koutsoumaris, C.C., Eptaimeros, K.G.: A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body. Acta Mech. 229, 3629–3649 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, Y., Yang, L., Zhang, L., Gao, Y.: Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading. Acta Mech. 229, 3501–3515 (2018). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Barati, M.R.: Vibration analysis of porous FG nanoshells with even and uneven porosity distributions using nonlocal strain gradient elasticity. Acta Mech. 229, 1183–1196 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Despotovic, N.: Stability and vibration of a nanoplate under body force using nonlocal elasticity theory. Acta Mech. 229, 273–284 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, H., Wang, C.M., Challamel, N.: Small length scale coefficient for Eringen’s and lattice-based continualized nonlocal circular arches in buckling and vibration. Compos. Struct. 165, 148–159 (2017). CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Wang, L., Hu, H.: Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B. 71, 1–7 (2005). Google Scholar
  19. 19.
    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, 89–106 (2009)MathSciNetGoogle Scholar
  20. 20.
    Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003). CrossRefGoogle Scholar
  21. 21.
    Hencky, H.: Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Der Eisenbau 11, 437–452 (1921)Google Scholar
  22. 22.
    Salvadori, M.G.: Numerical computation of buckling loads by finite differences. Trans. Am. Soc. Civ. Eng. 116, 590–624 (1951)Google Scholar
  23. 23.
    Challamel, N., Hache, F., Elishakoff, I., Wang, C.M.: Buckling and vibrations of microstructured rectangular plates considering phenomenological and lattice-based nonlocal continuum models. Compos. Struct. 149, 145–156 (2016). CrossRefGoogle Scholar
  24. 24.
    Hache, F., Challamel, N., Elishakoff, I., Wang, C.M.: Comparison of nonlocal continualization schemes for lattice beams and plates. Arch. Appl. Mech. 87, 1105–1138 (2017). CrossRefGoogle Scholar
  25. 25.
    Silverman, I.K.: Discussion on the paper of “Salvadori M.G., Numerical computation of buckling loads by finite differences”. Trans. Am. Soc. Civ. Eng. 116, 625–626 (1951)Google Scholar
  26. 26.
    Leckie, F.A., Lindberg, G.M.: The effect of lumped parameters on beam frequencies. Aeronaut. Quart. 14, 224–240 (1963)CrossRefGoogle Scholar
  27. 27.
    El Nashie, M.S.: Stress, Stability and Chaos in Structural Engineering: An Energy Approach. McGraw-Hill, London (1991)Google Scholar
  28. 28.
    Zhang, H., Wang, C.M.: Hencky bar-chain model for optimal circular arches against buckling. Mech. Res. Commun. 88, 7–11 (2018). CrossRefGoogle Scholar
  29. 29.
    Wang, C.M., Zhang, H., Gao, R.P., Duan, W.H., Challamel, N.: Hencky bar-chain model for buckling and vibration of beams with elastic end restraints. Int. J. Struct. Stab. Dyn. 15, 1540007 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, H., Wang, C.M., Challamel, N.: Buckling and vibration of Hencky bar-chain with internal elastic springs. Int. J. Mech. Sci. 119, 383–395 (2016). CrossRefGoogle Scholar
  31. 31.
    Zhang, H., Wang, C.M., Ruocco, E., Challamel, N.: Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation. Eng. Struct. 126, 252–263 (2016). CrossRefGoogle Scholar
  32. 32.
    Ruocco, E., Zhang, H., Wang, C.M.: Hencky bar-chain model for buckling analysis of non-uniform columns. Structures 6, 73–84 (2016). CrossRefGoogle Scholar
  33. 33.
    Zhang, H., Zhang, Y.P., Wang, C.M.: Hencky bar-net model for vibration of rectangular plates with mixed boundary conditions and point supports. Int. J. Struct. Stab. Dyn. 18, 03 (2017). MathSciNetGoogle Scholar
  34. 34.
    Wang, C.M., Zhang, Y.P., Pedroso, D.M.: Hencky bar-net model for plate buckling. Eng. Struct. 150, 947–954 (2017). CrossRefGoogle Scholar
  35. 35.
    Zhang, Y.P., Wang, C.M., Pedroso, D.M.: Hencky bar-net model for buckling analysis of plates under non-uniform stress distribution. Thin-Walled Struct. 122, 344–358 (2018). CrossRefGoogle Scholar
  36. 36.
    Zhang, Y.P., Wang, C.M., Pedroso, D.M., Zhang, H.: Extension of Hencky bar-net model for vibration analysis of rectangular plates with rectangular cutouts. J. Sound Vib. 432, 65–87 (2018). CrossRefGoogle Scholar
  37. 37.
    Challamel, N., Lerbet, J., Wang, C.M., Zhang, Z.: Analytical length scale calibration of nonlocal continuum from a microstructured buckling model. ZAMM J. Appl. Math. Mech./Zeitschrift für Angew. Math. und Mech. 94, 402–413 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Challamel, N., Zhang, Z., Wang, C.M.: Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams. J. Nanomech. Micromech. 5, A4014004 (2015). CrossRefGoogle Scholar
  39. 39.
    Wang, C.M., Zhang, Z., Challamel, N., Duan, W.H.: Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model. J. Phys. D Appl. Phys. 46, 345501 (2013). CrossRefGoogle Scholar
  40. 40.
    Zhang, Z., Challamel, N., Wang, C.M.: Eringen’s small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model. J. Appl. Phys. 114, 114902 (2013). CrossRefGoogle Scholar
  41. 41.
    Wang, C.M., Zhang, H., Challamel, N., Xiang, Y.: Buckling of nonlocal columns with allowance for selfweight. J. Eng. Mech. 142, 04016037 (2016). CrossRefGoogle Scholar
  42. 42.
    Wang, C.M., Zhang, H., Challamel, N., Duan, W.H.: On boundary conditions for buckling and vibration of nonlocal beams. Eur. J. Mech. A/Solids. 61, 73–81 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhang, Z., Wang, C.M., Challamel, N.: Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model. Int. J. Solids Struct. 51, 4307–4315 (2014). CrossRefGoogle Scholar
  44. 44.
    Zhang, Z., Wang, C.M., Challamel, N.: Eringen’s length-scale coefficients for vibration and buckling of nonlocal rectangular plates with simply supported edges. J. Eng. Mech. 141, 04014117 (2015). CrossRefGoogle Scholar
  45. 45.
    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
  46. 46.
    Timoshenko, S., Woinowshy Krieger, S.: Theory of Plates and Shells. Engineering Societies Monographs. McGraw-Hill, London (1959)Google Scholar
  47. 47.
    Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., Reddy, J.N.: Non-local elastic plate theories. Proc. R. Soc. A Math. Phys. Eng. Sci. 463, 3225–3240 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Marcus, H.: Die Theorie elastischer Gewebe, 2nd edn. Springer, Berlin (1932)zbMATHGoogle Scholar
  49. 49.
    Challamel, N., Reddy, J.N.: Reply to the comments of M.E. Golmakani and J. Rezatalab, Comment on “Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates” (by R. Aghababaei and J.N. Reddy, Journal of Sound and Vibration 326 (2009) 277–289), Journal of Sound Vibration, 333 (2014) 3831–3835. J. Sound Vib. 333, 5654–5656 (2014). CrossRefGoogle Scholar
  50. 50.
    Wifi, A.S., Wu, C.W., Obeid, K.A.: A simple discrete element mechanical model for the stability analysis of elastic structures. In: Kabil, Y.H., Said, M.E. (eds.) Current Advances in Mechanical Design and Production, pp. 149–156. Pergamon Press, Oxford (1989)CrossRefGoogle Scholar
  51. 51.
    Wang, C.M., Zhang, Y.Y., Ramesh, S.S., Kitipornchai, S.: Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D Appl. Phys. 39, 3904–3909 (2006). CrossRefGoogle Scholar
  52. 52.
    Challamel, N., Wang, C.M., Elishakoff, I.: Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis. Eur. J. Mech. A/Solids. 44, 125–135 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Triantafyllidis, N., Bardenhagen, S.: On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models. J. Elast. 33, 259–293 (1993). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Y. P. Zhang
    • 1
  • N. Challamel
    • 2
    Email author
  • C. M. Wang
    • 3
  • H. Zhang
    • 3
    • 4
  1. 1.School of Mechanical and Mining EngineeringUniversity of QueenslandSt LuciaAustralia
  2. 2.IRDL (CNRS UMR 6027), Centre de RechercheUniversité Bretagne SudLorient CedexFrance
  3. 3.School of Civil EngineeringUniversity of QueenslandSt LuciaAustralia
  4. 4.School of Mechatronical EngineeringBeijing Institute of TechnologyBeijingPeople’s Republic of China

Personalised recommendations