Skip to main content

Three-dimensional exact elastic analysis of nanoplates


This work investigates the application of three-dimensional nonlocal elasticity theory to elastic static analysis of nanoplates. Unlike all previous papers that considered two-dimensional Laplacian operator to stress components, this work uses the general three-dimensional nonlocal operator with thickness direction operator. The displacement field of nanoplate is assumed a function of three-dimensional coordinate x, y, z. The principle of virtual work is used to derive the governing equations. A solution procedure is developed for simply supported nanoplate. The solution along the thickness direction is derived using the characteristic equation and application of boundary conditions including free transverse shear stress and applied normal stress. The eigenvalue–eigenvector methodology is used to extract general solution along the transverse direction. The stress and deformation distribution along the transverse direction is presented with changes of significant parameters such as nonlocal parameter and aspect ratio.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Vel SS, Batra RC. Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J Solids Struct. 2003;40(25):7181–96.

    MATH  Google Scholar 

  2. 2.

    Zhong Z, Shang ET. Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate. Int J Solids Struct. 2003;40(20):5335–52.

    MATH  Google Scholar 

  3. 3.

    Vel SS, Batra RC. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vib. 2004;272(3–5):703–30.

    Google Scholar 

  4. 4.

    Arefi M, Rahimi GH, Khoshgoftar MJ. Exact solution of a thick walled functionally graded piezoelectric cylinder under mechanical, thermal and electrical loads in the magnetic field. Smart Struct Syst. 2012;9(5):427–39.

    Google Scholar 

  5. 5.

    Arefi M, Rahimi GH. Studying the nonlinear behavior of the functionally graded annular plates with piezoelectric layers as a sensor and actuator under normal pressure. Smart Struct Syst. 2012;9(2):127–43.

    Google Scholar 

  6. 6.

    Arefi M, Rahimi GH. Non linear analysis of a functionally graded square plate with two smart layers as sensor and actuator under normal pressure. Smart Struct Syst. 2011;8(5):433–47.

    Google Scholar 

  7. 7.

    Gao T, Changhe J, Dongzhou Z, Yanbin Y, Min W, Xiaoming C, Huajun L, Runze MH, Al XX. Surface morphology assessment of CFRP transverse grinding using CNT nanofluid minimum quantity lubrication. J Clean Prod. 2020;277:123328.

    Google Scholar 

  8. 8.

    Zhang Y, Changhe J, Dongkun ZD, Xiaowei Z. Experimental evaluation of the lubrication performance of MoS2/CNT nanofluid for minimal quantity lubrication in Ni-based alloy grinding. Int J Mach Tools Manuf. 2015;99:19–33.

    Google Scholar 

  9. 9.

    Yang Y, Chen H, Zou X, Shi X-L, Liu W-D, Feng L, Suo G, Hou X, Ye X, Zhang L, Sun C, Li H, Wang C, Chen Z-G. Flexible carbon-fiber/semimetal bi nanosheet arrays as separable and recyclable plasmonic photocatalysts and photoelectrocatalysts. ACS Appl Mater Interfaces. 2020;12:22.

    Google Scholar 

  10. 10.

    Malikan TM, Krasheninnikov M, Eremeyev VA. Torsional stability capacity of a nano-composite shell based on a nonlocal strain gradient shell model under a three-dimensional magnetic field. Int J Eng Sci. 2020;148:103210.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Gao N, Lu K. An underwater metamaterial for broadband acoustic absorption at low frequency. Appl Acoust. 2020;169:107500.

    Google Scholar 

  12. 12.

    Gao N, Wang B, Lu K, Hou H. Complex band structure and evanescent Bloch wave propagation of periodic nested acoustic black hole phononic structure. Appl Acoust. 2021;177:107906.

    Google Scholar 

  13. 13.

    Zhang K, Yang Z, Mao X, Chen X-L, Li H-H, Wang Y-Y. Multifunctional textiles/metal-organic frameworks composites for efficient ultraviolet radiation blocking and noise reduction. ACS Appl Mater Interfaces. 2020;12:49.

    Google Scholar 

  14. 14.

    Zhang K, Huo Q, Zhou Y-Y, Wang H-H, Li G-P, Wang Y-W, Wang Y-Y. Textiles/metal-organic frameworks composites as flexible air filters for efficient particulate matter removal. ACS Appl Mater Interfaces. 2019;11:19.

    Google Scholar 

  15. 15.

    Duan Z, Qingan L, Changhe D, Lan B, Xiufang Z, Yanbin Y, Min J, Dongzhou L, Runze L. Zhanqiang Milling force and surface morphology of 45 steel under different Al2O3 nanofluid concentrations. Int J Adv Manuf Technol. 2020;107(3):1277–96.

    Google Scholar 

  16. 16.

    Gao T, Changhe L, Yanbin Z, Min Y, Dongzhou J, Tan J, Yali H, Runze L. Dispersing mechanism and tribological performance of vegetable oil-based CNT nanofluids with different surfactants. Tribol Int. 2019;131:51–63.

    Google Scholar 

  17. 17.

    Kulikov GM, Plotnikova SV. Three-dimensional exact analysis of piezoelectric laminated plates via a sampling surfaces method. Int J Solids Struct. 2013;50(11–12):1916–29.

    Google Scholar 

  18. 18.

    Kulikov GM, Plotnikova SV. Three-dimensional exact analysis of laminated piezoelectric plates and shells. Adv Mater Res. 2013;745:1–12.

    Article  Google Scholar 

  19. 19.

    Jin G, Su Z, Shi S, Ye T, Gao S. Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Compos Struct. 2014;108:565–77.

    Google Scholar 

  20. 20.

    Messina A. Three-dimensional free vibration analysis of cross-ply laminated rectangular plates through 2D and exact models. Mech Adv Mater Struct. 2012;19:250–64.

    Google Scholar 

  21. 21.

    Alibeigloo A. Three-dimensional exact solution for functionally graded rectangular plate with integrated surface piezoelectric layers resting on elastic foundation. Mech Adv Mater Struct. 2010;17(3):183–95.

    Google Scholar 

  22. 22.

    Wang D, Xue C, Xuan F, Jilong T, Fengyuan L, Xinwei W, Guanlin L, Lei L, Johnny CH, Zhipeng W. Photoresponse improvement of mixed-dimensional 1D–2D GaAs photodetectors by incorporating constructive interface states. Nanoscale. 2021;13(2):1086–92.

    Google Scholar 

  23. 23.

    Zhang K, Qian H, Ying-Ying Z, Hong-Hong W, Gao-Peng L, Yao-Wu W. Textiles/metal–organic frameworks composites as flexible air filters for efficient particulate matter removal. ACS Appl Mater Interfaces. 2019;11(19):17368–74.

    Google Scholar 

  24. 24.

    Peng X-J, Hai-Ping H, Qian L, Kun S, Bao-Qi Z, Heng-Shan W, Hai-Tao T, Ying-Ming P. Photocatalyst-controlled and visible light-enabled selective oxidation of pyridinium salts. Sci China Chem. 2021:1–8.

  25. 25.

    Hu Y, Shengchuan W, Philip JW, Huatang C, Pei C, Yajun Z, Zhao S, Tomáš V, Pavel H. Corrosion fatigue lifetime assessment of high-speed railway axle EA4T steel with artificial scratch. Eng Fract Mech. 2021;245:107588.

    Google Scholar 

  26. 26.

    Chen A, Wang X, Wang Y, Yang D, Yao F, Zhang W, Wang B, Sewvandi GA, Yang D, Hu D. Additive manufacturing of piezoelectric materials. Adv Func Mater. 2020;30(52):2005141.

    Google Scholar 

  27. 27.

    Eisenberger M, Godoy LA. Navier type exact analytical solutions for vibrations of thin-walled shallow shells with rectangular planform. Thin Walled Struct. 2020;160:107356.

    Google Scholar 

  28. 28.

    Ghannadpour SAM, Moradi F, Tornabene F. Exact analytical solutions to the problem of relative post-buckling stiffness of thin nonlocal graphene sheets. Thin Walled Struct. 2020;151:106712.

    Google Scholar 

  29. 29.

    Farajpour A, Hairi Yazdi MR, Rastgoo A, Loghmani M, Mohammadi M. Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates. Compos Struct. 2016;140:323–36.

    Google Scholar 

  30. 30.

    Dastjerdi S, Akgöz B. New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment. Compos Struct. 2018;192:626–41.

    Google Scholar 

  31. 31.

    Gao N, Tang L, Deng J, Lu K, Hou H, Chen K. Design, fabrication and sound absorption test of composite porous matamaterial with embedding I-plates into porous polyurethane. Appl Acoust. 2021;175:107845.

    Google Scholar 

  32. 32.

    Yang X, Li Q, Lu E, Wang Z, Gong X, Yu Z, Guo Y, Wang L, Guo Y, Zhan W, Zhang J, Dai S. Taming the stability of Pd active phases through a compartmentalizing strategy toward nanostructured catalyst supports. Nat Commun. 2019;10:1611.

    Google Scholar 

  33. 33.

    Yan X, Huang X, Chen Y, Liu Y, Xi L, Zhang T, Lin H, Jia D, Zhong B, Wen G, Zhou Y. A theoretical strategy of pure carbon materials for lightweight and excellent absorption performance. Carbon. 2021;174:662–72.

    Google Scholar 

  34. 34.

    Shen C-L, Lou Q, Zang J-H, Liu K-K, Qu S-N, Dong L, Shan C-X. Near-infrared chemiluminescent carbon nanodots and their application in reactive oxygen species bioimaging. Adv Sci. 2020;7(8):1903525.

    Google Scholar 

  35. 35.

    Aydinlik S, Kiris A, Sumelka W. Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics—theory and validation. Eur Phys J Plus. 2021;136:169.

    Google Scholar 

  36. 36.

    Lazopoulos AK. On fractional peridynamic deformations. Arch Appl Mech. 2016;86:1987–94.

    Google Scholar 

  37. 37.

    Wang H, Du N. Fast solution methods for space-fractional diffusion equations. J Comput Appl Math. 2014;255:376–83.

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Xu Y, Zhou D. Two-dimensional analysis of simply supported piezoelectric beams with variable thickness. Appl Math Model. 2011;35(9):4458–72.

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Zhang K, Zhi Y, Xue M, Xue-Li C, Hai-Hong L, Yao-Yu W. Multifunctional textiles/metal-organic frameworks composites for efficient ultraviolet radiation blocking and noise reduction. ACS Appl Mater Interfaces. 2020;12(49):55316–23.

    Google Scholar 

  40. 40.

    Li W-H, Cun-Yao L, Huan-Yan X, Yang L, Wen-Yong H, Guang-Jun J, Zheng J, Hai-Tao T, Ying-Ming P, Yun-Jie D. Constructing mononuclear palladium catalysts by precoordination/solvothermal polymerization: recyclable catalyst for regioselective oxidative heck reactions. Angew Chem Int Ed. 2019;58(8):2448–53.

    Google Scholar 

  41. 41.

    Zuo C, Jiasong S, Jiaji L, Jialin Z, Anand A, Qian C. High-resolution transport-of-intensity quantitative phase microscopy with annular illumination. Sci Rep. 2017;7(1):1–22.

    Google Scholar 

  42. 42.

    Civalek Ö, Avcar M. Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Eng Comput. 2020.

    Article  Google Scholar 

  43. 43.

    Hadji L, Avcar M. Free vibration analysis of FG porous sandwich plates under various boundary conditions. J Appl Comput Mech. 2021;7(2):505–19.

    Google Scholar 

  44. 44.

    Moraveji Tabasi H, Eskandari Jam J, Malekzadeh Fard K, Heydari Beni M. Buckling and free vibration analysis of fiber metal-laminated plates resting on partial elastic foundation. J Appl Comput Mech. 2020;6(1):37–51.

    Google Scholar 

  45. 45.

    Huang W-Y, Guo-Qing W, Wen-Hao L, Ting-Ting L, Guang-Jun J, Shi-Cheng R, Miao J. Porous ligand creates new reaction route: bifunctional single-atom palladium catalyst for selective distannylation of terminal alkynes. Chem. 2020;6(9):2300–13.

    Google Scholar 

  46. 46.

    Liu G, Guimei R, Lei Z, Lei C, Chengtao W, Baoguo S. Antibacterial activity and mechanism of bifidocin A against Listeria monocytogenes. Food Control. 2017;73:854–61.

    Google Scholar 

  47. 47.

    Wang P, Ziqiang L, Qing X, Wei D, Xinchun Z, Huilong H. A passive anti-icing strategy based on a superhydrophobic mesh with extremely low ice adhesion strength. J Bion Eng. 2021;18(1):55–64.

    Google Scholar 

  48. 48.

    Zuo C, Qian C, Lei T, Laura W, Anand A. Transport of intensity phase retrieval and computational imaging for partially coherent fields: the phase space perspective. Opt Lasers Eng. 2015;71:20–32.

    Google Scholar 

  49. 49.

    Rahimi Z, Sumelka W, Yang X-J. A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams. Eur Phys J Plus. 2017;132(11):479.

    Google Scholar 

  50. 50.

    Onate AA, Onyeaju MC, Ikot AN, Ebomwonyi O. Eigen solutions and entropic system for Hellmann potential in the presence of the Schrödinger equation. Eur Phys J Plus. 2017;132:462.

    Google Scholar 

  51. 51.

    Bauchau OA, Han S. Three-dimensional beam theory for flexible multibody dynamics. J Comput Nonlinear Dyn. 2014;9(4):041011.

    Google Scholar 

  52. 52.

    Ma H. Rational approach for higher-order shear deformation beam theories. Compos Struct. 2020;251:112599.

    Google Scholar 

  53. 53.

    Lim AW. Three-dimensional vibration analysis of a cantilevered parallelepiped: exact and approximate solutions. J Acoust Soc Am. 1999;106:3375.

    Google Scholar 

  54. 54.

    Wang P, Tao L, Ziqiang W, Weidong X, Qing D, Wei HH. A superhydrophobic/electrothermal synergistically anti-icing strategy based on graphene composite. Compos Sci Technol. 2020;198:108307.

    Google Scholar 

  55. 55.

    Li X, Feng Y, Liu B, Yi D, Yang X, Zhang W, Chen G, Liu Y, Bai P. Influence of NbC particles on microstructure and mechanical properties of AlCoCrFeNi high-entropy alloy coatings prepared by laser cladding. J Alloy Compd. 2019;788:485–94.

    Google Scholar 

  56. 56.

    Lee SB, Lee CY, Hodges DH. On the mechanics of composite sandwich plates with three-dimensional stress recovery. Int J Eng Sci. 2020;157:103406.

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Abbasi S, Farhatnia F, Jazi SR. A semi-analytical solution on static analysis of circular plate exposed to non-uniform axisymmetric transverse loading resting on Winkler elastic foundation. Arch Civ Mech Eng. 2014;14:476–88.

    Google Scholar 

  58. 58.

    Zhu X, Fengyuan Z, Zhihong C, Xue H, Hao W, Dengkui T. Jilong enhancing performance of a GaAs/AlGaAs/GaAs nanowire photodetector based on the two-dimensional electron-hole tube structure. Nano Lett. 2020;20(4):2654–9.

    Google Scholar 

  59. 59.

    Sladek J, Sladek V, Hellmich Ch, Eberhardsteiner J. Analysis of thick functionally graded plates by local integral equation method. Commun Numer Method Eng. 2007;23:733–54.

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Wen PH, Sladek J, Sladek V. Three-dimensional analysis of functionally graded plates. Int J Numer Method Eng. 2011;87:923–42.

    MathSciNet  MATH  Google Scholar 

Download references


This work was supported by Science and Technology Program of Guangdong Province (2020B121201013); Science and Technology Special Fund Program of Guangdong Province (2020A0102009); Rural Science and Technology Commissioner Program of Guangdong Province (KTP20200278); National Natural Science Foundation of Guangdong Province (2021A1515012597).

Author information



Corresponding author

Correspondence to Yu Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, G., Zhang, Y. & Arefi, M. Three-dimensional exact elastic analysis of nanoplates. Archiv.Civ.Mech.Eng 21, 91 (2021).

Download citation


  • Nonlocal elasticity theory
  • Three-dimensional elasticity
  • Laplacian operator
  • Stress and deformation analysis