h-p finite element vibration analysis of side cracked rectangular nano-plates based on nonlocal elasticity theory

  • B. ZaouaguiEmail author
  • S. A. Belalia
  • A. Boukhalfa
Regular Article


The h-p version of the finite element method based on a quadrilateral p-element is applied to free vibration analysis of side cracked nano plates. The h-p version of the finite element method marries both the concepts of the conventional h-version and the p-version. The accuracy of the solution is sought by simultaneously refining the mesh and increasing the polynomial order in each element. The formulation takes into account shear deformation, rotary inertia, and non-local elasticity theory. The proposed model is compared to various analytical, numerical and experimental results to validate it’s accuracy and efficiency, computationally expensive preprocessing tasks are avoided due to hierarchic refining strategy that requires no re-meshing when changing crack parameters. The effects of plate aspect ratio, side-to-thickness ratio, non-local parameter, boundary conditions, crack length and crack angle on the free vibration analysis of side cracked nano plates is discussed and presented for the first time.


  1. 1.
    A.C. Eringen, J. Wegner, Appl. Mech. Rev. 56, B20 (2003)CrossRefGoogle Scholar
  2. 2.
    K.M. Liew, Yang Zhang, L.W. Zhang, J. Model. Mech. Mater., (2017)
  3. 3.
    R. Ansari, S. Sahmani, B. Arash, Phys. Lett. A 375, 53 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    S.C. Pradhan, A. Kumar, Compos. Struct. 93, 774 (2011)CrossRefGoogle Scholar
  5. 5.
    Parviz Malekzadeh, AliReza Setoodeh, Ali Alibeygi Beni, Compos. Struct. 93, 1631 (2011)CrossRefGoogle Scholar
  6. 6.
    C.Y. Wang, T. Murmu, S. Adhikari, Appl. Phys. Lett. 98, 153101 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    L.Y. Huang, Q. Han, Y.J. Liang, Nano 07, 1250033 (2012)CrossRefGoogle Scholar
  8. 8.
    Chen Liu, Liao-Liang Ke, Yue-Sheng Wang, Jie Yang, Sritawat Kitipornchai, Compos. Struct. 106, 167 (2013)CrossRefGoogle Scholar
  9. 9.
    Parviz Malekzadeh, Mohammad Shojaee, Compos. Struct. 95, 443 (2013)CrossRefGoogle Scholar
  10. 10.
    Shahrokh Hosseini-Hashemi, Mehdi Kermajani, Reza Nazemnezhad, Eur. J. Mech. - A/Solids 51, 29 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Chakraverty, Laxmi Behera, Physica E 56, 357 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    Abdelkrim Necira, Sid Ahmed Belalia, Abdelkrim Boukhalfa, Mech. Adv. Mater. Struct. (2018)
  13. 13.
    R. Ansari, M. Faghih Shojaei, A. Shahabodini, M. Bazdid-Vahdati, Compos. Struct. 131, 753 (2015)CrossRefGoogle Scholar
  14. 14.
    Mohammad Reza Barati, Hossein Shahverdi, J. Vib. Control 24, 4700 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ismahene Belkorissat, Mohammed Sid Ahmed Houari, Abdelouahed Tounsi, E.A. Bedia, S.R. Mahmoud, Steel Compos. Struct. 18, 1063 (2015)CrossRefGoogle Scholar
  16. 16.
    A. Daneshmehr, A. Rajabpoor et al., Int. J. Eng. Sci. 82, 84 (2014)CrossRefGoogle Scholar
  17. 17.
    Mohammad Rahim Nami, Maziar Janghorban, Compos. Struct. 111, 349 (2014)CrossRefGoogle Scholar
  18. 18.
    S. Natarajan, S. Chakraborty, M. Thangavel, Stephane Bordas, Timon Rabczuk, Comput. Mater. Sci. 65, 74 (2012)CrossRefGoogle Scholar
  19. 19.
    B.A. Szabo, A.K. Mehta, Int. J. Numer. Methods Eng. 12, 551 (1978)CrossRefGoogle Scholar
  20. 20.
    Barna Szabó, Ivo Babuška. Introduction to Finite Element Analysis: Formulation, Verification and Validation, Vol. 35 (John Wiley & Sons, 2011)Google Scholar
  21. 21.
    J.N. Reddy, Theory and Analysis of Elastic Plates and Shells, Second Edition, in Series in Systems and Control (Taylor & Francis, 2006)Google Scholar
  22. 22.
    Leon Stahl, B. Keer, Int. J. Solids Struct. 8, 69 (1972)CrossRefGoogle Scholar
  23. 23.
    K. Liew, K.C. Hung, M.K. Lim, Eng. Fract. Mech. 48, 393 (1994)ADSCrossRefGoogle Scholar
  24. 24.
    S. Sumi, T. Fujimoto, Trans. Jpn. Soc. Mech. Eng. Ser. A 53, 1124 (1987)CrossRefGoogle Scholar

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© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Computational Mechanics, Department of Mechanical Engineering, Faculty of TechnologyUniversity of TlemcenTlemcenAlgeria

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