Galerkin’s approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions

  • Behrouz KaramiEmail author
  • Maziar Janghorban
  • Abdelouahed Tounsi
Original Article


For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated. Also, it is the first time that the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotropic material. The properties of graded material are assumed vary exponentially through the z-direction. Nonlocal strain gradient theory is utilized to predicate the size-dependent buckling behavior of the nanoplate. The nanoplate is modeled by a higher order shear deformation refined plate theory in which any shear correction factor not used. Governing equations and boundary conditions are obtained using a virtual work of variational approach. To solve the buckling problem for different boundary conditions, Galerkin’s approach is utilized. Finally, the influences of different boundary conditions, small-scale parameters, geometry parameters and exponential factor are studied and discussed in detail. It is hoped that the present numerical results can help the engineers and designers to understand and predict the buckling response of FG anisotropic materials.


Functionally graded anisotropic material Nanoplate Nonlocal strain gradient theory Buckling Boundary conditions 



The author(s) received no financial support for the research, authorship, and/or publication of this article.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Marvdasht BranchIslamic Azad UniversityMarvdashtIran
  2. 2.Material and Hydrology Laboratory, Civil Engineering Department, Faculty of TechnologyUniversity of Sidi Bel AbbesSidi Bel AbbèsAlgeria

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