Engineering with Computers

, Volume 35, Issue 1, pp 215–228 | Cite as

Non-classical plate model for FGMs

  • Marzieh Alizadeh
  • A. M. FattahiEmail author
Original Article


In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was assumed that the material properties of FGM microplates vary along the direction of thickness and could be estimated using Mori–Tanaka homogenization technique. A variational approach based on Hamilton’s principle was employed to achieve motion equations and corresponding boundary conditions simultaneously. Unlike classical plate theory, the proposed plate model contained three material length scale parameters which allowed size effect to be taken into account effectively. Moreover, the proposed non-classical plate model could be divided into classical and modified couple stress plate models when all material length scale parameters were set at zero. To approve the applicability of the proposed size-dependent plate model, analytical solution for free vibration problem of a simply supported FGM microplate was obtained according to the Navier solution, where generalized displacements were stated as multiplication of undetermined functions with known trigonometric functions to identically satisfy simply supported boundary conditions for all edges. The results obtained by the proposed non-classical plate model were compared with those obtained from reduced ones relevant to different values of material property gradient index and dimensionless length scale parameter. It was found that the differences observed between the sets of predicted natural frequencies obtained from different types of plate models were more significant for lower values of dimensionless length scale parameter.


Microplates Free vibration Functionally graded materials Size effect 


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTabriz Branch, Islamic Azad UniversityTabrizIran

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