Microsystem Technologies

, Volume 24, Issue 8, pp 3549–3572 | Cite as

Thermal effect on bending, buckling and free vibration of functionally graded rectangular micro-plates possessing a variable length scale parameter

  • Reza AghazadehEmail author
  • Serkan Dag
  • Ender Cigeroglu
Technical Paper


Modified couple stress based model is presented to investigate statics, dynamics and stability of functionally graded micro-plates subjected to mechanical and thermal loadings. The features of FGM micro-plate including length scale parameter of modified couple stress theory assumed to be graded across the thickness by varying volume fractions of constituents. The governing equations of motion and boundary conditions are derived by means of Hamilton’s principle. Displacement field is expressed in a unified way capable of producing results on the base of Kirchhoff, Mindlin, and third order shear deformation theories. The system of equations is solved numerically by implementing differential quadrature method. Verification studies are carried out by comparing the results of special cases to those available in the literature. Further numerical results regarding static thermal bending, natural frequencies and critical buckling loads of micro-plates undergoing uniform temperature change are provided. Presented numerical results clearly illustrate size effect at micro-scale, impact of length scale parameter variations and influence of initial thermal displacements and stresses upon mechanical behavior of functionally graded rectangular micro-plates.

List of symbols


Area of mid-plane of micro-plate


Length of micro-plate


Width of micro-plate


Ceramic phase index


Young’s modulus


Alternating tensor


Shape function for plate theories


Thickness of micro-plate


Kinetic energy


Shear correction factor


Material length scale parameter


Metallic phase index


Higher order stress, work-conjugate to \(\chi_{ij}\)


Volume fraction exponent

\(n_{{x_{1} }}\), \(n_{{x_{2} }}\)

Direction cosines of unit normal of the boundary

\(N_{{x_{1} }}\), \(N_{{x_{2} }}\)

Number of grid points in \(x_{1}\), \(x_{2}\) directions


Stress resultants associated with \(\sigma_{ij}\)

\(P_{{x_{1} }}\), \(P_{{x_{2} }}\)

In-plane buckling loads

\(N_{{x_{1} }}^{0}\), \(N_{{x_{2} }}^{0}\), \(N_{{x_{1} x_{2} }}^{0}\)

Thermally induced initial in-plane forces


Critical buckling load


Stress-free state temperature


Strain energy

\(u_{1} ,\)\(u_{2}\), \(u_{3}\)

Displacements along \(x_{1} ,\) \(x_{2}\), \(x_{3}\) directions


Displacement of mid-plane along \(x_{1}\) direction


Volume fraction


Displacement of mid-plane along \(x_{2}\) direction


Work done by external forces


Displacement of mid-plane along \(x_{3}\) direction


Stress resultant associated with \(m_{ij}\)


Coefficient of thermal expansion


Length scale parameter ratio


Boundary curve enclosing mid-plane of micro-plate

\(\Delta T\)

Temperature change from \(T_{0}\)

\(\Delta T_{\text{cr}}\)

Critical buckling temperature difference


Kronecker delta


Strain tensor

\(\theta_{1}\), \(\theta_{2}\)

Transverse shear strains of any point on the mid-plane


Shear modulus


Poisson’s ratio


Mass density


Cauchy stress tensor

\(\phi_{1}\), \(\phi_{2}\)

Rotations of the transverse normal about \(x_{2}\), \(x_{1}\)


Symmetric curvature tensor




Natural frequency



This work was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through grant 213M606.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Aeronautical EngineeringUniversity of Turkish Aeronautical AssociationAnkaraTurkey

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