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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 Aghazadeh
  • Serkan Dag
  • Ender Cigeroglu
Technical Paper
  • 122 Downloads

Abstract

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

A

Area of mid-plane of micro-plate

a

Length of micro-plate

b

Width of micro-plate

c

Ceramic phase index

E

Young’s modulus

\(e_{ijk}\)

Alternating tensor

f

Shape function for plate theories

h

Thickness of micro-plate

K

Kinetic energy

\(k_{s}\)

Shear correction factor

\(l\)

Material length scale parameter

m

Metallic phase index

\(m_{ij}\)

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

n

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

\(M_{pq}^{i}\)

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

P

Critical buckling load

\(T_{0}\)

Stress-free state temperature

U

Strain energy

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

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

\(u\)

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

V

Volume fraction

\(v\)

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

W

Work done by external forces

\(w\)

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

\(Y_{pq}^{i}\)

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

\(\alpha\)

Coefficient of thermal expansion

β

Length scale parameter ratio

Γ

Boundary curve enclosing mid-plane of micro-plate

\(\Delta T\)

Temperature change from \(T_{0}\)

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

Critical buckling temperature difference

\(\delta_{ij}\)

Kronecker delta

\(\varepsilon_{ij}\)

Strain tensor

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

Transverse shear strains of any point on the mid-plane

\(\mu\)

Shear modulus

\(\nu\)

Poisson’s ratio

\(\rho\)

Mass density

\(\sigma_{ij}\)

Cauchy stress tensor

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

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

\(\chi_{ij}\)

Symmetric curvature tensor

Ω

Volume

ω

Natural frequency

Notes

Acknowledgements

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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