Skip to main content
SpringerLink
Log in

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

  • Technical Paper
  • Published:
Microsystem Technologies Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Abbreviations

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

References

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Middle East Technical University, 06800, Ankara, Turkey

    Reza Aghazadeh, Serkan Dag & Ender Cigeroglu

  2. Department of Aeronautical Engineering, University of Turkish Aeronautical Association, 06790, Ankara, Turkey

    Reza Aghazadeh

Authors
  1. Reza Aghazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serkan Dag
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ender Cigeroglu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reza Aghazadeh.

Appendix

Appendix

Governing equations:

$$\begin{aligned} \delta u: & \\ & - \frac{1}{4}A_{552} \frac{{\partial^{4} u}}{{\partial x_{2}^{4} }} - \frac{1}{4}A_{552} \frac{{\partial^{4} u}}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + A_{11} \frac{{\partial^{2} u}}{{\partial x_{1}^{2} }} + A_{55} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{4}A_{552} \frac{{\partial^{4} v}}{{\partial x_{1}^{3} \partial x_{2} }} \\ & + \frac{1}{4}A_{552} \frac{{\partial^{4} v}}{{\partial x_{1} \partial x_{2}^{3} }} + (A_{55} + A_{L11} )\frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + ( F_{11} - B_{11} )\frac{{\partial^{3} w}}{{\partial x_{1}^{3} }} + (F_{11} - B_{11} )\frac{{\partial^{3} w}}{{\partial x_{1} \partial x_{2}^{2} }} \\ & - \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{2}^{4} }} - \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + F_{11} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1}^{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} \\ & + \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{3} \partial x_{2} }} + \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L11} + F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }} \\ & - \frac{{\partial A_{T11} }}{{\partial x_{1} }} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} + (I_{3} - I_{1} )\frac{{\partial^{3} w}}{{\partial x_{1} \partial t^{2} }} + I_{3} \frac{{\partial^{2} \phi_{1} }}{{\partial t^{2} }}, \\ \end{aligned}$$
(66)
$$\begin{aligned} \delta v: & \\ & \frac{1}{4}A_{552} \frac{{\partial^{4} u}}{{\partial x_{1}^{3} \partial x_{2} }} + \frac{1}{4}A_{552} \frac{{\partial^{4} u}}{{\partial x_{1} \partial x_{2}^{3} }} + (A_{55} + A_{L11} )\frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{4}A_{552} \frac{{\partial^{4} v}}{{\partial x_{1}^{4} }} \\ & - \frac{1}{4}A_{552} \frac{{\partial^{4} v}}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + A_{55} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + A_{11} \frac{{\partial^{2} v}}{{\partial x_{2}^{2} }} + (F_{11} - B_{11} )\frac{{\partial^{3} w}}{{\partial x_{2}^{3} }} + (F_{11} - B_{11} )\frac{{\partial^{3} w}}{{\partial x_{1}^{2} \partial x_{2} }} \\ & + \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1}^{3} \partial x_{2} }} + \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L11} + F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} \\ & - \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{4} }} - \frac{1}{4}F_{472} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + F_{11} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{2}^{2} }} \\ & - \frac{{\partial A_{T11} }}{{\partial x_{2} }} = I_{0} \frac{{\partial^{2} v}}{{\partial t^{2} }} + (I_{3} - I_{1} )\frac{{\partial^{3} w}}{{\partial x_{2} \partial t^{2} }} + I_{3} \frac{{\partial^{2} \phi_{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
(67)
$$\begin{aligned} \delta w: & \\ & (B_{11} - F_{11} )\frac{{\partial^{3} u}}{{\partial x_{1}^{3} }} + (B_{11} - F_{11} )\frac{{\partial^{3} u}}{{\partial x_{1} \partial x_{2}^{2} }} + (B_{11} - F_{11} )\frac{{\partial^{3} v}}{{\partial x_{2}^{3} }} + (B_{11} - F_{11} )\frac{{\partial^{3} v}}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{4} w}}{{\partial x_{1}^{4} }} \\ & \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{4} w}}{{\partial x_{2}^{4} }} \\ & \quad + \left( { - 2D_{11} + 4F_{22} - 2F_{33} - 2A_{552} - \frac{1}{2}F_{552} + 2F_{572} } \right)\frac{{\partial^{4} w}}{{\partial x_{1}^{2} \partial x_{2}^{2} }} \\ & \quad + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} \phi_{1} }}{{\partial x_{1}^{3} }} \\ & \quad + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} \\ & \quad + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} \phi_{2} }}{{\partial x_{2}^{3} }} + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \quad + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} - P_{{x_{1} }} \frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} - P_{{x_{2} }} \frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + N_{{x_{1} }}^{0} \frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + N_{{x_{2} }}^{0} \frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + 2N_{{x_{1} x_{2} }}^{0} \frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} \\ & \quad - \frac{{\partial^{2} (B_{T11} - F_{T11} )}}{{\partial x_{1}^{2} }} - \frac{{\partial^{2} (B_{T11} - F_{T11} )}}{{\partial x_{2}^{2} }} \\ & = (I_{1} - I_{3} )\frac{{\partial^{3} u}}{{\partial x_{1} \partial t^{2} }} + (I_{1} - I_{3} )\frac{{\partial^{3} v}}{{\partial x_{2} \partial t^{2} }} + (2I_{4} - I_{2} - I_{5} )\frac{{\partial^{4} w}}{{\partial x_{1}^{2} \partial t^{2} }} + (2I_{4} - I_{2} - I_{5} )\frac{{\partial^{4} w}}{{\partial x_{2}^{2} \partial t^{2} }} \\ & \quad + I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} + (I_{4} - I_{5} )\frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial t^{2} }} + (I_{4} - I_{5} )\frac{{\partial^{3} \phi_{2} }}{{\partial x_{2} \partial t^{2} }}, \\ \end{aligned}$$
(68)
$$\begin{aligned} \delta \phi_{1} : & \\ & - \frac{1}{4}F_{472} \frac{{\partial^{4} u}}{{\partial x_{2}^{4} }} - \frac{1}{4}F_{472} \frac{{\partial^{4} u}}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + F_{11} \frac{{\partial^{2} u}}{{\partial x_{1}^{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{4}F_{472} \frac{{\partial^{4} v}}{{\partial x_{1}^{3} \partial x_{2} }} \\ & + \frac{1}{4}F_{472} \frac{{\partial^{4} v}}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L11} + F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1}^{3} }} \\ & + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1} \partial x_{2}^{2} }} + \left( { - k_{s} F_{55} - \frac{1}{4}F_{662} } \right)\frac{\partial w}{{\partial x_{1} }} - \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{2}^{4} }} \\ & - \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + \left( {F_{33} + \frac{1}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{1}^{2} }} + \left( {F_{44} + \frac{1}{2}F_{462} + F_{552} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} \\ & + \left( { - k_{s} F_{55} - \frac{1}{4}F_{662} } \right)\phi_{1} + \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{3} \partial x_{2} }} \\ & + \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L33} + F_{44} - \frac{1}{2}F_{462} - \frac{3}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }} \\ & - \frac{{\partial F_{T11} }}{{\partial x_{1} }} = I_{3} \frac{{\partial^{2} u}}{{\partial t^{2} }} + (I_{5} - I_{4} )\frac{{\partial^{3} w}}{{\partial x_{1} \partial t^{2} }} + I_{5} \frac{{\partial^{2} \phi_{1} }}{{\partial t^{2} }}, \\ \end{aligned}$$
(69)
$$\begin{aligned} \delta \phi_{2} : & \\ & \frac{1}{4}F_{472} \frac{{\partial^{4} u}}{{\partial x_{1}^{3} \partial x_{2} }} + \frac{1}{4}F_{472} \frac{{\partial^{4} u}}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L11} + F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{4}F_{472} \frac{{\partial^{4} v}}{{\partial x_{1}^{4} }} \\ & - \frac{1}{4}F_{472} \frac{{\partial^{4} v}}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + F_{11} \frac{{\partial^{2} v}}{{\partial x_{2}^{2} }} + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{2}^{3} }} \\ & + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1}^{2} \partial x_{2} }} + \left( { - k_{s} F_{55} - \frac{1}{4}F_{662} } \right)\frac{\partial w}{{\partial x_{2} }} + \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1}^{3} \partial x_{2} }} \\ & + \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{3} }} + \left( {F_{L33} + F_{44} - \frac{1}{2}F_{462} - \frac{3}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} - \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{4} }} - \frac{1}{4}F_{442} \frac{{\partial^{4} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} \\ & + \left( {F_{44} + \frac{1}{2}F_{462} + F_{552} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \left( {F_{33} + \frac{1}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{2}^{2} }} + \left( { - k_{s} F_{55} - \frac{1}{4}F_{662} } \right)\phi_{2} \\ & - \frac{{\partial (F_{T11} )}}{{\partial x_{2} }} = I_{3} \frac{{\partial^{2} v}}{{\partial t^{2} }} + (I_{5} - I_{4} )\frac{{\partial^{3} w}}{{\partial x_{2} \partial t^{2} }} + I_{5} \frac{{\partial^{2} \phi_{2} }}{{\partial t^{2} }}. \\ \end{aligned}$$
(70)

Corresponding boundary conditions:

$$\begin{aligned} \delta u & = 0{\text{ or}} \\ & \left( {\left( { - \frac{1}{8}A_{552} } \right)\frac{{\partial^{3} u}}{{\partial x_{1} \partial x_{2}^{2} }} + A_{11} \frac{\partial u}{{\partial x_{1} }} + \left( {\frac{1}{8}A_{552} } \right)\frac{{\partial^{3} v}}{{\partial x_{1}^{2} \partial x_{2} }} + A_{L11} \frac{\partial v}{{\partial x_{2} }}} \right. + (F_{11} - B_{11} )\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} \\& \quad + \left( {F_{L11} - B_{L11} - \frac{1}{8}F_{672} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + \left( { - \frac{1}{8}F_{472} } \right)\frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{2} }} + (F_{11} )\frac{{\partial \phi_{1} }}{{\partial x_{1} }} + \left( {\frac{1}{8}F_{472} } \right)\frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2} }} \\& \quad \left. { + \left( {F_{L11} - \frac{1}{8}F_{672} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} - A_{T11} } \right)n_{{x_{1} }} + \left( {\left( { - \frac{1}{4}A_{552} } \right)\frac{{\partial^{3} u}}{{\partial x_{2}^{3} }} + \left( { - \frac{1}{8}A_{552} } \right)\frac{{\partial^{3} u}}{{\partial x_{1}^{2} \partial x_{2} }} + A_{55} \frac{\partial u}{{\partial x_{2} }}} \right. \\& \quad + \left( {\frac{1}{8}A_{552} } \right)\frac{{\partial^{3} v}}{{\partial x_{1}^{3} }} + \left( {\frac{1}{4}A_{552} } \right)\frac{{\partial^{3} v}}{{\partial x_{1} \partial x_{2}^{2} }} + A_{55} \frac{\partial v}{{\partial x_{1} }} + \left( {2F_{47} - 2B_{55} + \frac{1}{8}F_{672} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} \\& \quad + \left( { - \frac{1}{4}F_{472} } \right)\frac{{\partial^{3} \phi_{1} }}{{\partial x_{2}^{3} }} + \left( { - \frac{1}{8}F_{472} } \right)\frac{{\partial^{3} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} \\& \left. {\quad + \left( {\frac{1}{8}F_{472} } \right)\frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{3} }} + \left( {\frac{1}{4}F_{472} } \right)\frac{{\partial^{3} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {F_{47} - \frac{1}{8}F_{672} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(71)
$$\begin{aligned} \delta \frac{\partial u}{{\partial x_{1} }} & = 0{\text{ or}} \\ & \left( {\frac{1}{8}A_{552} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{672} \frac{\partial w}{{\partial x_{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }}} \right. \\ & \quad \left. { - \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{672} \phi_{2} } \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(72)
$$\begin{aligned} \delta \frac{\partial u}{{\partial x_{2} }} & = 0{\text{ or}} \\ & \left( {\frac{1}{8}A_{552} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{672} \frac{\partial w}{{\partial x_{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }}} \right. \\ & \left. {\quad - \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{672} \phi_{2} } \right)n_{{x_{1} }} + \left( {\frac{1}{4}A_{552} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} - \frac{1}{4}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{4}F_{672} \frac{\partial w}{{\partial x_{1} }}} \right. \\ & \left. {\quad + \frac{1}{4}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} - \frac{1}{4}F_{672} \phi_{1} - \frac{1}{4}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(73)
$$\begin{aligned} \delta v & = 0{\text{ or}} \\ & \left( {\frac{1}{8}A_{552} \frac{{\partial^{3} u}}{{\partial x_{2}^{3} }} + \frac{1}{4}A_{552} \frac{{\partial^{3} u}}{{\partial x_{1}^{2} \partial x_{2} }} + A_{55} \frac{\partial u}{{\partial x_{2} }} - \frac{1}{8}A_{552} \frac{{\partial^{3} v}}{{\partial x_{1} \partial x_{2}^{2} }} - \frac{1}{4}A_{552} \frac{{\partial^{3} v}}{{\partial x_{1}^{3} }} + A_{55} \frac{\partial v}{{\partial x_{1} }}} \right. \\& \quad + \left( {2F_{47} - 2B_{55} + \frac{1}{8}F_{672} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{2}^{3} }} + \frac{1}{4}F_{472} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2} }} + \left( {F_{47} - \frac{1}{8}F_{672} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} \\& \left. {\quad - \frac{1}{4}F_{472} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{3} }} - \frac{1}{8}F_{472} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {F_{47} + \frac{1}{4}F_{672} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{1} }} \\& \quad + \left( {\frac{1}{8}A_{552} \frac{{\partial^{3} u}}{{\partial x_{1} \partial x_{2}^{2} }} + A_{L11} \frac{\partial u}{{\partial x_{1} }} - \frac{1}{8}A_{552} \frac{{\partial^{3} v}}{{\partial x_{1}^{2} \partial x_{2} }} + A_{11} \frac{\partial v}{{\partial x_{2} }}} \right. \\& \quad + \left( {F_{L11} - B_{L11} - \frac{1}{8}F_{672} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + (F_{11} - B_{11} )\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} \\& \left. {\quad + \frac{1}{8}F_{472} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {F_{L11} - \frac{1}{8}F_{672} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} - \frac{1}{8}F_{472} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2} }} + F_{11} \frac{{\partial \phi_{2} }}{{\partial x_{2} }} - A_{T11} } \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(74)
$$\begin{aligned} \delta \frac{\partial v}{{\partial x_{1} }} & = 0{\text{ or}} \\ & \left( { - \frac{1}{4}A_{552} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{4}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} - \frac{1}{4}F_{672} \frac{\partial w}{{\partial x_{2} }} - \frac{1}{4}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }}} \right. \\ & \left. {\quad + \frac{1}{4}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} - \frac{1}{4}F_{672} \phi_{2} } \right)n_{{x_{1} }} + \left( { - \frac{1}{8}A_{552} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{8}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{672} \frac{\partial w}{{\partial x_{1} }}} \right. \\ & \left. {\quad - \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{672} \phi_{1} + \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{2} }} , \\ \end{aligned}$$
(75)
$$\begin{aligned} \delta \frac{\partial v}{{\partial x_{2} }} & = 0{\text{ or}} \\ & \left( { - \frac{1}{8}A_{552} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{8}A_{552} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{672} \frac{\partial w}{{\partial x_{1} }}} \right. \\ & \quad \left. { - \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{672} \phi_{1} + \frac{1}{8}F_{472} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{1} }} = 0, \\ \end{aligned}$$
(76)
$$\begin{aligned} \delta w &= 0{\text{ or}} \hfill \\& \left( {(B_{11} - F_{11} )\frac{{\partial^{2} u}}{{\partial x_{1}^{2} }} + \left( {B_{55} - F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \left( {B_{55} + B_{L11} - F_{47} - F_{L11} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }}} \right. \hfill \\& \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1}^{3} }} \hfill \\& \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{\partial w}{{\partial x_{1} }} \hfill \\& \quad + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{1}^{2} }} + \left( { - F_{44} + F_{48} - \frac{1}{4}F_{462} - \frac{1}{2}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} \hfill \\& \quad + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\phi_{1} + \left( {F_{48} + F_{L22} - F_{44} - F_{L33} + \frac{1}{4}F_{462} + \frac{1}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }} \hfill \\& \left. {\quad - \frac{\partial w}{{\partial x_{1} }}P_{{x_{1} }} + \frac{{\partial \left( {F_{T11} - B_{T11} } \right)}}{{\partial x_{1} }}} \right)n_{{x_{1} }} + \left( {\left( {B_{55} + B_{L11} - F_{47} - F_{L11} + \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }}} \right. \hfill \\& \quad + \left( {B_{11} - F_{11} } \right)\frac{{\partial^{2} v}}{{\partial x_{2}^{2} }} + \left( {B_{55} - F_{47} - \frac{1}{4}F_{672} } \right)\frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} \hfill \\& \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{2}^{3} }} \hfill \\& \quad + \left( { - D_{11} + 2F_{22} - F_{33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{3} w}}{{\partial x_{1}^{2} \partial x_{2} }} \hfill \\& \quad + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\frac{\partial w}{{\partial x_{2} }} + \left( {F_{48} + F_{L22} - F_{44} - F_{L33} + \frac{1}{4}F_{462} + \frac{1}{4}F_{552} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} \hfill \\& \quad + \left( { - F_{44} + F_{48} - \frac{1}{4}F_{462} - \frac{1}{2}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \left( {F_{22} - F_{33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{2}^{2} }} \hfill \\& \left. {\quad + \left( {k_{s} F_{55} + \frac{1}{4}F_{662} } \right)\phi_{2} - \frac{\partial w}{{\partial x_{2} }}P_{{x_{2} }} + \frac{{\partial (F_{T11} - B_{T11} )}}{{\partial x_{2} }}} \right)n_{{x_{2} }} \hfill \\& = \left( {(I_{1} - I_{3} )\frac{{\partial^{2} u}}{{\partial t^{2} }} + (2I_{4} - I_{2} - I_{5} )\frac{{\partial^{3} w}}{{\partial x_{1} \partial t^{2} }} + (I_{4} - I_{5} )\frac{{\partial^{2} \phi_{1} }}{{\partial t^{2} }}} \right)n_{{x_{1} }} \hfill \\& \quad + \left( {(I_{1} - I_{3} )\frac{{\partial^{2} v}}{{\partial t^{2} }} + (2I_{4} - I_{2} - I_{5} )\frac{{\partial^{3} w}}{{\partial x_{2} \partial t^{2} }} + (I_{4} - I_{5} )\frac{{\partial^{2} \phi_{2} }}{{\partial t^{2} }}} \right)n_{{x_{2} }} , \hfill \\ \end{aligned}$$
(77)
$$\begin{aligned} \delta \frac{\partial w}{{\partial x_{1} }} &= 0{\text{ or}} \hfill \\ &\left( {( - B_{11} + F_{11} )\frac{\partial u}{{\partial x_{1} }} + ( - B_{L11} + F_{L11} )\frac{\partial v}{{\partial x_{2} }}} \right. \hfill \\& \quad + \left( {D_{11} + F_{33} - 2F_{22} + A_{552} + \frac{1}{4}F_{552} - F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} \hfill \\& \quad + \left( {D_{L11} - 2F_{L22} + F_{L33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} \hfill \\& \left. {\quad + \left( { - F_{L22} + F_{L33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} + (B_{T11} - F_{T11} )} \right)n_{{x_{1} }} + \left( {( - B_{55} + F_{47} )\frac{\partial u}{{\partial x_{2} }}} \right. \hfill \\& \quad ( - B_{55} + F_{47} )\frac{\partial v}{{\partial x_{1} }} + \left( {2D_{55} + 2F_{44} - 4F_{48} + 2A_{552} + \frac{1}{2}F_{552} - 2F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} \hfill \\& \left. {\quad + \left( {F_{44} - F_{48} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} + \left( {F_{44} - F_{48} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{2} }} = 0, \hfill \\ \end{aligned}$$
(78)
$$\begin{aligned} \delta \frac{\partial w}{{\partial x_{2} }} &= 0{\text{ or}} \hfill \\& \left( {( - B_{55} + F_{47} )\frac{\partial u}{{\partial x_{2} }} + ( - B_{55} + F_{47} )\frac{\partial v}{{\partial x_{1} }}} \right. \hfill \\& \quad + \left( {2D_{55} + 2F_{44} - 4F_{48} + 2A_{552} + \frac{1}{2}F_{552} - 2F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} \hfill \\& \left. {\quad + \left( {F_{44} - F_{48} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} + \left( {F_{44} - F_{48} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{1} }} \hfill \\& \quad + \left( {\left( { - B_{L11} + F_{L11} } \right)\frac{\partial u}{{\partial x_{1} }} + ( - B_{11} + F_{11} )\frac{\partial v}{{\partial x_{2} }}} \right. \hfill \\& \quad + \left( {D_{L11} - 2F_{L22} + F_{L33} - A_{552} - \frac{1}{4}F_{552} + F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} \hfill \\& \quad + \left( {D_{11} + F_{33} - 2F_{22} + A_{552} + \frac{1}{4}F_{552} - F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} + \left( { - F_{L22} + F_{L33} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} \hfill \\& \left. {\quad + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} + (B_{T11} - F_{T11} )} \right)n_{{x_{2} }} = 0, \hfill \\ \end{aligned}$$
(79)
$$\begin{aligned} \delta \phi_{1} & = 0{\text{ or}} \hfill \\ & \left( { - \frac{1}{8}F_{472} \frac{{\partial^{3} u}}{{\partial x_{1} \partial x_{2}^{2} }} + F_{11} \frac{\partial u}{{\partial x_{1} }} + \frac{1}{8}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1}^{2} \partial x_{2} }} + F_{L11} \frac{\partial v}{{\partial x_{2} }} + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} } \right.} \right. \hfill \\ & \left. {\quad - \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + \left( { - F_{L22} + F_{L33} - \frac{1}{8}F_{462} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} - \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{2} }} \hfill \\& \quad \left. { + \left( {F_{33} + \frac{1}{4}F_{552} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} + \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2} }} + \left( {F_{L33} - \frac{1}{8}F_{462} - \frac{1}{4}F_{552} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} - F_{T11} } \right)n_{{x_{1} }} \hfill \\& \quad + \left( { - \frac{1}{4}F_{472} \frac{{\partial^{3} u}}{{\partial x_{2}^{3} }} + - \frac{1}{8}F_{472} \frac{{\partial^{3} u}}{{\partial x_{1}^{2} \partial x_{2} }} + F_{47} \frac{\partial u}{{\partial x_{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1}^{3} }} + \frac{1}{4}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1} \partial x_{2}^{2} }} + F_{47} \frac{\partial v}{{\partial x_{1} }}} \right. \hfill \\& \quad + \left( {2F_{44} - 2F_{48} + \frac{1}{8}F_{462} + \frac{1}{2}F_{552} - F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{4}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{2}^{3} }} - \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2} }} \hfill \\& \quad + \left( {F_{44} + \frac{1}{4}F_{462} + F_{552} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} \hfill \\& \left. {\quad + \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{3} }} + \frac{1}{4}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {F_{44} - \frac{1}{8}F_{462} - \frac{1}{2}F_{552} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{2} }} = 0, \hfill \\ \end{aligned}$$
(80)
$$\begin{aligned} \delta \frac{{\partial \phi_{1} }}{{\partial x_{1} }} & = 0{\text{ or}} \hfill \\& \left( {\frac{1}{8}F_{472} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{462} \frac{\partial w}{{\partial x_{2} }}} \right. \hfill \\& \left. {\quad + \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{462} \phi_{2} } \right)n_{{x_{2} }} = 0, \hfill \\ \end{aligned}$$
(81)
$$\begin{aligned} \delta \frac{{\partial \phi_{1} }}{{\partial x_{2} }} &= 0{\text{ or}} \hfill \\ & \left( {\frac{1}{8}F_{472} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{462} \frac{\partial w}{{\partial x_{2} }} + \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} - \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{462} \phi_{2} } \right)n_{{x_{1} }} \hfill \\& \quad + \left( {\frac{1}{4}F_{472} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} - \frac{1}{4}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }}} \right. \hfill \\& \left. {\quad + \left( { - \frac{1}{4}F_{462} } \right)\frac{\partial w}{{\partial x_{1} }} + \left( {\frac{1}{4}F_{442} } \right)\frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} + \left( { - \frac{1}{4}F_{462} } \right)\phi_{1} + \left( { - \frac{1}{4}F_{442} } \right)\frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{2} }} = 0, \hfill \\ \end{aligned}$$
(82)
$$\begin{aligned} \delta \phi_{2} &= 0{\text{ or}} \\ &\left( {\frac{1}{8}F_{472} \frac{{\partial^{3} u}}{{\partial x_{2}^{3} }} + \frac{1}{4}F_{472} \frac{{\partial^{3} u}}{{\partial x_{1}^{2} \partial x_{2} }} + F_{47} \frac{\partial u}{{\partial x_{2} }} - \frac{1}{4}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1}^{3} }} - \frac{1}{8}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1} \partial x_{2}^{2} }} + F_{47} \frac{\partial v}{{\partial x_{1} }}} \right. \\ & \quad + \left( {2F_{44} - 2F_{48} + \frac{1}{8}F_{462} + \frac{1}{2}F_{552} - F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{2}^{3} }} + \frac{1}{4}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1}^{2} \partial x_{2} }} \\& \quad + \left( {F_{44} - \frac{1}{8}F_{462} - \frac{1}{2}F_{552} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{2} }} - \frac{1}{4}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{3} }} - \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \quad \left. { + \left( {F_{44} + \frac{1}{4}F_{462} + F_{552} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{1} }}} \right)n_{{x_{1} }} + \left( {\frac{1}{8}F_{472} \frac{{\partial^{3} u}}{{\partial x_{1} \partial x_{2}^{2} }} + F_{L11} \frac{\partial u}{{\partial x_{1} }} - \frac{1}{8}F_{472} \frac{{\partial^{3} v}}{{\partial x_{1}^{2} \partial x_{2} }}} \right. \\ & \quad + F_{11} \frac{\partial v}{{\partial x_{2} }} + \left( { - F_{22} + F_{33} + \frac{1}{4}F_{552} - \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} \\ & \quad + \left( { - F_{L22} + F_{L33} - \frac{1}{8}F_{462} - \frac{1}{4}F_{552} + \frac{1}{2}F_{572} } \right)\frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{1} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \left. {\quad + \left( {F_{L33} - \frac{1}{8}F_{462} - \frac{1}{4}F_{552} } \right)\frac{{\partial \phi_{1} }}{{\partial x_{1} }} - \frac{1}{8}F_{442} \frac{{\partial^{3} \phi_{2} }}{{\partial x_{1}^{2} \partial x_{2} }} + \left( {F_{33} + \frac{1}{4}F_{552} } \right)\frac{{\partial \phi_{2} }}{{\partial x_{2} }} - F_{T11} } \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(83)
$$\begin{aligned} \delta \frac{{\partial \phi_{2} }}{{\partial x_{1} }} & = 0{\text{ or}} \\ & \left( { - \frac{1}{4}F_{472} \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{4}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} - \frac{1}{4}F_{462} \frac{\partial w}{{\partial x_{2} }} - \frac{1}{4}F_{442} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{1} \partial x_{2} }} + \frac{1}{4}F_{442} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1}^{2} }} - \frac{1}{4}F_{462} \phi_{2} } \right)n_{{x_{1} }} \\ & \quad + \left( { - \frac{1}{8}F_{472} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{462} \frac{\partial w}{{\partial x_{1} }}} \right. \\ & \quad \left. { - \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{462} \phi_{1} + \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{2} }} = 0, \\ \end{aligned}$$
(84)
$$\begin{aligned} \delta \frac{{\partial \phi_{2} }}{{\partial x_{2} }} & = 0{\text{ or}} \\ & \left( { - \frac{1}{8}F_{472} \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{472} \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \frac{1}{8}F_{462} \frac{\partial w}{{\partial x_{1} }}} \right. \\ & \quad \left. { - \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{1} }}{{\partial x_{2}^{2} }} + \frac{1}{8}F_{462} \phi_{1} + \frac{1}{8}F_{442} \frac{{\partial^{2} \phi_{2} }}{{\partial x_{1} \partial x_{2} }}} \right)n_{{x_{1} }} = 0, \\ \end{aligned}$$
(85)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghazadeh, R., Dag, S. & Cigeroglu, E. Thermal effect on bending, buckling and free vibration of functionally graded rectangular micro-plates possessing a variable length scale parameter. Microsyst Technol 24, 3549–3572 (2018). https://doi.org/10.1007/s00542-018-3773-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00542-018-3773-x

Search

Navigation