# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 2 Governing Equation of Thermal Buckling  
In this work, the FGM rectangular plate of length \(a\), width \(b\) and thickness \(h\) is investigated as shown in Figure 1.  
CPT is used with the following displacement fields  
\[ \begin{array}{rcl}u&=&u_{0}-z\frac{\partial w}{\partial x}\\ \
u&=&\
u_{0}-z\frac{\partial w}{\partial y}\\ w&=&w_{0}\end{array}  \tag{1}\]  
where \(u_{0}\) and \(v_{0}\) are the in-plane displacements on the midplane of the plate along the \(x\) and \(y\) direction, respectively; \(w\) is the midplane deflection. The strains associated with the displacement fields in Equation (1) are  
\[\left[ \begin{array}{c}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \varepsilon_{xy}\end{array} \right]=\left[ \begin{array}{c}\frac{\partial u _{0}}{\partial x}-z\frac{\partial^{2}w}{\partial x^{2}}\\ \frac{\partial v_{0}}{\partial y}-z\frac{\partial^{2}w}{\partial y^{2}}\\ \frac{\partial u_{0}}{\partial y}+\frac{\partial v_{0}}{\partial x}-2z\frac{ \partial^{2}w}{\partial x\partial y}\end{array} \right] \tag{2}\]  
The constitutive relation for the FGM plate subjected to thermo-mechanical loadings is  
\[\left[ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{array} \right]=\frac{E}{1-\
u^{2}}\left[ \begin{array}{ccc}1 &\
u&0\\ \
u&1&0\\ 0&0&\frac{1-\
u}{2}\end{array} \right]\left(\left[ \begin{array}{c}\varepsilon _{xx}\\ \varepsilon_{yy}\\ \varepsilon_{xy}\end{array} \right]-\Delta T\left[ \begin{array}{c}\alpha\\ \alpha\\ 0\end{array} \right]\right) \tag{3}\]  
The stress resultants can be written as  
\[ \begin{array}{c}\left(N_{xx},N_{yy},N_{xy}\right)=\int_{-h/2}^{h/2}(\sigma_ {xx},\sigma_{yy},\sigma_{xy})\mathrm{d}z\\ \left(M_{xx},M_{yy},M_{xy}\right)=\int_{-h/2}^{h/2}z\big{(}\sigma_{xx},\sigma_ {yy},\sigma_{xy}\big{)}\mathrm{d}z\end{array}  \tag{4}\]  
Substituting Equations (1)-(3) into Equation (4) yields  
Figure 1: Functionally Graded Material (FGM) thin plate and Cartesian coordinates. \[\left[ \begin{array}{c}N_{xx}\\ M_{xx}\end{array} \right]=\left[ \begin{array}{c}\int_{-h/2}^{h/2}\left\{\frac{E}{ 1-\
u^{2}}\left[\frac{\partial u_{0}}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}}+\
u\left(\frac{\partial v_{0}}{\partial y}-z\frac{\partial^{2} w}{\partial y^{2}}\right)\right]\right\}\mathrm{d}z-N_{T}\\ \int_{-h/2}^{h/2}\left\{\frac{Ez}{1-\
u^{2}}\left[\frac{\partial u_{0}}{ \partial x}-z\frac{\partial^{2}w}{\partial x^{2}}+\
u\left(\frac{\partial v_{0 }}{\partial y}-z\frac{\partial^{2}w}{\partial y^{2}}\right)\right]\right\} \mathrm{d}z-M_{T}\end{array} \right]\]  
\[\left[ \begin{array}{c}N_{yy}\\ M_{yy}\end{array} \right]=\left[ \begin{array}{c}\int_{-h/2}^{h/2}\left\{\frac {E}{1-\
u^{2}}\left[\frac{\partial v_{0}}{\partial y}-z\frac{\partial^{2}w}{ \partial y^{2}}+\
u\left(\frac{\partial u_{0}}{\partial x}-z\frac{\partial^{2 }w}{\partial x^{2}}\right)\right]\right\}\mathrm{d}z-N_{T}\\ \int_{-h/2}^{h/2}\left\{\frac{Ez}{1-\
u^{2}}\left[\frac{\partial v_{0}}{ \partial y}-z\frac{\partial^{2}w}{\partial y^{2}}+\
u\left(\frac{\partial u_{0 }}{\partial x}-z\frac{\partial^{2}w}{\partial x^{2}}\right)\right]\right\} \mathrm{d}z-M_{T}\end{array} \right] \tag{5}\]  
\[\left[ \begin{array}{c}N_{xy}\\ M_{xy}\end{array} \right]=\left[ \begin{array}{c}\int_{-h/2}^{h/2}\left[\frac {E}{2(1+\
u)}\left(\frac{\partial u_{0}}{\partial y}+\frac{\partial v_{0}}{ \partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}\right)\right]\mathrm{ d}z\\ \int_{-h/2}^{h/2}\left[\frac{Ez}{2(1+\
u)}\left(\frac{\partial u_{0}}{ \partial y}+\frac{\partial v_{0}}{\partial x}-2z\frac{\partial^{2}w}{\partial x \partial y}\right)\right]\mathrm{d}z\end{array} \right]\]  
where  
\[N_{T}=\int_{-h/2}^{h/2}\frac{E\alpha T}{1-\
u}\mathrm{d}z \tag{6}\]  
\[M_{T}=\int_{-h/2}^{h/2}\frac{E\alpha\Delta T}{1-\
u}\mathrm{d}z\]  
From Equation (5), it can be seen that the in-plane loading and bending of the plate couple together because there is in-plane stretching in the geometrical midplane due to the materials' asymmetry with respect to the midplane.