## 2 Theory and Formulation
#### Nonlinear Temperature Changes
\begin{array}{l}\delta u_{0}:\frac{\partial N_{xx}}{\partial x}+\frac{ \partial N_{yy}}{\partial y}=I_{0}\frac{\partial^{2}u_{0}}{\partial t^{2}}+I_ {1}\frac{\partial^{2}\partial x}{\partial t^{2}}-I_{3}c_{1}\Big{(}\frac{ \partial^{2}\partial x}{\partial t^{2}}+\frac{\partial^{3}w_{0}}{\partial t^ {2}\partial x}\Big{)}\\ \delta v_{0}:\frac{\partial N_{yy}}{\partial y}+\frac{\partial N_{xy}}{ \partial x}=I_{0}\frac{\partial^{2}v_{0}}{\partial t^{2}}+I_{1}\frac{ \partial^{2}\partial y}{\partial t^{2}}-I_{3}c_{1}\Big{(}\frac{\partial^{2} \partial y}{\partial t^{2}}+\frac{\partial^{3}w_{0}}{\partial t^{2}\partial y }\Big{)}\\ \delta w_{0}:c_{1}\frac{\partial^{2}P_{xx}}{\partial x}+c_{1}\frac{\partial^{2 }P_{yy}}{\partial y^{2}}+2c_{1}\frac{\partial^{2}P_{xy}}{\partial x}+\frac{ \partial\partial x_{yz}}{\partial x}-3c_{1}\frac{\partial S_{yz}}{\partial x }+\frac{\partial Q_{yz}}{\partial y}-3c_{1}\frac{\partial S_{yz}}{\partial y} \\ +\frac{N_{xx}}{R_{1}}+\frac{N_{yy}}{R_{2}}-N_{1}^{T}\frac{\partial^{2}\partial x _{yz}}{\partial x^{2}}-N_{2}^{T}\frac{\partial^{2}\partial w_{0}}{\partial x^ {2}}=c_{1}I_{3}\frac{\partial^{3}u_{0}}{\partial x^{2}}+c_{1}I_{4}\frac{ \partial^{3}\partial v_{0}}{\partial t^{2}}-I_{6}c_{1}2\Big{(}\frac{\partial ^{3}\partial_{0}}{\partial x\partial^{2}}+\frac{\partial^{4}w}{\partial t^ {2}\partial x^{2}}\Big{)}\\ +c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial y\partial t^{2}}+c_{1}I_{4}\frac{ \partial^{2}\partial y}{\partial y\partial t^{2}}-I_{6}c_{1}2\Big{(}\frac{ \partial^{3}\partial y}{\partial y\partial t^{2}}+\frac{\partial^{4}w}{ \partial t^{2}\partial y^{2}}\Big{)}+\left(I_{0}\frac{\partial^{2}w_{0}}{ \partial t^{2}}\right)\\ \delta\phi_{x}:\frac{\partial M_{xx}}{\partial x}-c_{1}\frac{\partial P_{xx}} {\partial x}+\frac{\partial M_{yy}}{\partial y}-c_{1}\frac{\partial P_{yy}} {\partial y}-Q_{xz}+3c_{1}S_{xz}=\\ +I_{1}\frac{\partial^{2}v_{0}}{\partial t^{2}}+I_{2}\frac{\partial^{2}\phi_{x }}{\partial t}-I_{4}c_{1}\Big{(}\frac{\partial^{2}\phi_{x}}{\partial t^{2}}+ \frac{\partial^{3}w}{\partial t^{2}\partial x}\Big{)}\\ -c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial t^{2}}-c_{1}I_{4}\frac{\partial^{2 }\phi_{y}}{\partial t^{2}}+I_{6}c_{1}2\Big{(}\frac{\partial^{2}\phi_{x}}{ \partial t^{2}}+\frac{\partial^{3}w}{\partial t^{2}\partial y}\Big{)}\\ \delta\phi_{y}:\frac{\partial M_{yy}}{\partial y}-c_{1}\frac{\partial P_{yy}} {\partial y}+\frac{\partial M_{yy}}{\partial x}-c_{1}\frac{\partial P_{yy}} {\partial x}-Q_{yz}+3c_{1}S_{yz}=\\ I_{1}\frac{\partial^{2}v_{0}}{\partial t^{2}}+I_{2}\frac{\partial^{2}\phi_{y }}{\partial t^{2}}-I_{4}c_{1}\Big{(}\frac{\partial^{2}\phi_{y}}{\partial t^{2}}+ \frac{\partial^{3}w}{\partial t^{2}\partial y}\Big{)}\\ -c_{1}I_{3}\frac{\partial^{2}v_{0}}{\partial t^{2}}-c_{1}I_{4}\frac{\partial^{2 }\phi_{y}}{\partial t^{2}}+I_{6}c_{1}2\Big{(}\frac{\partial^{2}\phi_{y}}{ \partial t^{2}}+\frac{\partial^{3}w}{\partial t^{2}\partial y}\Big{)}\end{array}  \tag{13}\]  
All boundary condition equations of the doubly curved panel can be divided as follows:  
Simply boundary conditions:  
\[ \begin{array}{l}at\ x=0,a\rightarrow\ M_{xx}-c_{1}P_{xx}=0,\ N_{xx}=0,\ \ \phi_{y}=0,\ \ W=0,\ \ \ V=0,\\ at\ y=0,b\rightarrow\ M_{yy}-c_{1}P_{yy}=0,\ N_{yy}=0,\ \ \phi_{x}=0,\ \ W=0,\ \ \ U=0, \end{array} \]  
Clamped boundary conditions:  
\[ \begin{array}{l}at\ x=0,a\rightarrow\ \phi_{y}=0,\ \ \phi_{x}=0,U=0,W=0,\ \ \ \ \ V=0,\\ at\ y=0,b\rightarrow\ \phi_{y}=0,\ \ \phi_{x}=0,U=0,W=0,\ \ \ \ \ V=0, \end{array} \]  
Free boundary conditions:  
\[ \begin{array}{l}at\ x=0,a\rightarrow\ N_{xx}=0,\ M_{xx}-c_{1}P_{xx}=0,\ \ N_{xy}=0,\ \ M_{xy}-c_{1}P_{xy}=0,\\ -c_{1}P_{xx}+c_{1}\frac{\partial^{2}P_{xx}}{\partial x}-2c_{1}\frac{\partial P_{ xy}}{\partial y}+(Q_{xz}-3c_{1}S_{xz})=0\end{array} \]  
Furthermore, the following relations represent the weighting coefficients of higher-order derivatives:  
\[