## 2 Materials and Methods
### Axisymmetric Annular Plate Governing Equations
\begin{array}{ccccc}A_{11}&0&(\lambda B_{11}-\beta E_{11})&0&[( \lambda-1)B_{11}-\beta E_{11}]&0&0\\ 0&A_{11}&0&(\lambda B_{11}-\beta E_{11})&0&[(\lambda-1)B_{11}-\beta E_{11}]&0\\ B_{11}&0&(\lambda D_{11}-\beta F_{11})&0&[(\lambda-1)D_{11}-\beta F_{11}]&0\\ 0&B_{11}&0&(\lambda D_{11}-\beta F_{11})&0&[(\lambda-1)D_{11}-\beta F_{11}]&0\\ E_{11}&0&(\lambda F_{11}-\beta H_{11})&0&[(\lambda-1)F_{11}-\beta H_{11}]&0&0\\ 0&E_{11}&0&(\lambda F_{11}-\beta H_{11})&0&[(\lambda-1)F_{11}-\beta H_{11}]&0 \\ 0&0&0&0&0&\left(\lambda\frac{\lambda\mu}{\kappa}-3\beta\frac{D\mu}{\kappa} \right)\\ 0&0&0&0&0&\left(\lambda\frac{D\mu}{\kappa}-3\beta\frac{D\mu}{\kappa}\right) \end{array} \right] \tag{10}\]\[\mathbf{B}=\left( \begin{array}{cccccc}\frac{\partial}{\partial r}+\frac{u}{r}&\mu \frac{\partial}{\partial r}+\frac{1}{r}&0&0&0&0&0\\ 0&0&\frac{\partial}{\partial r}+\frac{u}{r}&\mu\frac{\partial}{\partial r}+ \frac{1}{r}&0&0&1\\ 0&0&0&0&\frac{\partial^{2}}{\partial r^{2}}+\frac{u}{r}\frac{\partial}{ \partial r}&\mu\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{ \partial r}&\frac{\partial}{\partial r}\end{array} \right)^{T} \tag{11}\]  
\[\mathbf{f}=\left( \begin{array}{cccccc}u&\varphi&w\end{array} \right)^{T} \tag{12}\]  
The coefficients of each component in the coefficient matrix \(\mathbf{A}\) are  
\[\left(A_{11},B_{11},D_{11},E_{11},F_{11},H_{11}\right)=\int_{-h/2}^{h/2}\frac{ E(z,T)}{1-\mu^{2}}\Big{(}1,z,z^{2},z^{3},z^{4},z^{6}\Big{)}\mathrm{d}z \tag{13}\]  
\[\left(A_{44},D_{44},F_{44}\right)=\int_{-h/2}^{h/2}\frac{E(z,T)}{2(1+\mu)} \Big{(}1,z^{2},z^{4}\Big{)}\mathrm{d}z \tag{14}\]  
Substituting expression (7) into the nonlocal theory expression (3), the internal force relationship between the classical theory and the nonlocal theory can be obtained  
\[\Big{[}1-(\varepsilon_{0}a)^{2}\
abla^{2}\Big{]}\mathbf{s}=\mathbf{s}^{{}^{\prime}}= \mathbf{ABf} \tag{15}\]  
\[\mathbf{s}=\left(N_{r}N_{\theta}M_{r}M_{\theta}P_{r}P_{\theta}Q_{r}R_{r}\right)^{T} \tag{16}\]  
\[\left\{ \begin{array}{l}\left(N_{r}M_{r}P_{r}\right)=\int_{-h/2}^{h/2}\sigma_ {r}\big{(}1,z,z^{3}\big{)}\mathrm{d}\;z\\ \left(N_{\theta}M_{\theta}P_{\theta}\right)=\int_{-h/2}^{h/2}\sigma_{\theta} \big{(}1,z,z^{3}\big{)}\mathrm{d}\;z\\ \left(Q_{r}R_{r}\right)=\int_{-h/2}^{h/2}\tau_{\mathrm{r}}\big{(}1,z^{2} \big{)}\mathrm{d}\;z\end{array} \right. \tag{17}\]  
where \(N_{r}\), \(N_{\theta}\) represent nonlocal axial force; \(M_{r}\), \(M_{\theta}\) represent nonlocal bending moment; \(Q_{r}\) is the nonlocal shear force; \(P_{r}\), \(P_{\theta}\) are higher order nonlocal bending moments; \(R_{r}\) is a higher order nonlocal shear force;  
Strain energy  
\[U=\int_{V}\big{(}\sigma^{\prime}r_{e}\varepsilon_{r}+\sigma^{\prime}{}_{ \theta}\varepsilon_{\theta}+\tau^{\prime}{}_{r\mathrm{z}}\gamma_{r\mathrm{z}} \big{)}\mathrm{d}V \tag{18}\]  
Kinetic energy  
\[T=\int_{V}\frac{1}{2}\rho\big{(}z,T\big{)}\Bigg{[}\left(\frac{\partial u_{r}}{ \partial t}\right)^{2}+\left(\frac{\partial u_{z}}{\partial t}\right)^{2} \Bigg{]}\mathrm{d}V \tag{19}\]  
External force potential energy  
\[H=-\frac{1}{2}\int_{r_{1}}^{r_{2}}\int_{0}^{2\pi}rN^{RT}\bigg{(}\frac{\partial w }{\partial r}\bigg{)}^{2}\mathrm{d}\theta\mathrm{d}r \tag{20}\]  
where \(N^{RT}=N^{R}+N^{T}\), \(N^{R}\) and \(N^{T}\) are the radial internal force generated due to the rotation and the radial internal force caused by temperature changes, respectively. The internal force produced by the temperature change is  
\[N^{T}=-\int_{-h/2}^{h/2}\frac{E(z,T)}{1-\mu}\alpha(z,T)\Delta T\mathrm{d}z \tag{21}\]  
\(N^{R}\) can be solved by the plane stress problem.