## 2 Materials and Methods
### Axisymmetric Annular Plate Governing Equations  
There are different theories for the strain in the thickness direction of the plates; the commonly used ones are the Kirchhoff plate, Mindlin plate, and Reddy plate strain theory. Kirchhoff plate theory and Mindlin plate theory assume that the cross-section of the plate remains flat after bending deformation, while the Kirchhoff plate theory ignores the influence of shear deformation and cross-section rotation around the neutral axis. The Reddy plate theory assumes that the cross-section of the plate is cubic after bending  
Figure 1: Schematic diagram of the cross-section rotating functionally graded nano-annular plate. and deformation. The displacement modes of the three plate theories can be uniformly expressed as  
\[\left\{ \begin{array}{l}u_{r}(r,z,t)=u(r,t)+z\big{[}(\lambda-1)-\beta z^{2}\big{]} \frac{\partial w(r,t)}{\partial r}+z(\lambda-\beta z^{2})\varphi(r,t)\\ u_{z}(r,z,t)=w(r,t)\end{array} \right. \tag{4}\]  
where \(u_{r}(r,z,t)\) and \(u_{z}(r,z,t)\) represent the respective radial displacement and lateral displacement of any point in the annular plate; \(u(r,t)\) and \(w(r,t)\) represent the respective radial displacement and lateral displacement of any point in the midplane of the annular plate; \(\varphi(r,t)\) is the rotation angle of the surface normal of the annular plate, and \(t\) is the time variable. When \(\lambda=\beta=0\), expression (4) is the displacement expression of [PERSON] plate theory; when \(\lambda=1,\beta=0\), expression (4) is the displacement expression of [PERSON] plate theory; when \(\lambda=1,\beta=4/\left(3h^{2}\right)\), expression (4) represents the displacement expression for the [PERSON] plate theory. From the geometric relationship, the relationship between the strain and displacement of the annular plate is  
\[\left( \begin{array}{c}\varepsilon_{r}\\ \varepsilon_{\theta}\\ \gamma_{rz}\end{array} \right)=\left[ \begin{array}{ccc}\frac{\lambda}{ \partial r}&\frac{1}{r}&\frac{\lambda}{\partial r}\\ 0&0&\frac{\partial}{\partial r}\end{array} \right]^{\text{T}}\left( \begin{array} []{c}u_{r}\\ u_{z}\end{array} \right) \tag{5}\]  
where \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) represent the respective radial strain and hoop strain of the annular plate; \(\gamma_{rz}\) is the shear strain. Since the [PERSON] plate theory ignores the shear strain, the corresponding shear stress and shear force are no longer considered. According to the physical equations, the expressions of classical stress can be derived as  
\[\boldsymbol{\sigma}^{{}^{\prime}}=\frac{E(z,T)}{1-\mu^{2}} \begin{bmatrix}1& \mu&0\\ \mu&1&0\\ 0&0&1/2\kappa\end{bmatrix} \left( \begin{array}{c}\varepsilon_{r}\\ \varepsilon_{\theta}\\ \gamma_{rz}\end{array} \right) \tag{6}\]  
where \(\kappa\) is the shear correction factor; take \(\text{k}=1\) for [PERSON] plate theory; take \(\kappa=12/\pi^{2}\) for [PERSON] plate theory; for [PERSON] plate theory, \(\kappa\rightarrow\infty\), then \(\tau^{\prime}{}_{rz}=0\). From the expressions (6), the internal force of the annular plate under the classical theory can be derived  
\[\boldsymbol{s}^{{}^{\prime}}=\boldsymbol{A}\boldsymbol{B}\boldsymbol{f} \tag{7}\]  
where \(\boldsymbol{s}^{{}^{\prime}},\boldsymbol{A},\boldsymbol{B},\boldsymbol{f}\) are the internal force vector, coefficient matrix, operator matrix, displacement vector, respectively. The specific representations are  
\[\boldsymbol{s}^{{}^{\prime}}=\left(N^{\prime}{}_{r}N^{\prime}{}_{\theta}M^{ \prime}{}_{r}M^{\prime}{}_{\theta}P^{\prime}{}_{r}P^{\prime}{}_{\theta}Q^{ \prime}{}_{r}R^{\prime}{}_{r}\right)^{\text{T}} \tag{8}\]  
\[\left\{ \begin{array}{l}(N^{\prime}{}_{r}M^{\prime}{}_{r}P^{\prime}{}_{r})= \int_{-h/2}^{h/2}\sigma^{\prime}{}_{r}(1,z,z^{3})\,\text{d}\,z\\ (N^{\prime}{}_{\theta}M^{\prime}{}_{\theta}P^{\prime}{}_{\theta})=\int_{-h/2}^ {h/2}\sigma^{\prime}{}_{\theta}(1,z,z^{3})\,\text{d}\,z\\ (Q^{\prime}{}_{r}R^{\prime}{}_{r})=\int_{-h/2}^{h/2}\tau^{\prime}{}_{rz}\big{(} 1,z^{2}\big{)}\,\text{d}\,z\\ \end{array} \right. \tag{9}\]  
\[A=\left[