## 2 Model Description and Theoretical Analysis
\tag{1}\]  
where \(\sigma=-s_{11}^{D}/s_{12}^{D}\) denotes Poisson's ratio of piezoelectric materials at a constant electrical displacement, \(s_{11}^{D}\) and \(s_{12}^{D}\) denote the compliance coefficients of piezoelectric materials, \(\vec{\beta}_{33}=\beta_{33}^{T}/(1-k_{p}^{2})\), \(\beta_{33}^{T}\) denotes the dielectric impermeability at a constant stress field, \(k_{p}\) denotes the plane electromechanical coupling coefficient, \(E_{3}\) denotes the electric field component, \(D_{3}\) denotes the electric displacement component, \(T_{1}\) and \(T_{2}\) denote the stress components of the composite layer, \(S_{1}\) and \(S_{2}\) denote the strain components, and \(g_{31}\) denotes the piezoelectric coefficient. In the metal substrate segment, the strain and stress Equations [27] for the metal substrate are  
\[\left\{ \begin{array}{l}T_{1}^{\prime}=\frac{E_{m}}{1-\sigma_{m}^{2}}(S_{1}+ \sigma_{m}S_{2})\\ T_{2}^{\prime}=\frac{E_{m}}{1-\sigma_{m}^{2}}(\sigma_{m}S_{1}+S_{2})\end{array}  \right., \tag{2}\]  
Figure 2: Configuration of the trilaminar bending ring with unequal diameters. (**a**) the cross-sectional view; (**b**) the top view. where \(T_{1}^{\prime}\) and \(T_{2}^{\prime}\) are the stress components of the metal substrate, \(E_{m}\) is [PERSON]'s modulus of the metal substrate, and \(\sigma_{m}\) is [PERSON]'s ratio of the metal substrate. Considering [PERSON]'s hypotheses and axisymmetric problem hypotheses, the strain components can be reduced to the following form:  
\[\left\{ \begin{array}{l}S_{1}=-z\frac{\partial^{2}w}{\partial r^{2}}\\ S_{2}=-z\frac{1}{r}\frac{\partial w}{\partial r}\end{array} \right., \tag{3}\]  
For the composite layer segment, the bending moment per unit width can be expressed as  
\[\left\{ \begin{array}{l}M_{r}=\int_{-\frac{1}{2}m}^{\frac{\hbar m}{2}}T_{1}^ {\prime}zdz+\int_{\frac{1}{2}m}^{\frac{\hbar}{2}}T_{1}zdz+\int_{-\frac{1}{2} }^{-\frac{\hbar m}{2}}T_{1}zdz\\ M_{\theta}=\int_{-\frac{1}{2}m}^{\frac{\hbar m}{2}}T_{2}^{\prime}zdz+\int_{ \frac{1}{2}}^{\frac{\hbar}{2}}T_{2}zdz+\int_{-\frac{1}{2}}^{-\frac{\hbar m}{ 2}}T_{2}zdz\end{array} \right., \tag{4}\]  
where \(M_{r}\) and \(M_{\theta}\) are the bending moments in the \(r\) direction and \(z\) direction respectively. Substitution of Equations (1) and (3) into Equation (4) leads to  
\[\left\{ \begin{array}{l}M_{r}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})} \left(\frac{\partial^{2}w}{\partial r^{2}}+\sigma_{m}\frac{1}{r}\frac{\partial w }{\partial r}\right)-\frac{h^{3}-h_{m}^{3}}{12s_{11}^{2}(1-\sigma^{2})}\left( \frac{\partial^{2}w}{\partial r^{2}}+\frac{r}{r}\frac{\partial w}{\partial r} \right)-\frac{g_{31}(h^{2}-h_{m}^{2})}{4s_{11}^{2}(1-\sigma)}D_{3}\\ M_{\theta}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}\left(\sigma_{m}\frac{ \partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r} \right)-\frac{h^{3}-h_{m}^{3}}{12s_{11}^{2}(1-\sigma^{2})}\left(\sigma\frac{ \partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r} \right)-\frac{g_{31}(h^{2}-h_{m}^{2})}{4s_{11}^{2}(1-\sigma)}D_{3}\end{array}  \right., \tag{5}\]  
For the metal substrate segment, the bending moment per unit width is  
\[\left\{ \begin{array}{l}M_{r}^{\prime}=\int_{-\frac{1}{2}m}^{\frac{\hbar m}{2 }}T_{1}^{\prime}zdz\\ M_{\theta}^{\prime}=\int_{-\frac{1}{2}m}^{\frac{\hbar m}{2}}T_{2}^{\prime}zdz \end{array} \right., \tag{6}\]  
Insertion of Equations (2) and (3) into Equation (6) leads to  
\[\left\{ \begin{array}{l}M_{r}^{\prime}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^ {2})}\left(\frac{\partial^{2}w}{\partial r^{2}}+\sigma_{m}\frac{1}{r}\frac{ \partial w}{\partial r}\right)\\ M_{\theta}^{\prime}=\frac{-E_{m}h_{m}^{3}}{12(1-\sigma_{m}^{2})}\left(\sigma_ {m}\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r }\right)\end{array} \right., \tag{7}\]  
For a thin plate, its moment of inertia can be ignored.