## 3 Energy Method
### Strain Energy  
The strain energy for the nanoplate could be obtained as  
\[U=\frac{1}{2}\int_{A}\int_{-h/2}^{h/2}\left(\sigma_{xx}\varepsilon_{xx}+\sigma_{ yy}\varepsilon_{yy}+K_{s}\sigma_{xz}\gamma_{xz}+K_{s}\sigma_{yz}\gamma_{yz}+ \sigma_{xy}\gamma_{xy}\right)\mathrm{d}A\mathrm{d}z. \tag{10}\]  
In the above equation, \(K_{s}\) stands for the shear correction factor that is here set to 5/6, as typically assumed in the literature for nanoplates and single-layer graphene sheets [20].  
The combination of Equations (6), (7), and (10) yields  
\[\begin{array}{ll}U=\frac{1}{2}\int\limits_{A}&\left(N_{xx}\frac{\partial u_{ 0}}{\partial x}+M_{xx}\frac{\partial\varphi_{x}}{\partial x}+N_{yy}\frac{ \partial\varphi_{0}}{\partial y}+M_{yy}\frac{\partial\varphi_{y}}{\partial y} +N_{xy}\left(\frac{\partial u_{0}}{\partial y}+\frac{\partial\varphi_{0}}{ \partial x}\right)\right.\\ &\left.+M_{xy}\Big{(}\frac{\partial\varphi_{x}}{\partial y}+\frac{\partial \varphi_{y}}{\partial x}\Big{)}+Q_{xx}\Big{(}\frac{\partial u_{0}}{\partial x }+\varphi_{x}\Big{)}+Q_{yz}\Big{(}\frac{\partial u_{0}}{\partial y}+\phi_{y }\Big{)}\right)\mathrm{d}A\end{array} \tag{11}\]  
where  
\[\left\{\begin{array}{ll}N_{a\beta}\\ M_{a\beta}\end{array}\right\}=\int_{-\frac{b}{2}}^{\frac{b}{2}}\quad\sigma_{ a\beta}^{nl}\left\{\begin{array}{ll}1\\ z\end{array}\right\}\mathrm{d}z\qquad\alpha=x,\;y\quad\beta=x,\;y \tag{12}\]  
\[Q_{az}=K_{s}\int_{-\frac{b}{2}}^{\frac{b}{2}}\quad\sigma_{xz}^{nl}\mathrm{d}z \qquad\alpha=x,\;y\;. \tag{13}\]  
\(N_{a\beta}\) denotes the resultant force, \(M_{a\beta}\) describes the resultant moment, and \(Q_{az}\) indicates the resultant shear force.  
By substitution of Equations (3), (6), and (7) into Equations (12) and (13), the stress resultant-displacement relations are determined as  
\[\begin{array}{ll}N_{xx}-\mu\
abla^{2}(N_{xx})=A_{11}\frac{\partial u_{0}}{ \partial x}+A_{12}\frac{\partial\varphi_{0}}{\partial y}-N_{xx}^{T}\;,\\ N_{yy}-\mu\
abla^{2}\big{(}N_{yy}\big{)}=A_{12}\frac{\partial u_{0}}{ \partial x}+A_{22}\frac{\partial\varphi_{0}}{\partial y}-N_{yy}^{T}\;,\\ N_{xy}-\mu\
abla^{2}\big{(}N_{xy}\big{)}=A_{66}\Big{(}\frac{\partial u_{0}}{ \partial y}+\frac{\partial\varphi_{0}}{\partial x}\Big{)}\;,\\ \\ M_{xx}-\mu\
abla^{2}(M_{xx})=D_{11}\frac{\partial\varphi_{x}}{\partial x}+D_{12 }\frac{\partial\varphi_{y}}{\partial y}-M_{xx}^{T}\;,\\ M_{yy}-\mu\
abla^{2}\big{(}M_{yy}\big{)}=D_{12}\frac{\partial\varphi_{x}}{ \partial x}+D_{22}\frac{\partial\varphi_{y}}{\partial y}-M_{yy}^{T}\;,\\ M_{xy}-\mu\
abla^{2}\big{(}M_{xy}\big{)}=D_{66}\Big{(}\frac{\partial\varphi_{x}}{ \partial y}+\frac{\partial\varphi_{0}}{\partial x}\Big{)}\;,\\ \\ Q_{yz}-\mu\
abla^{2}\big{(}Q_{yz}\big{)}=J_{44}\Big{(}\phi_{y}+\frac{\partial u _{0}}{\partial y}\Big{)}\;,\\ Q_{xz}-\mu\
abla^{2}\big{(}Q_{xz}\big{)}=J_{55}\Big{(}\phi_{x}+\frac{\partial u _{0}}{\partial x}\Big{)}\;.\end{array} \tag{16}\]  
More specifically, the extensional stiffness \(A_{ij}\), the bending stiffness \(D_{ij}\), and the shear stiffness \(J_{ii}\) of the SLGS, are given as  
\[\big{(}A_{ij},\;D_{ij}\big{)}=\int_{-h/2}^{h/2}\;C_{ij}\Big{(}1,\;z^{2}\Big{)} \mathrm{d}z\qquad\qquad(i,\;j=1,2,6)\;, \tag{17}\]  
\[J_{ii}=K_{s}\int_{-h/2}^{h/2}\;C_{ii}\;\mathrm{d}z\;=\;K_{s}h\;C_{ii}\qquad(i=4,5)\;. \tag{18}\]  
Additionally, \(\Big{(}N_{xx}^{T},N_{yy}^{T}\Big{)}\) and \(\Big{(}M_{xx}^{T},M_{yy}^{T}\Big{)}\) denote the thermally induced forces and moments due to a uniform temperature variation, respectively, determined by  
\[\left\{\begin{array}{l}N_{xx}^{T}\\ N_{yy}^{T}\end{array}\right\}=\int_{-\frac{b}{2}}^{\frac{b}{2}}\;\left\{\begin{array} []{l}A_{11}\alpha_{11}+A_{12}\alpha_{22}\\ A_{12}\alpha_{11}+A_{22}\alpha_{22}\end{array}\right\}\Delta T\;\mathrm{d}z, \tag{19}\]\[\left\{\begin{array}{c}M_{xx}^{T}\\ M_{yy}^{T}\end{array}\right\}=\int_{-\frac{\hbar}{2}}^{\frac{\hbar}{2}}\quad \left\{\begin{array}{c}A_{11}\alpha_{11}+A_{12}\alpha_{22}\\ A_{12}\alpha_{11}+A_{22}\alpha_{22}\end{array}\right\}\Delta T\,z\,\mathrm{d}z. \tag{20}\]