# Buckling and Free Vibrations of
## 2 Formulation of the Problem
### Constitutive Nonlocal Equations and Variational Formulation  
The brief overview of nonlocal theories has been presented in the Introduction. We have focused on the classical (local) and modified couple-stress formulations as well as the stress and strain gradient theories. Now, on the base of the local continuum, the constitutive nonlocal equations for the nanoplate are presented. Generally, both strain and stress gradients can influence the constitutive response of materials. The local (classical) constitutive relations for a shear deformable nanoplate (orthotropic) are in the following form: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Classical Plate** & **First Order Shear** & **Higher Order Shear** & **Higher Order Shear** \\  & **Theory (CPT)** & **Deformation Theory** & **Deformation Theory** & **Deformation Theory** \\  & **(FSDT)** & **(HSDT)-[PERSON] [67] & **(HSDT)-[PERSON] [67]** & **(HSDT)-[PERSON]** \\ \hline \multirow{2}{*}{\(F(z)\)} & \multirow{2}{*}{0} & \(z\); the second term in & \multirow{2}{*}{\(z\left(1-\frac{z^{2}}{\beta^{2}}\right)\)} & \multirow{2}{*}{\(z\exp\left(-\frac{z^{2}}{\beta^{2}}\right)\)} \\  & & Equation (10) is & & \\ \cline{1-1}  & & eliminated & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Function \(F(z)\) for variants of plate theories. Figure 2: Variations of the function F(\(z\)) through the plate thickness \(t\). * local stress theory \[\left\{ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{13}\end{array} \right\}=\left[ \begin{array}{ccccc}\text{C}_{11}&\text {C}_{12}&0&0&0\\ \text{C}_{12}&\text{C}_{22}&0&0&0\\ 0&0&\text{C}_{66}&0&0\\ 0&0&0&\text{C}_{44}&0\\ 0&0&0&0&\text{C}_{55}\end{array} \right]\times\left\{ \begin{array}{c}\varepsilon _{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\\ \varepsilon_{23}\\ \varepsilon_{13}\end{array} \right\}\] (6)
* local strain theory \[\left\{ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{13}\end{array} \right\}=\left[ \begin{array}{ccccc}\text{C}_{11}& \text{C}_{12}&0&0&0\\ \text{C}_{12}&\text{C}_{22}&0&0&0\\ 0&0&\text{C}_{66}&0&0\\ 0&0&0&\text{C}_{44}&0\\ 0&0&0&0&\text{C}_{55}\end{array} \right]\times\left(1-\left(\varepsilon_{0} \theta\right)^{2}\
abla^{2}\right)\left\{ \begin{array}{c}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\\ \varepsilon_{23}\\ \varepsilon_{13}\end{array} \right\}\] (7)  
\(\
abla^{2}\) represents the classical 2-D Laplacian operator. In the Cartesian system of coordinates the nonlocal functional of the strain energy of orthotropic plates becomes:  
\[U_{L}=\frac{1}{2}\!\!\int\limits_{\Omega}\int\limits_{-t/2}^{t/2}\left\{ \sigma^{NLoc}\right\}\!\left\{\varepsilon\right\}\!dxdydz \tag{8}\]  
where \(\Omega\) denotes the space occupied by the nanoplate, \(\left\{\varepsilon\right\}\) is the local strain tensor and \(\left\{\sigma^{NLoc}\right\}\) is the nonlocal stress tensor. The nonlocal stress tensor \(\left\{\sigma^{NLoc}\right\}\) is defined as follows:  
* the nonlocal stress theory \[\left\{\sigma\right\}=[C]\!\left\{\varepsilon\right\}=(1-\left(e_{0}\theta \right)^{2}\
abla^{2})\!\left\{\sigma^{NLoc}\right\}=\ \Re\!\left\{\sigma^{NLoc}\right\}\] (9)
* the nonlocal strain theory \[\left\{\sigma^{NLoc}\right\}=[C](1-\left(e_{0}\theta\right)^{2}\
abla^{2}) \!\left\{\varepsilon\right\}\] (10) where \(\left\{\sigma\right\}\) is the local stress tensor, \([C]\) is the stiffness matrix and \(\Re\) is an operator. The components of the strain tensor are functions of the kinematic variables (4), (5) so that inserting the definitions (9) or (10) in Equation (8) one can find that the functional \(U_{L}\) becomes a function of the kinematic variables \(u\), \(v\), \(w\), \(\psi_{1}\), \(\psi_{2}\) in the right-hand side of (4) only.