# Buckling and Free Vibrations of
## 4 Numerical Results and Discussion
### Buckling Loads-Uniform, Uni-Axial Compression along the \(x\)-axis (\(N_{y}^{0}=0\))  
#### 4.1.1 Nonlocal Stress Theory  
Taking into account the uniform, uni-axial compression loading along the \(x\)-axis (assuming \(N_{y}^{0}=0\)) the following relation for the nonlocal stress theory are considered:  
* CPT: \[N_{x}^{0}=\frac{D\left(\alpha_{m}^{2}+\beta_{n}^{2}\right)^{2}}{\alpha_{m}^{2 }\left[1+e_{0}^{2}\theta^{2}(\alpha_{m}^{2}+\beta_{n}^{2})\right]}\] (23)
* FSDT and HSDT: The determinant of the coefficient matrix derived from Equation (20) must be equal to zero, i.e.,: \[\det\left| \begin{array}{ccc}K_{33}&K_{33}&K_{33}\\ K_{43}&K_{44}&K_{45}\\ K_{53}&K_{54}&K_{55}\end{array} \right|=0\] (24)  
The determinant is cut off to three coupled displacement terms (3-\(w\), 4-\(\psi_{1}\), 5-\(\psi_{2}\)) and in this way, the coefficients vary in the range 3-5. \[ \begin{array}{l}K_{33}^{FSDT}=\left(\alpha_{m}^{2}+\beta_{n}^{2}\right)- \frac{N_{y}^{0}}{K_{44}}\alpha_{m}^{2}\left[1+e_{0}^{2}\theta^{2}\left(\alpha _{m}^{2}+\beta_{n}^{2}\right)\right],\;K_{33}^{FSDT}=-\alpha_{m},\;K_{35}^{FSDT }=-\beta_{n}\\ K_{43}^{FSDT}=-K_{34}^{FSDT},K_{44}^{FSDT}=\;1-\frac{D}{K_{44}}\alpha_{m}^{2 }\left[1+\frac{[1-\
u]}{2}\right],K_{45}^{FSDT}=-\frac{D}{K_{44}}\alpha_{m} \beta_{n}\frac{(1+\
u)}{2},\\ K_{53}^{FSDT}=K_{35}^{FSDT},K_{54}^{FSDT}=K_{45}^{FSDT},\;K_{55}^{FSDT}=\;1 -\frac{D}{K_{44}}\beta_{n}^{2}\left[1+\frac{[1-\
u]}{2}\right],\\ K_{33}^{HSDT}=D\left(\alpha_{m}^{2}+\beta_{n}^{2}\right)-N_{y}^{0}\alpha_{m} ^{2}\left[1+e_{0}^{2}\theta^{2}\left(\alpha_{m}^{2}+\beta_{n}^{2}\right) \right]\\ K_{43}^{HSDT}=-\xi_{1}\alpha_{m}\left(\alpha_{m}^{2}+\beta_{n}^{2}\right),K_{35}^{HSDT}=-\xi_{1}\beta_{n}\left(\alpha_{m}^{2}+\beta_{n}^{2}\right),\\ K_{43}^{HSDT}=K_{34}^{HSDT},K_{44}^{HSDT}=\xi_{2}\left(\alpha_{m}^{2}+\frac{1- \
u}{2}\beta_{n}^{2}\right)+\xi_{3},K_{45}^{HSDT}=\xi_{2}\frac{1-\
u}{2} \alpha_{m}\beta_{n},\\ K_{53}^{HSDT}=K_{35}^{HSDT},K_{54}^{HSDT}=K_{45}^{HSDT},K_{55}^{HSDT}=\xi_{2}\left(\frac{1-\
u}{2}\alpha_{m}^{2}+ \beta_{n}^{2}\right)+\xi_{3}\\ D=\frac{E^{3}}{12(1-\
u^{2})},\xi_{1}=\frac{E}{1-\
u^{2}}\int\limits_{-t/2}^{t/2 }{zF(z)dz},\xi_{2}=\frac{E}{1-\
u^{2}}\int\limits_{-t/2}^{t/2}{F^{2}(z)dz}, \xi_{3}=\frac{E}{2(1+\
u)}\int\limits_{-t/2}^{t/2}{\left[\frac{dF(z)}{dz} \right]^{2}dz}\end{array}  \tag{26}\]  
Computing the values of the parameters \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\) one can verify easily that for the third order polynomial and exponential approximations of the functions \(F(z)\) (see Equation (4), Table 1 and Figure 2) the differences of their values are negligibly small. In order to compare various effects of constitutive relation formulations (the nonlocal stress and strain theory) as well as of kinematical relations (Equation (4)) the buckling loads will be further referred to the value:  
\[\overline{N}=\frac{D\big{(}\alpha_{m}^{2}+\beta_{n}^{2}\big{)}^{2}}{\alpha_{m}^{ 2}} \tag{27}\]  
It corresponds to the buckling loads evaluated with the use of the classical plate theory. In addition, the results of the analysis carried out in Refs [74; 75; 76] show that for the \(L_{x}/L_{y}>1\) the buckling mode \(n=1\) and \(m\geq 1\). The presented plots are just restricted to the abovementioned limits. Figure 3 demonstrates the classical results--the formulations of 2-D kinematical relations have a significant effect on the values of buckling loads which increase with the growth of the \(t/L_{x}\) ratio. The decrease of buckling loads for more accurate kinematical formulations in the thickness (z-direction) is generally accepted. However it is essential to point out that the percentage error between classical and higher order plate theories is also a function of mechanical properties of nanoplates, particularly in view of the values of \(G_{xz}/E\) and \(G_{yz}/E\) ratios, where \(G_{xz}\) and \(G_{yz}\) are the transverse (out-of-plane) shear moduli--see, e.g., Muc [76].