## 4 Results and Discussion  
Firstly, the validation of the present model is validated. In Table 1, the natural frequencies (\(\left(\omega_{n}L^{2}/h_{t}\right)\sqrt{12\rho_{\max}/C_{\max}}\)) with consideration of the nonlocal strain gradient are compared with those in Ref. [70]. For comparison purposes, the electric constants and mass densities of the core layer and piezoelectric layers are assumed to be the same, and the piezoelectricity and flexoelectricity are neglected. The lumped mass is set as zero, and the axially load is released. The length and the thickness of the beam are assumed to be \(L=50\) nm and \(h_{t}=1\) nm, respectively. Results show good agreements with Ref. [70]. In addition, in Table 1, it can be seen that the results obtained in the present study are slightly higher than those in Ref. [70] by 0.019%. The reason why the results in the present study are slightly higher than those in Ref. [70], is that the rotational inertia is neglected in the present study, which is considered in Ref. [70]. In Figure 4, the dimensionless natural frequency (\((\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\max}/C_{\max}}\)) with the dimensionless axial load (\(PL^{2}/(\pi^{2}bB_{11})\)) are compared with those in Ref. [13]. The material properties are \(E=126 GPa\), \(e_{311}=-6.5C/m^{2}\), \(\mu_{3113}=-1.0\times 10^{-7}C/m\), \(a_{33}=1.8\times 10^{-8}C/(V\cdot m)\). Results show good agreements with Ref. [13]. It seems that the natural frequency decreases with the increase in the compressive load \(P\) at the prebuckling state and reaches zero when \(P=P_{cr}\), and increases with the increase in the compressive load \(P\) at the postbuckling state. A similar phenomenon is also found in the experimental work [71]. In the text bellow, for the convenience of discussion, some values of material constants and geometric dimensions are provided (unless otherwise specified). The material constants of the piezoelectric material are \(c_{11P}=126 GPa\), \(\rho_{p}=7800 kg/m^{3}\), \(e_{311}=6.5C/m^{2}\), \(\mu_{3113}=0.1\mu C/m\), and \(a_{33}=1.3\times 10^{-8}C/(V\cdot m)\). The maximum elastic constant of the core layer is \(C_{\max}=200 GPa\), the maximum mass density of the core layer is \(\rho_{\max}=8700\) kg/m\({}^{3}\). The geometric dimension of the beam is fixed to be \(L:b:h_{t}:h_{p}=100\):\(10\):\(1\):\(0.1\). The nonlocal parameter is set as \(e_{0}a=0.1L\), the strain gradient constant is \(l=0.2L\). In Figure 5, effects of the lumped mass on the natural frequency (\((\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\max}/C_{\max}}\)) for distribution \(1\) are displayed. The porosity is \(\kappa_{0}=0\) (uniform distribution without porosity), and the nonlocal parameter and strain gradient constant are \(e_{0}a=0.1L\), and \(l=0.2L\). It seems that at both the prebuckling state and postbuckling state, the lumped mass decreases the natural frequency. In this case, when \(m_{s}=0.1I_{0}L\), \(m_{s}=0.5I_{0}L\), and \(m_{s}=I_{0}L\), the natural frequencies are reduced to 94.7%, 79.6% and 68.1% of those without lumped mass, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \(e_{0}a\) **(nm)** & & \(I=0\) & \(I=0.5h\) & \(L=h\) \\ \hline
0 & [PERSON] et al. In Figure 6, effects of the nonlocal parameter and strain gradient constant on the natural frequency \(\big{(}(\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\text{max}}/C_{\text{max}}}\big{)}\) and buckling load for distribution 1 are displayed. The porosity is \(\kappa_{0}\) (uniform distribution without porosity). It seems that at the prebuckling state, the resonance frequency decreases because the natural frequency decreases as the axial load increases; this is opposite at the postbuckling state. When the nonlocal strain gradient is considered, the critical buckling load is dependent on the values of the nonlocal parameter \(e_{0}a\) and strain gradient constant \(l\). The nonlocal parameter \(e_{0}a\) diminishes the critical buckling load, while the strain gradient constant \(l\) enlarges the critical buckling load.