## 5 Conclusions  
In this study, the nonlinear electromechanical behavior of the piezoelectric energy harvester model of FG porous sandwich nanobeams under compressive axial loading is investigated based on the NSGT. The porosity distribution of three different types of FG layers for the Euler-Bernoulli beam theory is discussed. The effects of different types of porosity distributions, porosity coefficients, length scale parameters, nonlocal parameters, flexoelectricity, excitation frequencies, lumped mass and axial loads on the vibration frequency, and voltage output of nanobeams are investigated. The results show that:  
The porous distribution, the porosity coefficient of porous materials, the excitation frequencies and the axial load have a large effect on the vibration frequency and voltage output of the sandwiched piezoelectric nanobeams. The axial load and excitation frequency also have an effect on the vibration amplitude of the beam. In the critical buckling state, the vibration amplitude of the beam decreases by nearly 91.1% compared with one without the axial load. The lumped mass reduces the natural frequency of the structure. When the lumped mass equals the mass of the beam, the natural frequency decreases by 32.8%. When the NSGT are considered, the critical buckling load depends on the values of the nonlocal parameters and the strain gradient constant. In addition, the peak value of the RMS voltage output of the postbuckling process is significantly larger than that of the prebuckling process, which provides a new idea to improve the mechanical electrical conversion efficiency of the structure. Furthermore, the size-dependence of the flexoelectric effect on the performance of the energy harvester is demonstrated, indicating that the flexoelectric effect can significantly increase the RMS voltage output of the energy harvester. The results have some reference value for the design and application of piezoelectric energy harvesters.  
Conceptualization, formal analysis, investigation, methodology, project administration, supervision, writing--original draft and writing--review and editing, S.Z.; formal analysis, investigation, methodology, software and writing--review and editing, Z.P.; and conceptualization, methodology, project administration and validation, K.W.; conceptualization, project administration and supervision, B.W.; project administration, software, supervision and validation,  
Figure 18: Effect of the ratio of the piezoelectric layer thickness to the total beam thickness on the dimensionless RMS voltage output (\(h_{t}=3\times 10^{-4}m\), \(m_{s}=0\), \(P=0\), \(R=1k\Omega\), \(\kappa_{0}=0.5\), \(W_{0}=0.01h_{t}\) for distribution 1).  
J.W.; writing--original draft, T.L. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the National Natural Science Foundation of China under Grants 51965041, 51975266, and 11972137, the Foundation of Jiangxi Educational Committee under Grant GJJ190522 and the Jiangxi Provincial Natural Science Foundation under Grant 20202 BAB211005.  
Not applicable.  
The authors declared no potential conflict of interest with respect to the research, authorship, and/or publication of this article.