# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 4 Numerical Results and Discussion
### Power Density FRFs
When the load resistances locate between 10 M\(\Omega\) and 1000 M\(\Omega\), the optimum external frequencies should be greater than the short circuit resonance frequency and smaller than the open circuit resonance frequency. Figure 8: (**a**) Variation of the power density FRFs with load resistance for excitations at the short and open circuit resonance frequencies of the first order mode (\(h=0.3\)\(\upmu\)m). (**b**) Variation of the optimal external frequencies with the load resistances. Figure 9a gives the variation of the peak power density FRFs with beam thickness for the different flexoelectric coefficients. With decreasing thickness, the peak power density FRFs increases first and then decreases. When the thickness was small, the relatively large induced potential due to the large strain gradient opposes the mechanical bending of the beam, which would reduce the beam bending. When the thickness was large, the induced potential could not increase sufficiently to overcome the increasing volume due to small strain gradient in elastic deformation. For a particular flexoelectric coefficient, there was an optimal size for the flexoelectric energy harvesting, which could achieve the maximum power density. This optimal size increases with increasing flexoelectric coefficient. Our previous work [33] obtained the thickness \(h_{0}\) for the maximum induced electric potential in the flexoelectric sensors:  
\[h_{0}=\sqrt{\frac{12\mu_{3113}^{2}}{c_{1111}\kappa_{33}+e_{311}^{2}}}. \tag{38}\]  
Figure 9: (**a**) Variation of the peak power density FRFs and (**b**) variation of the corresponding optimal resistance with beam thicknesses at different flexoelectric coefficients. When flexoelectric coefficients were \(1.15\times 10^{-2}\), \(2.3\times 10^{-2}\), and \(4.6\times 10^{-2}\)\(\mu\)C/m, the thicknesses \(h_{0}\) were about 0.07, 0.14, and 0.28 \(\mu\)m, respectively. Figure 9a shows that the optimal thickness for the maximum power density in the flexoelectric energy harvesting was about four times as long as \(h_{0}\) for the induced electric potential in the flexoelectric sensor. Figure 9b shows the optimal resistance for the peak power density as a function of the beam thickness for different flexoelectric coefficients. When the thickness was large, the optimal resistance of the flexoelectric energy harvesting was nearly independent of the thickness. It is interesting that the optimal resistance (about 110 M\(\Omega\)) corresponding to the large thickness was almost independent on the flexoelectric coefficient. For a particular flexoelectric coefficient, there was a critical thickness, below which the optimal resistance increases with decreasing thickness. The critical thickness decreases with decreasing flexoelectric coefficient. Therefore, Figure 9 could provide optimal design parameters in terms of beam thickness and load resistance for flexoelectric energy harvesting. It should be noted that, although the small structure considered in the present study may not be applicable for the ambient vibration energy harvesting because of its super high frequency, it is possible to design large size PVDF energy harvesting with giant flexoelectricity [55] for the ambient vibration frequency because the optimal thickness for the maximum power density is proportional to the flexoelectric coefficient.