# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 2 Electromechanical System and Mathematical Formation
\right., \tag{6}\]  
where \(\sigma_{113}\) is defined as the higher order stress or the moment stress [25; 36], which is induced by the electric field due to the flexoelectric effect, \(c_{1111}\) is an elastic modulus, \(e_{311}\) is a piezoelectric coefficient, \(\mu_{3113}\) is a flexoelectric coefficient, and \(\kappa_{33}\) is a dielectric coefficient. In the absence of free body charges, [PERSON]'s law of electrostatics leads to  
\[\frac{\partial^{2}\varPhi}{\partial x_{3}^{2}}=\frac{e_{311}}{\kappa_{33}} \varepsilon_{11,3}, \tag{7}\]  
where \(\varPhi=\varPhi\ (x_{1},x_{3},t)\) is the electric potential of the beam and is related to the electric field by \(E_{3}=-\frac{\partial\varPhi}{\partial x_{3}}\). Considering the surface electrical boundary conditions \(\varPhi\big{(}x_{1},\ x_{3}=\frac{h}{2},t\big{)}=\psi_{1}(x_{1},\ t)\) and \(\varPhi\big{(}x_{1},\ x_{3}=-\frac{h}{2},t\big{)}=\psi_{2}(x_{1},\ t)\), the dynamic electric potential \(\varPhi\) and dynamic electric field \(E_{3}\) can be obtained from Equation (7) as  
\[\varPhi(x_{1},\ x_{3},t)=\frac{e_{311}}{2\kappa_{33}}\frac{\partial^{2}w}{ \partial x_{1}^{2}}\bigg{(}\frac{h^{2}}{4}-x_{3}^{2}\bigg{)}+\bigg{(}\frac{1}{ 2}+\frac{x_{3}}{h}\bigg{)}\varpsi(x_{1},\ t)+\psi_{2}(x_{1},\ t), \tag{8}\]  
\[E_{3}=-\frac{e_{311}}{\kappa_{33}}\varepsilon_{11}-\frac{\psi(x_{1},\ t)}{h}, \tag{9}\]  
where \(\psi(x_{1},\ t)=\psi_{1}(x_{1},\ t)-\psi_{2}(x_{1},\ t)\) is the potential difference or voltage between the both surface electrodes. Substituting Equation (9) into Equation (6), the electric displacement, stress, and higher order stress could be expressed as  
\[\left\{ \begin{array}{c}D_{3}=\mu_{3113}\varepsilon_{11,3}-\kappa_{33}\frac {\psi}{h}\\ \sigma_{11}=\bigg{(}c_{1111}+\frac{e_{311}^{2}}{\kappa_{33}}\bigg{)}\varepsilon _{11}+e_{311}\frac{\psi}{h}\\ \sigma_{113}=\frac{e_{311}e_{311}}{\kappa_{33}}\varepsilon_{11}+\mu_{3113} \frac{\psi}{h}\end{array} \right.. \tag{10}\]Then, by Equation (3), the electric Gibbs free energy density is obtained as follows [36]:  
\[g_{1}=\frac{1}{2}\bigg{(}c_{1111}+\frac{e_{311}^{2}}{\kappa_{33}}\bigg{)}\xi_{11} ^{2}+\frac{e_{311}\mu_{3113}}{\kappa_{33}}\varepsilon_{11}\varepsilon_{11,3}+ \mu_{3113}\varepsilon_{11,3}\frac{\psi}{h}-\frac{1}{2}\kappa_{33}\frac{\psi^{2 }}{h^{2}}. \tag{11}\]  
Using Equations (5) and (6), the variational expression of general electric Gibbs free energy can be written as [36]  
\[ \begin{split}&\delta\int_{0}^{T}\mathrm{d}t\int_{V}\delta_{1} \mathrm{d}V=\int_{0}^{T}\int_{0}^{L}\bigg{\{}\bigg{[}G_{\mathrm{P}}\frac{ \partial^{2}w}{\partial x_{1}^{2}}-\mu_{3113}B\frac{\partial^{2}\psi}{\partial x _{1}^{2}}\bigg{]}\delta w-\bigg{[}\mu_{3113}B\frac{\partial^{2}w}{\partial x_{ 1}^{2}}+\kappa_{33}\frac{\psi B}{h}\bigg{]}\delta\psi\bigg{\}}\mathrm{d}x_{1} \mathrm{d}t+\\ &\int_{0}^{T}\bigg{\{}G_{\mathrm{P}}\frac{\partial^{2}w}{ \partial x_{1}^{2}}-\mu_{3113}B\psi\bigg{\}}\delta\bigg{(}\frac{\partial w}{ \partial x_{1}}\bigg{)}\bigg{|}x_{1}=L\mathrm{d}t-\int_{0}^{T}\bigg{\{}G_{ \mathrm{P}}\frac{\partial^{2}w}{\partial x_{1}^{2}}\bigg{\}}\delta w\bigg{|}x _{1}=L\mathrm{d}t,\end{split}  \tag{12}\]  
where \(G_{\mathrm{P}}=\frac{B\mu^{3}}{12}\bigg{(}c_{1111}+\frac{e_{311}^{2}}{\kappa_{ 33}}\bigg{)}\) is the effective bending rigidity of the piezoelectric beam. It should be noted that the present effective bending rigidity is quite different from the previous result in [37], where the effective bending rigidity was derived from an internal energy density function. The effective bending rigidity in [37] depends on the flexoelectricity of the material and becomes negative when the beam thickness is small (\(\sim\)several nanometers). In contrast, the effective bending rigidity \(G_{\mathrm{P}}\) in the present model remains positive, independent of the flexoelectric coefficients. Substituting Equation (12) into Equation (2), the generalized Hamilton's variational equation of the flexoelectric beam can be rewritten as  
\[