# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 2 Electromechanical System and Mathematical Formation  
The present study focuses on the vibration responses of a flexoelectric cantilever beam with length \(L\), width \(B\), thickness \(h\), and a tip mass \(M_{\text{t}}\) as shown in Figure 1, in which the beam is coated by conductive electrodes on both upper and lower surfaces. We assumed the electrode layers were thin so that the effect of their stiffness on mechanical responses of the system was negligible. The coordinate system \((x_{1},x_{2},\ x_{3})\) is shown in Figure 1. The motion of the base for the flexoelectric beam was in the \(x_{3}\) direction. With a transverse base excitation \(w_{\text{b}}\ (t)\), the cantilever beam vibrates in the bending mode. As a result of the dynamic strain gradient caused by vibration responses, the flexoelectric beam produces an alternating potential difference across the surface electrodes. These electrodes were connected to an external resistance (\(R\)) to quantify the electric potential and power output. Here, the internal electrical resistance of the flexoelectric beam was not taken into consideration since it can be regarded as a resistance connected in parallel to the external electrical load [37; 41]. The generalized Hamilton's variational principle for flexoelectric materials can be written as [32]  
\[\delta\int_{0}^{T}[K-G+W]\text{d}t=0, \tag{1}\]  
where \(K,G,\ \text{and}\ W\) are the total kinetic energy, total electric Gibbs free energy, and external work, respectively. In this study, the tip mass was regarded as a particle and the rotary inertia of the tip mass was not considered, which has less influence. Assuming no external body forces and electric field, for the flexoelectric cantilever beam showing in Figure 1, Equation (1) could be rewritten as  
\[\delta\int_{0}^{T}\text{d}t\int_{V}\left[\frac{1}{2}\rho\big{|}\dot{w}^{m} \big{|}^{2}-g_{1}\right]\text{d}V+\delta\int_{0}^{T}\frac{1}{2}M_{\  
where \(\sigma_{ij}\) is the classical Cauchy stress tensor, \(\varepsilon_{ij}\) is the strain tensor, \(\sigma_{ijk}\) is the higher-order stress tensor, \(D_{i}\) is the electric displacement vector, and \(E_{i}\) is the electric field vector. In this paper, the kinematics of the classical Euler-Bernoulli beam model was adopted to analyze the bending vibration of the flexoelectric beam for energy harvesting. The relative displacement field \(\mathbf{u}\) in the Euler-Bernoulli model is  
\[\mathbf{u}=\left\{-x_{3}\frac{\partial w}{\partial x_{1}},0,w\right\}, \tag{4}\]  
where the displacement in the \(x_{2}\) direction is set to be zero as in the plane strain elasticity. With Equation (4) for infinitesimal deformation, the only non-zero strain component and its gradients are  
\[\varepsilon_{11}=-x_{3}\frac{\partial^{2}w}{\partial x_{1}^{2}},\ \ \varepsilon_{11,3}=-\frac{ \partial^{2}w}{\partial x_{1}^{2}},\ \ \varepsilon_{11,1}=-x_{3}\frac{\partial^{3}w}{\partial x_{1}^{3}}, \tag{5}\]  
where the stain gradient \(\varepsilon_{11,3}\) is essentially the bending curvature of the beam and \(\varepsilon_{11,1}\) is proportional to the gradient of curvature. The latter is typically small compared to the former in the Euler-Bernoulli beam model and may be neglected for a slender beam. For the same reason, only one component of the electric field, \(E_{3}\), was considered due to the relatively small electric field \(E_{1}\) in the length direction. According to the constitutive equations of the flexoelectric beams, the non-zero stress, higher-order stress, and electric displacement are [25; 32; 36]  
\[\left\{ \begin{array}{c}\sigma_{11}=c_{1111}\varepsilon_{11}-\varepsilon_{31 1}E_{3}\\ \sigma_{113}=-\mu_{3113}E_{3}\\ D_{3}=\kappa_{33}E_{3}+e_{311}\varepsilon_{11}+\mu_{3113}\varepsilon_{11,3} \end{array}