## 2 Mathematical Formulation  
Shown in Figure 1 is the sketch of a piezoelectric energy harvester with a simply supported nanobeam settled on a base. The beam is composed of three layers, i.e., a core layer manufactured of functionally graded porous materials, and two piezoelectric layersattached perfectly on the core layer and no slip between layers. The adhesive thickness of the bonding interface is neglected. The elastic constant of the piezoelectric layer and core layer are denoted as \(C_{11p}\) and \(C_{11c}(z)\), respectively. The mass density of the piezoelectric layer and core layer are denoted as \(\rho_{p}\) and \(\rho_{c}\), respectively. The height of two piezoelectric layers is \(h_{p}\), and of the core layer is \(h_{c}\). The length and the width of three layers are the same, which is denoted as \(L\), \(b\), respectively. The height of the entire beam is \(h_{t}=2h_{p}+h_{c}\). There is a lumped mass \(m_{s}\) attached on the midspan of the nanobeam to improve the energy capture efficiency of the energy harvester. A compressive load is applied axially on the nanobeam. The base is applied a displacement excitation \(w_{b}(t)=W_{0}\cos(\omega t)\). The two piezoelectric layers are connected in series in circuit, and connected to an external resistor \(R\). The core layer is manufactured of functionally graded porous materials. In this study, three typical porosity distributions are considered [38; 67; 68]. The elastic properties and mass density are described by Equation (1) for distribution 1, Equation (2) for distribution 2, and Equation (3) for distribution 3. \[\left\{ \begin{array}{l}C_{11c}(z)=C_{\max}(1-\chi_{0}\alpha)\\ \rho_{c}(z)=\rho_{\max}\sqrt{1-\chi_{0}\alpha}\\ \end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{l}C_{11c}(z)=C_{\max}\Big{(}1-\chi_{0}\cos\frac{\pi z}{ \hbar_{c}}\Big{)}\\ \rho_{c}(z)=\rho_{\max}\Big{(}1-\chi_{m}\cos\frac{\pi z}{\hbar_{c}}\Big{)}\\ \end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}C_{11c}(z)=C_{\max}\Big{[}1-\chi_{0}\cos\Big{(}\frac{ \pi z}{2\hbar_{c}}+\frac{\pi}{4}\Big{)}\Big{]}\\ \rho_{c}(z)=\rho_{\max}\Big{[}1-\chi_{m}\cos\Big{(}\frac{\pi z}{2\hbar_{c}}+ \frac{\pi}{4}\Big{)}\Big{]}\\ \end{array} \right. \tag{3}\]  
where \(C_{\max}\) (\(\rho_{\max}\)) and \(C_{\min}\) (\(\rho_{\min}\)) are the maximum value and minimum value of elastic constant (mass density) for porosity distributions, respectively. \(\chi_{0}\) and \(\chi_{m}\) represent the porosity coefficients which are expressed as \(\chi_{0}=1-C_{\min}/C_{\max}\) and \(\chi_{m}=1-\rho_{\min}/\rho_{\max}\), respectively. The relationship between \(\chi_{0}\) and \(\chi_{m}\) is expressed as \(\chi_{m}=1-\sqrt{1-\chi_{0}}\) for the open-cell model [68]. A diagrammatic sketch of those three distributions is shown in Figure 2. For the convenience of comparing discussion, the total mass of the entire beam with those three types of distributions is assumed to be equal, which provides \(\alpha=\frac{1}{\chi_{0}}-\frac{1}{\chi_{0}}\big{(}\frac{2}{\pi}\sqrt{1-\chi_ {0}}-\frac{2}{\pi}+1\big{)}^{2}\). Figure 1: Sketch of a piezoelectric energy harvester with a simply supported nanobeam settled on base. The approximate process of the governing equations and boundary conditions of the model obtained is shown in Figure 3.  
Figure 3: Mathematical modeling process diagram. Figure 2: Material constants of three typical distributions. Based on the Euler-Bernoulli beam assumption, the displacement field of the beam are:  
\[u_{0}(x,z,t)=u(x,t)-z\frac{\partial w(x,t)}{\partial x} \tag{4}\]  
\[w_{0}(x,z,t)=w(x,t)+w_{b}(t) \tag{5}\]  
Considering the von [PERSON] geometric nonlinearity, the strain fields can be expressed as:  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}+\frac{1}{2}\bigg{(}\frac{ \partial w}{\partial x}\bigg{)}^{2}-z\frac{\partial^{2}w}{\partial x^{2}} \tag{6}\]  
It is assumed that the electric field only exists in the thickness direction which is expressed as \(E_{Z}=-V/h_{p}\) for both piezoelectric layers.