## 2 Piezoelectric Graphene/Fibre-Reinforced Nanocomposite Beam
### Constitutive Equations  
In this paper, it is assumed that the top and bottom layers are made of the isotropic piezoelectric material and the host beam is made of the multi-phase GPL/fibre-reinforced polymer orthotropic composite material. Therefore, for the analysis of a (1D) beam problem, the constitutive equations can be rewritten in the following form:  
_for piezoelectric layers:_  
\[\mathbf{\sigma}^{(p)}=\left\{\begin{matrix}\sigma_{x}\\ \tau_{xz}\end{matrix}\right\}^{(p)}=\left[\begin{matrix}\tilde{Q}_{11}^{(p)}&0 \\ 0&k_{sc}\tilde{Q}_{55}^{(p)}\end{matrix}\right]\left\{\begin{matrix}\varepsilon_ {x}\\ \gamma_{xz}\end{matrix}\right\}-\left\{\begin{matrix}\tilde{\varepsilon}_{31}^{(p )}\\ 0\end{matrix}\right\}E_{z}^{(p)} \tag{11}\]  
\[D_{z}^{(p)}=\tilde{\varepsilon}_{31}^{(p)}\varepsilon_{x}+\tilde{\xi}_{33}^{(p )}E_{z}^{(p)} \tag{12}\]  
_for GPLFR layers:_  
\[\mathbf{\sigma}^{(k)}=\left\{\begin{matrix}\sigma_{x}\\ \tau_{xz}\end{matrix}\right\}^{(k)}=\left[\begin{matrix}\tilde{Q}_{11}^{(k)}&0 \\ 0&k_{sc}\tilde{Q}_{55}^{(k)}\end{matrix}\right]\left\{\begin{matrix} \varepsilon_{x}\\ \gamma_{xz}\end{matrix}\right\} \tag{13}\]  
where the superscripts \(p\) and \(k\) are referred to as the piezoelectric and the elastic plies, respectively. In Equations (11)-(13), \(\sigma_{x},\tau_{xz}\) denote the normal and shear stress, respectively, and \(\varepsilon_{x},\gamma_{xz}\) denote normal and shear strain, respectively. Also, the reduced stiffness coefficients are denoted as \(\tilde{Q}_{11},\tilde{Q}_{55}\). The transverse electric displacement is represented by \(D_{z}\), and the electric field is represented by \(E_{z}\). The piezoelectric constant is denoted by \(\tilde{\varepsilon}_{31}^{(p)}\) and the electric permittivity constant by \(\tilde{\xi}_{53}^{(p)}\). Lastly, the shear correction coefficient is represented by \(k_{sc}\) and, in this work, is assumed to be equal to \(\frac{5}{6}\). Notably, in this paper, the employed piezoelectric material is assumed to be of beam type in which only the electric field through the thickness direction of the piezoelectric patch is dominant. Furthermore, it is assumed that the electric potential varies linearly in the thickness direction. Therefore, the electric field of each piezoelectric layer can be expressed as follows:  
\[E_{z}^{(p)}=-\frac{v_{p}}{h_{p}}\equiv B_{v}v_{p} \tag{14}\]  
where \(h_{p},v_{p}\) are the thickness and the difference of electric potential between the electrodes covering the surface of each side of the piezoelectric layer.  
Finally, for the case of cross-ply composite configuration, the parameters \(\tilde{Q}_{11}^{(k)}\) and \(\tilde{Q}_{55}^{(k)}\) can be expressed as:  
\[\tilde{Q}_{11}^{(k)}=\tilde{Q}_{11}^{(k)}-\frac{\tilde{Q}_{12}^{(k)^{2}}}{ \tilde{Q}_{22}^{(k)}},\quad\tilde{Q}_{55}^{(k)}=\tilde{Q}_{55}^{(k)} \tag{15}\]  
with  
\[\tilde{Q}_{11}^{(k)}=Q_{11}^{(k)}\cos^{4}\theta+2(Q_{12}^{(k)}+2Q_{66}^{(k)}) \sin^{2}\theta\cos^{2}\theta+Q_{22}^{(k)}\sin^{4}\theta, \tag{16}\]  
\[\tilde{Q}_{55}^{(k)}=Q_{55}^{(k)}\cos^{2}\theta+Q_{44}^{(k)}\sin^{2}\theta\]  
and  
\[\begin{split}& Q_{11}^{(k)}=\frac{E_{11}^{(k)}}{1-v_{12}^{(k)} },\quad Q_{12}^{(k)}=\frac{v_{12}^{(k)}E_{22}^{(k)}}{1-v_{12}^{(k)}v_{21}^{(k)} },\quad Q_{22}^{(k)}=\frac{E_{22}^{(k)}}{1-v_{12}^{(k)}v_{21}^{(k)}},\\ & Q_{66}^{(k)}=G_{12}^{(k)},\qquad Q_{44}^{(k)}=k_{sc}G_{23}^{(k) },\qquad Q_{55}^{(k)}=k_{sc}G_{13}^{(k)}\end{split} \tag{17}\]  
where \(E_{11}^{(k)},E_{22}^{(k)}\) are the longitudinal and transverse moduli, \(v_{12}^{(k)},v_{21}^{(k)}\) are the [PERSON]'s ratios, \(G_{12}^{(k)},G_{13}^{(k)},G_{23}^{(k)}\) are the shear moduli of the \(k^{th}\) layer. For the case of isotropic material,two mater l constants are required, namely [PERSON]'s modulus \(E\) and Poisson ratio \(v\). In addition \(Q_{11}=Q_{22},Q_{12}=Q_{21}\) and \(Q_{66}=G\) apply with \(G=E/(2(1+v))\).