# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 2 Lumped-Parameter Modeling of Piezoelectric Metamaterial
### Semi-Analytical Model of Unit-Cell
\begin{array}{c}(E _{b}l_{b}+E_{p}l_{p})w^{\prime\prime}\delta w^{\prime}\left|\genfrac{}{}{0.0 pt }{}{}{}{}^{l_{b}/2}{-r_{b}/2}-(E_{b}l_{b}+E_{p}l_{p})w^{\prime\prime}\delta w \left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{b}/2}{-r_{b}/2}+\int_{-l_{p}/2}^{l_{b}/2}( E_{b}l_{b}+E_{p}l_{p})w^{\prime\prime}\delta wdx\\ -2\int_{-l_{b}/2}^{l_{b}/2}F_{p}h_{31}D^{\prime\prime}\delta w\left|H(x+\frac{l _{p}}{2})-H(x-\frac{l_{p}}{2})\right|dx-2F_{p}h_{31}(D^{\prime}\delta w^{\prime \prime}\left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{p}/2}{-r_{b}/2}-D^{\prime}\delta w \left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{p}/2}{-r_{b}/2}\right)\\ -2\int_{-l_{b}/2}^{l_{b}/2}F_{p}h_{31}w^{\prime\prime}\delta D\left|H(x+\frac{l _{p}}{2})-H(x-\frac{l_{p}}{2})\right|dx\\ +2\int_{-l_{p}/2}^{l_{b}/2}\beta_{33}A_{p}D\delta D\left|H(x+\frac{l_{p}}{2})- H(x-\frac{l_{p}}{2})\right|dx\end{array} \right]dt \tag{7}\]  
where \(H(x)\) is the Heaviside function. As the length and width of the piezoelectric transducer may be smaller than those of the host beam substrate, terms associated with the piezoelectric transducer are multiplied by \(\left[H(x+l_{p}/2)-H(x-l_{p}/2)\right]\) hereafter. The variation of the virtual work done by forces from adjacent unit-cells (applied at the boundaries), can be expressed as [35]. \[\delta W=(E_{b}l_{b}+E_{p}l_{p})w^{\prime\prime}\delta w^{\prime}\left| \genfrac{}{}{0.0 pt}{}{}{}^{l_{b}/2}{-l_{b}/2}-(E_{b}l_{b}+E_{p}l_{p})w^{\prime \prime}\left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{b}/2}{-l_{b}/2}-2F_{p}h_{31}(D^{ \prime}\delta w^{\prime}\left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{p}/2}{-l_{p}/2}-D^ {\prime\prime}\delta w\left|\genfrac{}{}{0.0 pt}{}{}{}^{l_{p}/2}{-l_{p}/2}\right)\right. \tag{8}\]  
Substituting Equations (6)-(8) into Equation (1), the partial differential equations that govern the motion of the unit-cell, coupled with the piezoelectric transducers are obtained,  
\[ \begin{array}{l}\left(\rho_{b}A_{b}+2\rho_{p}A_{p}\left[H(x+\frac{l_{p}}{2}) -H(x-\frac{l_{p}}{2})\right]\right)\bar{w}(x,t)\\ +(E_{b}l_{b}+E_{p}l_{p})w^{\prime\prime}-2F_{p}h_{31}D^{\prime\prime}\left[H(x +\frac{l_{p}}{2})-H(x-\frac{l_{p}}{2})\right]=W_{c}(x,t)\end{array}  \tag{9a}\] \[(F_{p}h_{31}w^{\prime\prime}+\beta_{33}A_{p}D)\left[H(x+\frac{l_{p}}{2})-H(x- \frac{l_{p}}{2})\right]=W_{s}(x,t) \tag{9b}\]  
where \(W_{c}(x,t)\) is the virtual work per unit length of the substrate/transducer, due to the mechanical damping. In Equation (9b), the effect of the shunt circuit, i.e., the inductive load, is included as virtual work done to the unit-cell. Let \(W_{s}(x,t)\), represent such virtual work per unit length of the transducer. The actual virtual work done by the shunt circuit thus is \(\int_{-l_{b}/2}^{l_{b}/2}W_{s}(x,t)dx\). To analyze the dynamic behavior of the piezoelectric metamaterial, one unit-cell is considered, where the Bloch-Floquet theory is employed. Specifically, the mechanical displacement and the electrical displacement are both assumed to have the following representations [36, 37],  
\[w(x,t)=\overline{q}e^{i(k_{x}x-\omega t)}, \tag{10a}\] \[D(x,t)=\overline{D}e^{i(k_{x}x-\omega t)} \tag{10b}\]  
where \(k_{x}\) is the wavenumber of the transverse wave involved, \(\omega\) is its frequency, \(\overline{q}\) and \(\overline{D}\) are, respectively, the amplitudes of mechanical and electrical displacements. Furthermore, let \(\overline{D}_{1}\) and \(\overline{D}_{2}\) denote, respectively, the electrical displacements of the top and bottom transducers. Essentially, these displacement representations reflect that the wave propagates in the piezoelectric metamaterial that is spatially periodic. Our goal here is to develop a lumped-parameter model of the unit-cell, which can be used to elucidate the underlying physics of the piezoelectric metamaterial. Consider an arbitrary wavenumber \(k_{x}\). Equation (10a,b) are substituted into Equation (9a,b), which are further integrated over the cell length \(l_{b}\).