# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 2 Lumped-Parameter Modeling of Piezoelectric Metamaterial
### Semi-Analytical Model of Unit-Cell  
In this section, Euler-Bernoulli beam theory is used as the basis to analyze the integrated system. The behavior of the metamaterial can be assessed through analyzing the dynamics of a unit-cell. For each unit-cell, a perfect bonding condition between the piezoelectric transducers and the substrate is assumed. Let \(l\), \(b\), \(h\), and \(\rho\) denote, respectively, the length, width, thickness, and mass density. Throughout this research, subscripts '\(b\)' and '\(p\)' are employed to indicate variables related to the host beam substrate and the piezoelectric transducer, respectively. The modeling starts from the extended [PERSON]'s principle,  
\[\int_{t_{1}}^{t_{2}}{(\delta T-\delta U+\delta W)dt}=0 \tag{1}\]  
where \(T\) is the kinetic energy, \(U\) is the potential energy, and \(W\) is the virtual work. For the unit-cell, the kinetic energy is  
\[T=\frac{1}{2}\rho_{b}h_{b}\int_{-l_{b}/2}^{l_{b}/2}{\int_{-l_{b}/2}^{l_{b}/2}{ \left(\dot{w}(x,t)\right)}^{2}dxdy}+\rho_{p}h_{p}\int_{-l_{p}/2}^{l_{p}/2}{ \int_{l_{b}/2}^{\left(\dot{w}(x,t)\right)}^{2}dxdy} \tag{2}\]  
where \(w(x,t)\) is the displacement of the transverse wave. The directions of the coordinate system are shown in Figure 1, and the origin is located at the center of the unit-cell being analyzed. The variation of the kinetic energy, takes the form  
\[\delta T=\rho_{b}A_{b}\int_{-l_{b}/2}^{l_{p}/2}{\ddot{w}(x,t)\delta wdx}+2\rho _{p}A_{p}\int_{-l_{p}/2}^{l_{p}/2}{\ddot{w}(x,t)\delta wdx} \tag{3}\]  
where \(A_{b}\) and \(A_{p}\) are, respectively, the cross-sectional area of the host beam substrate and that of one piezoelectric transducer. The linear constitutive relation of the piezoelectric transducer is [34]. \[\tau_{p}=E_{p}\varepsilon_{p}-h_{31}D \tag{4a}\] \[E=-h_{31}\varepsilon_{p}+\beta_{33}D \tag{4b}\]  
where \(\tau_{p}\), \(\varepsilon_{p}\), \(D\), and \(E\) represent, respectively, the stress, stain, electrical displacement (charge/area), and electrical field (voltage/length) of the piezoelectric transducer, and \(E_{p}\), \(h_{31}\), and \(\beta_{33}\) are the Young's modulus at constant electrical displacement, the piezoelectric constant, and the dielectric constant of the transducer at constant strain. Let \(E_{b}\) denote the Young's modulus of the host beam substrate. The elastic energy of the unit-cell comes from the host substrate and the two transducers, and the  
Figure 1: Unit-cell of piezoelectric metamaterial. electrical energy comes from the two transducers. The variation of the elastic and electrical energies can be derived as  
\[ \begin{array}{l}\delta U=\int_{-l_{b}/2}^{l_{b}/2}E_{b}l_{b}w^{\prime\prime} \delta w^{\prime\prime}dx+\int_{-l_{p}/2}^{l_{p}/2}E_{p}l_{p}w^{\prime\prime} \delta w^{\prime\prime}dx-2\int_{-l_{p}/2}^{l_{p}/2}F_{p}h_{31}D\delta w^{ \prime\prime}dx\\ -2\int_{-l_{p}/2}^{l_{p}/2}F_{p}h_{31}w^{\prime\prime}\delta Ddx+2\int_{-l_{p}/ 2}^{l_{p}/2}\beta_{33}A_{p}D\delta Ddx\end{array}  \tag{5}\]  
where \(I_{b}\) and \(I_{p}\) represent, respectively, the second moment of inertia of the substrate and that of one piezoelectric transducer, and \(F_{p}\) represents the first moment of area of each transducer. Using the variation principle and setting the initial conditions as \(\delta w(t_{1})=\delta w(t_{2})=0\), one can derive  
\[\int_{t_{1}}^{t_{2}}\delta Tdt=\int_{t_{1}}^{t_{2}}\left[\int_{-l_{b}/2}^{l_{b} /2}\rho_{b}A_{b}\bar{w}(x,t)\delta wdx+2\int_{-l_{b}/2}^{l_{b}/2}\rho_{P}A_{p} \bar{w}(x,t)\left[H(x+\frac{l_{p}}{2})-H(x-\frac{l_{p}}{2})\right]\delta wdx \right]dt \tag{6}\]  
\[\int_{t_{1}}^{t_{2}}\delta Udt=\int_{t_{1}}^{t_{2}}\left[