# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 2 Mechanisms of Actively Tunable Elastic Metasurface
### Derivations of the Governing Equation  
The constitutive equation of piezoelectric materials can be expressed as follows:  
\[ \begin{array}{l}S_{\rm{i}}=s_{\rm{i}}\bar{l}_{\rm{j}}-d_{\rm{in}}E_{\rm{n}} \\ D_{\rm{m}}=d_{\rm{mj}}T_{\rm{j}}+\epsilon_{\rm{mn}}E_{\rm{n}}\end{array}  \tag{1}\]  
where \(S\) is mechanical strain, \(T\) is mechanical stress, \(D\) is electrical displacement, \(E\) is the electric field, \(s_{\rm{ij}}\) is the flexibility coefficient under constant electric field conditions, \(d\) is the piezoelectric stress constant, and \(\epsilon_{\rm{mn}}\) is the dielectric constant. The correspondence between the numerical subscripts and the coordinate axes is 1\(\rightarrow\)x, 2\(\rightarrow\)y, 3\(\rightarrow\)z, 4\(\rightarrow\)yz, 5\(\rightarrow\)xz, 6\(\rightarrow\)xy. In the piezoelectric sheet used in this study, the polarization surface is perpendicular to the \(z\)-axis, and the action of the electric field in the \(z\)-axis direction is mainly considered, ignoring the electric field in the \(x\)-axis and \(y\)-axis directions, which implies that \(E_{1}=E_{2}=0\). The piezoelectric sheet is constrained in the \(x\)-\(y\) plane, with the stress relation \(T_{3}=T_{4}=T_{5}=0\). The piezoelectric material is transverse isotropic in the \(x\)-\(y\) plane, and it can be shown that \(s_{\rm{11}}=s_{\rm{22}}\). Consider a piezoelectric material; its constitutive equations can be expressed as [42]  
\[ \begin{array}{l}S_{\rm{1}}=s_{\rm{11}}T_{\rm{1}}-d_{\rm{31}}E_{\rm{3}}\\ D_{\rm{3}}=d_{\rm{31}}T_{\rm{1}}+\varepsilon_{\rm{33}}E_{\rm{3}}\end{array}  \tag{2}\]  
Figure 1: (**a**) A 3D model of the adaptive unit cell with a negative proportional feedback control strategy. (**b**) A 2D model of the 3D adaptive unit cell in (**a**). It is worth noting that the potential \(D_{3}\) is constant at the electrode with area \(A_{\rm s}\)[49; 50]. According to the classical laminated beam theory, the equivalent density \(\rho_{\rm a}\) and the equivalent stiffness \(D_{\rm a}\) of the laminated composite beam with the piezoelectric material can be written as [40]  
\[\rho_{\rm a}=\rho_{\rm s}h_{\rm s}+2\rho_{\rm p}h_{\rm p} \tag{3}\]  
\[D_{\rm a}=\frac{E_{\rm s}h_{\rm s}^{3}}{12(1-v_{\rm s}^{2})}+2c_{11}(A_{\rm s} I_{1}^{2}+\frac{w_{1}h_{\rm p}^{3}}{12}) \tag{4}\]  
where \(\rho_{\rm s}\), \(E_{\rm s}\), and \(v_{\rm s}\) are the mass density, Young's modulus, and Poisson's ratio of the substrate material, respectively; \(\rho_{\rm p}\) is the density of the piezoelectric material; \(w_{1}\) is the width of the adaptive unit cell; \(h_{\rm s}\) is the thickness of the base beam; and \(I_{1}=(h_{\rm p}+h_{\rm s})/2\). Following [PERSON]'s beam theory, the vibration equation of a composite beam with a piezoelectric actuator and sensors is  
\[D_{\rm a}\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho_{\rm a}h\frac{ \partial^{2}w(x,t)}{\partial t^{2}}+\frac{d_{31}I_{1}}{h_{\rm p}}\frac{ \partial^{2}Y_{\rm a}}{\partial x^{2}}-\frac{d_{31}I_{1}}{h_{\rm p}}\frac{ \partial^{2}Y_{\rm s}}{\partial x^{2}}=0 \tag{5}\]  
where \(Y_{\rm a}\) is the external voltage applied by the control system and \(Y_{\rm s}\) is the sensed voltage generated by the deformation of the sensor. The sensed voltage of the piezoelectric sensor \(Y_{\rm s}\) produced by the deformation of the composite base beam can be given by [50]  
\[Y_{\rm s}=-\frac{d_{31}I_{1}}{\varepsilon_{33}w_{1}}\frac{\partial^{2}w}{ \partial x^{2}} \tag{6}\]  
where \(\varepsilon_{33}\) is the dielectric constant of the piezoelectric material and \(d_{31}\) is the piezoelectric stress constant of the piezoelectric material. The metasurface adaptive unit cells were designed using a negative proportional feedback control strategy. The sensing voltage \(Y_{\rm s}\) caused by the deformation of the piezoelectric sensor is measured and fed back to the piezoelectric actuator as an external control voltage.