# Piezoelectric Electro-Mechanical Impedance (EMI) Based Structural Crack Monitoring
## 3 Finite Element Analysis of Piezoelectric Impedance  
To verify the above proposed method, the finite element analysis (FEA) is adopted to simulate the piezoelectric impedance and to find out the influence of crack and crack size on the peak frequency in the piezoelectric admittance signature, i.e., the local resonance frequency of the coupled system. The host structure is an aluminum alloy beam with a dimension of \(250\times 30\times 5\) mm. A PZT patch, with a size of \(8\times 7\times 1\) mm, is bonded on the center of the beam surface with epoxy resin adhesive. The adhesive layer thickness is kept 0.125 mm. The properties of the piezoelectric patch are listed in  
Figure 1: Simplified piezoelectric electro-mechanical impedance (EMI) Model. Table 1[60], and the material constants of the adhesive layer and the aluminum alloy beam are listed in Table 2.  
Cracks are simulated by setting rectangular grooves on a beam. The width of each rectangular groove is 0.5 and the depth is 1 mm. Only the length of groove is changed to simulate the different crack sizes. The length of the groove is set as 0, 6 and 12 mm, respectively. The location of the piezoelectric patch and the crack on the beam is shown in Figure 2. In the numerical analysis, free boundaries are set at both ends of the beam. The system damping ratio is set to 0.005. The finite element model of the cracked beam with piezoelectric patch is built, as shown in Figure 3a. The model is meshed by a mapping grid with an element size of 2 mm, except where the crack generates and the piezoelectric patch is bonded. For the region where the crack is located, the mapping mesh with the element size of 0.25 mm is used. In the vicinity of the crack, the free mesh is used to gradually transfer the specialized element size of 0.25 mm to the overall model element size of 2 mm. The mesh at the crack is detailed in Figure 3b. The piezoelectric patch is discretized with an element size of 0.5 mm by mapping, and the adhesive layer under the piezoelectric patch is discretized by free mesh with a 1 mm element size gradually changing to a 2 mm element size. In the simulation, the modal analysis is firstly carried out to obtain the resonance frequency, and then the piezoelectric harmonic analysis is performed. In the piezoelectric harmonic analysis, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Density & \(\rho\) & 7850 kg/m\({}^{3}\) \\ Dielectric constant & \(\varepsilon_{33}^{T}\) & \(3.01\times 10^{-8}\) F/m \\ Piezoelectric constant & \(d_{31}\) & \(-274\times 10^{-12}\) C/N \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of piezoelectric patch. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Density /kg-m\({}^{-3}\)** & **Young’s Modulus /GPa** & **Poisson Ratio** \\ \hline Beam & 2700 & 70 & 0.3 \\ Adhesive layer & 2000 & 0.8 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material coefficients of the beam and the adhesive layer. Figure 3: Overall (**a**) and local (**b**) finite element model of cracked beam and lead–zirconate–titanate (PZT) patch. Figure 2: Schematic diagram of piezoelectric patch and crack location. the top and bottom surface piezoelectric degrees of the piezoelectric patch is coupled, and the voltage of 1 V and 0 V are applied on the top and bottom surfaces of the piezoelectric patch respectively to simulate 1 V AC voltage. The selected frequency range in the harmonic analysis is near the modal resonant frequencies obtained from the modal analysis. The time-varying charge on the piezoelectric surface is obtained, and then the piezoelectric impedance, piezoelectric admittance and other information are calculated. Comparing the results of the modal analysis with the piezoelectric admittance analysis, it is found that the modal resonant frequencies correspond approximately to the peak frequencies in the admittance signature, indicating that some of the peak admittance frequencies are the resonant frequencies of the coupled system.