# Piezoelectric Electro-Mechanical Impedance (EMI) Based Structural Crack Monitoring
## 2 Principle of Piezoelectric Electro-Mechanical Impedance Method  
Because of its strong piezoelectric effect, a PZT (lead-zirconate-titanate) type piezoelectric transducer has been successfully used in SHM and is adopted in this research. For a PZT patch of length \(l_{a}\), width \(b_{a}\), and thickness \(h_{a}\) is bonded on a structure, based on the assumption of elastic connection (represented by a pair of stiffness springs \(2K_{\text{str}}\)) between piezoelectric material and host structure, as shown in Figure 1, [PERSON] and [PERSON] [57] proposed an EMI model and the piezoelectric admittance expression is given as:  
\[Y(\omega)=i\omega C\left[1-\kappa_{31}^{2}\left(1-\frac{1}{\varphi\cdot\cot( \varphi)+K_{\text{str}}/K_{\text{PZT}}}\right)\right] \tag{1}\]  
where \(\omega\) is the angular frequency of excitation signal, \(\rho\) is the density of the PZT patch, \(\kappa_{31}^{2}\) is the electromechanical coupling factor, \(\varphi=\gamma\cdot l_{a}/2\), while \(\gamma=\omega/c\) is the wave number, \(c=\sqrt{1/\rho s_{11}^{E}}\) is the wave speed, \(s_{11}^{E}\) is the mechanical compliance at zero field, \(C=l_{a}b_{a}\overline{\varepsilon}_{33}^{T}l_{a}^{-1}\) is the conventional capacitance of the PZT patch, \(K_{\text{str}}\) is the stiffness of the host structure, \(K_{\text{PZT}}=A_{a}/\left(s_{11}^{E}l_{a}\right)\) is the stiffness of the PZT patch, \(A_{a}=h_{a}\times b_{a}\) is the cross-sectional area of PZT patch, and \(\overline{\varepsilon}_{33}^{T}\) is dielectric constant at zero stress. The expressions in Equations (1) cover the complete frequency spectrum and encompass both structure and sensor dynamics, overcoming low-frequency structure-focused analysis of [PERSON]'s model [58]. Equation (1) reveals that the change of admittance Y of the coupled system is only related on the host structural stiffness \(K_{str}\) while the parameters of piezoelectric material and host structural material are determined. In order to find out the effect of the stiffness of the host structure on the admittance signal, Equation (1) is derived with \(\omega\) simultaneously on both sides. Here, we expand the cotangent function by a Taylor series and substitute the first two terms in the Taylor series into Equation (1):  
\[\frac{dY}{d\omega}=iC\left[1-\chi_{31}^{2}+\chi_{31}^{2}\frac{1+\frac{K_{str}} {K_{PZT}}+\frac{l_{2}^{2}}{12\epsilon^{2}}\omega^{2}}{\left(1+\frac{K_{str}}{K_ {PZT}}-\frac{l_{2}^{2}}{12\epsilon^{2}}\omega^{2}\right)^{2}}\right] \tag{2}\]  
Some peak values in the admittance signature represent the resonant frequency of the host structure. In this case, the peak frequency is a certain order resonance frequency, \(\omega_{n}\), of the coupled system of the host structure and the piezoelectric patch. Thus, in the position of the resonance frequency, \(\omega_{n}\), the derivative of piezoelectric admittance signal equals zero, i.e., \(dY/d\omega_{n}=0\). In Equation (2), both \(c\) and \(K_{PZT}\) are complex. Considering that the imaginary part of c and \(K_{PZT}\) depend on the material damping and the effect of material damping is small, the imaginary part is neglected in order to simplify the analysis process. Meanwhile, let \(\chi_{31}^{2}=a+bi\) in Equation (2). By extracting the imaginary part of Equation (2) and making it equal to zero, the expressions about \(K_{str}\) and \(\omega_{n}\) can be obtained as  
\[\left(\frac{l_{n}^{2}}{12\epsilon^{2}}\right)^{2}\omega_{n}^{4}-\frac{l_{n}^{2 }}{12\epsilon^{2}}(3+K_{str}/K_{PZT})\omega_{n}^{2}+(1+K_{str}/K_{PZT})\cdot K _{str}/K_{PZT}=0 \tag{3}\]  
Solving Equation (3) results in,  
\[\omega_{n}^{2}=6c^{2}\frac{3+2K_{str}/K_{PZT}-\sqrt{9+8K_{str}/K_{PZT}}}{l_{n} ^{2}} \tag{4}\]  
Equation (4) shows that the peak frequency \(\omega_{n}\) in the piezoelectric admittance signature depends on the stiffness of the host structure. When a crack occurs and propagates in the host structure, the local stiffness of the host structure will reduce.