# Study on Corrosion Monitoring of Reinforced Concrete Based on Longitudinal Guided Ultrasonic Waves
## 3 Finite Element Simulation of Corroded Reinforcement
### Results of Finite Element Simulation
#### 3.2.2 Damage Index  
Corrosion products generated by the reinforcement in concrete cause cracks, rust pits, and other defects. It causes different reflection, scattering, diffraction, etc. when the GUW propagates in the reinforcement under different corrosion degrees. During initial corrosion, the volume of the corrosion products increases and then penetrates into concrete, resulting in a denser interface transition zone, which can mitigate the gradient difference of impedance and allow for more energy to be transmitted into the concrete [38]. Cracking leads to the contact between the reinforcement and air, and the impedance difference between the reinforcement and air is larger than that between the reinforcement and concrete, thereby reducing the energy leak from the reinforcement. The damage index (DI) is a quantitative analysis of the response and monitoring of GUWs to corrosion damage by associating corrosion damage with some characteristic of the response signal of GUWs. To study the influence of corrosion damage on the propagation of GUWs in structures, five damage indexes characterizing the corrosion degree, such as the time domain cross-correlation (TDC), time domain difference (TDD), spectrum amplitude difference (SAD), mean square difference (MSD), and normalized correlation moment (NCM), are proposed in this paper based on existing research [39; 40]. Figure 7: Waveforms of GUW signals at different corrosion degrees (100 kHz). 1. DI-TDC The TDC is based on time domain signal correlation and focuses on obtaining the time domain waveform of the signal over time. \[\text{DI-TDC}=1-\frac{\int_{t_{0}}^{t_{1}}\left(\text{b}(t)-\mu_{b(t)}\right) \left(m(t)-\mu_{m(t)}\right)\text{d}t}{\sigma_{b(t)}\sigma_{m(t)}}\,\] (5) where \(t\) is time, \(b(t)\) is the reference signal corresponding to the uncorroded state, and \(m(t)\) is the detection signal corresponding to different corrosion degrees. \(t_{0}\) and \(t_{1}\) are the start time and stop time corresponding to the time window of the detection signal, respectively. \(\mu_{b(t)}\) and \(\mu_{m(t)}\) are the average values of the signal, while \(\sigma_{b(t)}\) and \(\sigma_{m(t)}\) are the variances of the signal. (2) DI-TDD The principle of the TDD is that the detection signal is first normalized by \(\alpha\), which is the scale factor of the reference signal, then calculated from the minimum of the mean squared difference between the normalized detection signal and the reference signal. Therefore, the difference between the signals reflected by the TDD is independent of the signal amplitude. \[\text{DI-TDD}=\int_{t_{0}}^{t_{1}}\tilde{d}^{2}(t)\text{d}t=\int_{t_{0}}^{t_{1 }}\tilde{d}^{2}(t)\text{d}t\,\] (6) where \(\tilde{d}(t)=\widetilde{D}(t)-\alpha b(t)\); \(\widetilde{D}(t)=m(t)/\sqrt{\int_{t_{0}}^{t_{1}}m^{2}(t)\text{d}t}\); \(\widetilde{D}(t)=m(t)/\sqrt{\int_{t_{0}}^{t_{1}}m^{2}(t)\text{d}t}\). (3) DI-SAD The SAD is based on the amplitude of the signal to measure the difference in the frequency response amplitude of the signal, and is calculated as follows: \[\text{DI-SAD}=\sqrt{\frac{\int_{\omega_{0}}^{\omega_{1}}\left(\left|b(\omega) \right|-\left|m(\omega)\right|\right)^{2}\text{d}\omega}{\int_{\omega_{0}}^{ \omega_{1}}\left(\left|b(\omega)\right|\right)^{2}\text{d}\omega}}\,\] (7) where \(b(\omega)=\int_{t_{0}}^{t_{1}}  
where \(n\) is the order number of the correlation moment, which may not be an integer. \(r_{xy}(\tau)\) is the correlation between the signals \(x(t)\) and \(y(t)\), and the signal time domain length is [\(t_{0}\), \(t_{1}\)], where \(\tau\) is a lag parameter. As can be seen from Equation (10), the NCM is negative when the detected signal amplitude is greater than the reference signal and takes on a value in the range [0, 1] when the signal amplitude decreases due to damage.