# Evaluation of Bol Corrosion Degree Based on Non-Destructive Testing and Neural Network
## 3 Propagation Mechanism and Signal Processing of Guided Waves in Anchor Bolts  
### Propagation of Ultrasonic Guided Waves in Anchor Bolts  
In assuming that the concrete is an isotropic elastic medium and the cylindrical body is axially symmetric and infinitely long, the axis of the cylinder coincides with the \(z\)-axis. In this configuration, guided waves can propagate along the cylinder and form three primary wave modes: a longitudinal wave \(L\), transverse wave \(T\), and torsional wave \(F\). The longitudinal wave \(L\) propagates along the axis of the cylinder (i.e., the \(z\)-axis), with its vibration direction being parallel to the direction of wave propagation. The transverse wave \(T\) propagates radially or tangentially along the cylinder, with its vibration direction perpendicular to the direction of wave propagation. The torsional wave \(F\) is a wave mode that rotates the cylinder around its axis without significant radial or axial displacement. A schematic of guided wave propagation is presented in Figure 5, and the propagation equations are given in (10) to (15) [17]. For longitudinal waves in a cylinder, the displacement field propagates along the axis of the cylinder. The wave velocity of the longitudinal wave is given by the following equation:  
\[c_{L}=\sqrt{((\lambda+2\mu)/\rho)} \tag{10}\]  
where \(\rho\) represents density, \(\lambda\) and \(\mu\) are Lame constants, and \(C_{L}\) denotes the longitudinal wave velocity in the cylindrical body. The dispersion equation for longitudinal waves in a cylindrical body is given by the following equation:  
\[k^{2}+(\omega^{2}/c_{L}^{2})=0 \tag{11}\]  
where \(k\) is the wave number and \(\omega\) is the angular frequency. For transverse waves in a cylinder, the displacement field propagates perpendicular to the axis of the cylinder. The wave velocity of the transverse wave is given by the following equation:  
\[c_{T}=\sqrt{\mu/\rho} \tag{12}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Corrosion Time (h)** & **Remaining Mass (g)** & **Mass Loss** & **Corrosion Percentage** & **Corresponding Actual Corrosion Time (d)** \\ \hline
0 & 3477.8 & 0 & 0 & 0 \\
12 & 3452.7 & 25.1 & 0.72\% & \(\geq\)200 \\
24 & 3424.2 & 53.6 & 1.54\% & \(\geq\)400 \\
36 & 3394.6 & 83.2 & 2.39\% & \(\geq\)600 \\
48 & 3369.4 & 108.4 & 3.12\% & \(\geq\)800 \\
60 & 3345.2 & 132.6 & 3.81\% & \(\geq\)1000 \\
72 & 3322.6 & 155.2 & 4.46\% & \(\geq\)1200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Corrosion results. Figure 5: Propagation of guided waves in anchor bolts. The dispersion equation for transverse waves in a cylindrical body is as follows:  
\[k^{2}+(\omega^{2}/c_{T}^{2})(1+(J_{n}^{\prime}(kr))^{2}/(J_{n}(kr))^{2})=0 \tag{13}\]  
where \(J_{n}\) and \(J_{n}^{\prime}\) are the derivatives of the first-kind and second-kind Bessel functions, respectively, and \(r\) is the radius of the cylinder. For torsional waves in a cylinder, the displacement field forms a helical pattern and propagates along the axis of the cylinder. The wave velocity of the torsional wave is given by the following equation:  
\[c_{F}=\sqrt{(\mu/(\rho+4\mu/3))} \tag{14}\]  
The dispersion equation for torsional waves in a cylindrical body is given by  
\[k^{4}+(\omega^{2}/c_{S}^{2})(k^{2}+1.5k^{2}(J_{n}^{\prime}(kr)/J_{n}(kr))^{2})=0 \tag{15}\]  
During the propagation of guided waves, there are three modes of waves: L(0,\(m\)), T(0,\(m\)), and F(1,\(m\)). Here, '\(m\)' represents the mode order, while '0' and '1', respectively, denote symmetric and asymmetric wave fields.