# Study on Corrosion Monitoring of Reinforced Concrete Based on Longitudinal Guided Ultrasonic Waves
## 2 Theoretical Background  
### Corrosion Process Model  
To facilitate research and analysis, the following assumptions are made:  
1. The concrete surrounding the reinforcement is a thick-walled cylinder with its wall thickness equal to the cover thickness.
2. The thickness of the corrosion products around the reinforcement is the same, that is, the reinforcement is uniformly corroded, and the pressure exerted by the corrosion products on the concrete around the reinforcement is uniform.
3. The effects of external factors are not taken into account in the calculations; only the expansion of the corrosion products induced by stresses and strains is considered.  
Figure 1 shows that the time required for cover cracking is divided into three time intervals [26]:  
1. \(T_{free}\) interval (determined by the \(A_{r1}\) surface): as shown in Figure 1a, corrosion products do not induce pressure on the concrete at the beginning of reinforcement corrosion until the porous zone (\(\delta_{0}\)) is filled with corrosion products.
2. \(T_{stress}\) interval (associated with the \(A_{r2}\) surface): Corrosion products gradually penetrate into the concrete and density the transition zone at the interface between the reinforcement and the concrete. Once the corrosion products fill the porous zone, they exert pressure on the surrounding concrete (Figure 1b). When the pressure is so high that the concrete cannot withstand it, initial cracks will develop in the concrete, as shown in Figure 1c.
3. \(T_{ingness}\) interval (corresponding to the \(A_{r3}\) surface): When initial cracks induced by internal pressure appear in the concrete, corrosion continues to expand and reduce the cross-section of the reinforcement. The corrosion products accumulated in the cracks will further lead to cover cracking, as shown in Figure 1d.  
As corrosion continues to expand, the initial cracks can evolve into cover cracking, and even lead to concrete crushing and spalling in severe cases, as shown in Figure 1e, which indicates that corrosion may exceed the acceptable damage threshold. Therefore, monitoring the initial corrosion and initial cracks can help to determine the safety of the concrete structures.  
One of the effective parameters for calculating the cover cracking time is the ratio of the volume of the corrosion products to that of the consumed reinforcement in the  
Figure 1: The process of development of corrosion of reinforcement. Note: where \(C\) is the thickness of concrete cover, \(D\) is the diameter of reinforcement, \(\delta_{0}\) is porous zone of thickness, \(\delta_{C}\) the amount of concrete displacement, \(a\) is the crack length, \(R_{1}\) and \(R_{2}\), respectively, are the internal and exterior cylinder radii.  
corrosion process (\(n\)). In this paper, the main type of corrosion product is Fe\({}_{2}\)O\({}_{3}\); it is therefore assumed that the parameter (\(n\)) is determined to be 2.20 according to Ref. [27].  
\[A_{loss}=\frac{A_{r}}{n}=\frac{A_{r1}+A_{r2}+A_{r3}}{n}\, \tag{1}\]  
where \(A_{loss}\) is the reinforcement area loss, and \(A_{r}\) is the cross-sectional area of the rust causing cover cracking.  
[PERSON]'s law has been used to estimate the time by [PERSON] et al. [26]. To obtain the total cover cracking time, it is necessary to calculate it using Equation (2).  
\[T=1879.2\frac{m_{l}D}{i_{cor}}\, \tag{2}\]  
where \(m_{l}=100A_{loss}/A_{st}\); \(m_{l}\) is the mass loss percentage of reinforcement; \(A_{loss}\) is the reinforcement area loss; \(A_{st}\) is the original cross-sectional area of reinforcement; \(i_{cor}\) is the corrosion current density (A/cm\({}^{2}\)).