# Corrosion-Fatigue Evaluation of Uncoated Weathering Steel Bridges
## 4 Model of Corrosion-Fatigue  
### Corrosion-Fatigue Progress  
Since UWS bridges are exposed to the corrosion and fatigue action simultaneously, it is critical to incorporate the coupling effect of the two processes in solving the service life. At the early stage, pitting flaws will be generated on the surface of components by the corrosion. The size of these flaws increases with the time, as suggested by the [PERSON]'s law [22; 39]. Meanwhile, the pitting process could be further escalated by the load-induced stress range [23; 35]. Once the pitting flaw reaches the critical size, the fatigue process becomes the major driving force in the crack propagation while the chemical effect could be ignored [11]. Generally, the range of SIF could be used to determine the propagation rate of cracks [43]. Meanwhile, failure of members could be assumed when the crack penetrated through the thickness of any plate in the member. Based on the above analysis, C-F life can be divided into two stages, the C-F crack initiation and the C-F crack propagation, which is also suggested in [44]. Figure 4 illustrates the whole C-F process. According to the above discussion, the C-F model can be established. At the beginning of the C-F process, the pitting flaw keeps growing under the corrosion only, as described by [PERSON]'s law. The corrosion process could be further accelerated by the vehicle-induced cyclic stress [23; 35]. Thus, modification has been made in Equation (4) to incorporate the influence of cyclic stress, as shown in Equation (7):  
\[a=Bm_{asp}^{2/3}l_{p}^{1/3}t^{1/3}C_{p}^{a}, \tag{7}\]  
where \(C_{p}\) is the constant related to the material and \(\sigma_{a}\) is the stress range. Figure 3: Pitting current in different corrosivity categories. As aforementioned, the pitting flaw could be regarded as an equivalent crack. As suggested by [PERSON] et al. [40], the stress intensity factor (SIF) could be solved for the equivalent crack based on the stress range, pitting flaw property and geometry of plates, as shown in Equation (8):  
\[\Delta K=(S_{t}+HS_{b})\sqrt{\pi\frac{a}{Q}}F(\frac{a}{t},\frac{a}{c},\frac{c}{b },\varnothing), \tag{8}\]  
where \(a\) is the pitting depth; \(c\) is the pitting width; \(t\) it the thickness of the plate; \(b\) is the width of the whole plate; \(\varnothing\) is the angle of crack boundary; \(S_{t}\) and \(S_{b}\) are uniform-tension stress and bending stress; \(H\) is boundary-correction factor and \(Q\) and \(F\) are functions associated with \(a,c,t,b\) and \(\varnothing\). Due to the fatigue limit that exists in metals, the fatigue effect has little influence on the crack propagation when the crack size is under a specific critical size. Once the pitting flaw grows to the critical size, the fatigue effect will join as a parallel driving force in the crack propagation. The fatigue-induced propagation could be solved according to the fracture mechanics. For instance, the well-known Paris law [43] can be applied, as shown in Equation (9):  
\[\frac{da}{dN}=C(\Delta K)^{m}, \tag{9}\]  
where \(a\) is the crack size; \(N\) is the numbers of loading cycles; \(C\) and \(m\) are the material-related constant and power index, respectively and \(\Delta K\) is the range of the SIF. As proposed in [39], the competition criterion is introduced that the two effects, i.e., corrosion and fatigue, will compete in terms of the crack growth rate. As a result, the C-F process will be dominated by the driving force with the higher rate. It can be found from Equations (7)-(9) that, with the time passing, the growth rate of pitting corrosion is decreasing while that of the fatigue is increasing. As a result, the fatigue will gradually replace the corrosion as the major driving force in the propagation as aforementioned. Figure 4: Schematic stages of corrosion-fatigue (C-F). Based on the above analysis, the time period before the growth rate of fatigue surpasses that of pitting corrosion could be regarded as the C-F crack initiation stage, in which the crack growth is only determined by the corrosion process.