# Corrosion-Fatigue Evaluation of Uncoated Weathering Steel Bridges
## 4 Model of Corrosion-Fatigue
### Corrosion-Fatigue Model  
In the C-F crack initiation stage, [PERSON]'s law is more suitable for this process, which considers many factors such as the material property, temperature, corrosion system and time. In the C-F crack  
Figure 5: C-F initiation life of selected materials under various stress ranges: (**a**) 2014-T3 aluminum alloy and (**b**) 12% Cr stainless steel. propagation stage, Paris law needs different propagation parameters \(C\), \(m\) and the threshold value for a short crack and long crack respectively, which makes the evaluation rather complicated. Thus, the NASGRO equation requiring only a few parameters is preferable for the evaluation on this process, while it can describe both a short crack and long crack within the same pattern effectively. Based on the above analysis, the whole C-F life of the structural steel for bridges could be solved by combining the initiation life and propagation life, as shown in Equation (14):  
\[t_{C-F}=\frac{a_{c}^{3}}{B^{3}m_{asp}^{2}I_{p}\mathrm{C}_{p}^{3\alpha_{a}}}+t_ {c}+\frac{1}{f}\int_{a_{i}}^{h}da/\left\{D\Big{[}\left(\Delta K-\Delta K_{thr} \right)/\sqrt{1-K_{max}/A}\Big{]}^{m}\right\}, \tag{14}\]  
where \(t_{C-F}\) is the C-F life of the material. As shown in Equation (14), the whole C-F life consists of three parts. In the first part, the pitting corrosion is the only driving force since the condition is not met for the fatigue-induced propagation. The material constant \(B\) is similar in different kinds of steel so that its influence on the C-F life of different steel is not significant. Meanwhile, the constant \(C_{p}\) is highly dependent on the category of steel that a small variation in the constant can lead to a great difference in the initiation life. Besides, with the increase in the environment-related pitting current, the initiation life can be remarkably decreased. It is also indicated by Equation (14) that, the higher stress range further accelerates the pitting process described in the first part. In the last part, the fatigue effect outcompetes the corrosion effect and becomes the major driving force in crack propagation. Fatigue limit and fracture toughness are dependent on the fracture mechanical property of the material. Meanwhile, a higher frequency of cyclic loads can reduce the propagation life directly. Within the integral operation, the parameters \(\Delta K\) and \(K_{max}\) related to the stress range and stress ratio can also influence the propagation. Besides, the constants \(D\) and \(m\) are almost the same for structural steel, as given in [12]. For the geometric factor \(a_{i}\), it depends on the stress range, stress ratio, plate thickness, the rate of pitting corrosion and the rate of fatigue crack propagation. In the middle part, it is assumed that during the time period \(t_{c}\), both the pitting corrosion and the fatigue effect serve as the driving force in crack propagation parallelly and compete with each other. Thus, the time period \(t_{c}\) at this part will be affected by the above factors in both the corrosion and fatigue process. In sum, the C-F life of steel is a highly complicated process, which is affected by many crucial factors. Further verification has been carried out for the whole C-F life, through the experimental data of a type of precipitation hardening steel [10]. The material properties of the steel are listed in Table 3, with the fatigue limit recommended by [PERSON] [16]. The experiment data has been obtained under three different temperatures, i.e., 25 \({}^{\circ}\)C, 75 \({}^{\circ}\)C and 150 \({}^{\circ}\)C. Based on that, the pitting current can be respectively calculated under the three testing temperature based on Equation (14). As a result, the pitting currents are determined as, \(1.8065\times 10^{-8}\) C/s, \(1.8991\times 10^{-11}\) C/s and \(8.7407\times 10^{-11}\) C/s under 25 \({}^{\circ}\)C, 75 \({}^{\circ}\)C and 150 \({}^{\circ}\)C, respectively.