# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 2 Corrosion Fatigue Lifespan Model Based on EIFS Theory  
### EIFS Theory  
The key of fatigue lifespan prediction lies in determining the crack initiation point. In the case of small-sized specimens, the crack propagation lifespan is usually considered negligible compared to the overall fatigue lifespan. Thus, the entire fatigue lifespan is nearly equal to the crack initiation lifespan. However, the crack initiation lifespan is only a part of the overall lifespan for large-sized structures. In this scenario, the crack initiation lifespan is closely related to the critical size of crack nucleation. At present, there are three main methods for determining the critical size. One method is to empirically define a crack size as the critical size, which is usually between 0.25 and 1 mm for metals [30]. Another method is to use non-destructive testing to detect the crack nucleation size [31]. However, current non-destructive testing methods are unable to truly detect the size of the crack nucleation. If the empirically defined or non-destructive testing crack nucleation size is used, the predicted results tend to be overly optimistic. Therefore, the EIFS theory is introduced as a third method for determining the critical size of crack nucleation, which has been proven to be a practical method for evaluating crack initiation and fatigue lifespan [32]. The method of EIFS was initially proposed to address the issues in fracture mechanics-based fatigue lifespan prediction models. It is well known that the surface of any specimen is never perfectly smooth and always contains various types of defects, including intentional defects and corrosion defects as well as accidental scratches and impact-induced defects. All these surface defects can be equivalently treated as hypothesized cracks with known initial sizes, which are referred to as EIFS. The next question is how to determine EIFS. Surface-nucleated cracks generally exhibit typical short crack characteristics. It is well known that the crack growth behavior of short cracks is significantly different from that of long cracks [33; 34; 35]. Figure 1 illustrates the crack growth behavior of short cracks and long cracks in the near-threshold region, where \(a\), \(N\), and \(\mathrm{d}a\)/\(\mathrm{d}N\) are the crack length, loading cycle number and the crack growth rate. In addition, \(\Delta K\), \(\Delta K_{th}\) and \(K_{p}\) represent the stress intensity factor range, threshold value of the material for crack initiation and critical stress intensity factor. It can be observed that in the near-threshold region, the fluctuating small crack growth curve deviates significantly from the long crack growth curve. When the stress intensity  
Figure 1: Schematic diagram of crack propagation. factor reaches the critical stress intensity factor, small cracks will develop toward long cracks under the influence of a pitting pit. Hence, the usage of a long crack growth threshold to calculate EIFS is unreasonable. Since it is challenging to measure the crack growth curve of small cracks, the EIFS theory employs the extension of the linear portion of the Paris law as a surrogate for the short crack growth curve. Typically, the crack extension threshold for short cracks is taken as the Paris equation's slope extended to \(10^{-10}\) mm/cycle of the stress intensity factor range. The feasibility of this approach has been demonstrated in the literature [36]. Based on the EIFS theory [27], the initiation lifespan of corrosion fatigue cracks, also known as pitting, is considered as part of crack propagation, as shown in Figure 2. Once the initial equivalent defect is determined, the crack initiation lifespan of the corroded steel wire is equivalent to the integration from this equivalent initial crack depth to the critical crack depth under the long crack extension law. As the crack size decreases, the propagation rate of a long crack approaches an asymptotic trend, which is known as the crack threshold.