## 3 Results and Discussion
### Fatigue Lives Predictions  
The previous analysis showed for the specimens in the as-welded condition, that at high stress levels, due to the presence of small imperfections, several fatigue starts are activated and coalesce rapidly, presenting conditions to neglect the fatigue initiation stage and therefore for the application of the fracture mechanics approach. The general procedure consists, first, in identifying the failure modes of the specimens with characteristic types of cracks and subsequently estimating the fatigue lives, N, according to well-known relationships of Paris law and SIFR, \(\Delta\)K:  
\[\mathrm{N}=\int_{a_{i}}^{a_{i}}\frac{da}{\Delta\,\,\Delta K^{n}} \tag{1}\]  
\[\Delta\mathrm{K}=Y\,\mathrm{M}_{k}\,\Delta\sigma\sqrt{\pi a} \tag{2}\]  
Figure 10a-d shows four fractured specimens of the as-welded condition with details of the fatigue starts and schemes of crack types associated to the failure modes. Therefore, for the S2 series, a continuous surface crack corresponds to a failure due to multiple small imperfections along the weld root, for the S1 series, semi-elliptical surface cracks correspond to the failure due to the undercuts and for the S3 series, a corner crack corresponds to a failure in the underfill near at the border of the specimen. Failure is assumed to occur when, from an initial crack size, the final crack size is reached that corresponds to a critical fracture condition. For the S1 series when the semi-elliptical cracks (in Figure 10b,c) grow from the initial aspect ratios, (a\({}_{i}\)/c\({}_{i}\)), until they coalesce and cover the entire width, allows the determination of the fatigue lives of the specimens. In the case of the S3 series, when the initial crack depth grows to the thickness of the specimen (Figure 10d) and for the S2 series specimens, from a very small initial crack depth to the depth as observed in the fractured surface (Figure 10a). Table 5 contains geometric data of the cracks for five specimens (the four shown in Figure 10 and an additional specimen similar to the one shown in Figure 10a), parameters used to estimate the fatigue lives, the respective experimental results and the relations between the experimental lives and the corresponding estimated values, meanwhile, Table 6 corresponds to predictions. It should be clarified how the crack sizes shown in Tables 5 and 6 were established: the estimates were measured on the fractured surfaces of the specimens (as illustrated in the Figure 10c,d), meanwhile, predictions are based on the sizes of the imperfections reported in [27]. In both cases, where applicable, the sizes are the mean values of the imperfections. The following observations point out aspects of the crack sizes of each series. For the S2 series estimations, since there were multiple small imperfections along the width of the specimens, 12.5 mm was assumed as the initial and final lengths. Meanwhile for predictions, the final crack sizes, \(\alpha_{\text{f}}\) and \(c_{\text{f}}\), were determined as described as follows. The crack depth \(\alpha_{\text{f}}\) for the straight crack corresponding to the S2 series, was assumed as half the plate thickness (1.5 mm), as is recommended in [8], and the final crack length \(c_{\text{f}}\) was assumed to be 12.5 mm, because it corresponded to an extended crack. The value, \(\alpha_{\text{f}}\) of 3 mm in the S3 series, corresponds to the failure critical condition assumed for the corner  
Figure 10: Fatigue failure modes with details and corresponding types of cracks. (**a**) extended crack, (**b**) semi-elliptical multicracks (**c**) semi-elliptical crack and (**d**) corner crack. crack, meanwhile, the value 4.81 mm, corresponds to the same aspect ratio a/c as in Table 5. The a\({}_{\text{f}}\) = 0.96 mm and c\({}_{\text{f}}\) = 1.56 mm values, for the S1 series, were set considering that the critical condition occurs when the semi-elliptical cracks grow until their crack length ends meet and cover the width of the specimen as observed in Figure 10b,c.