# Fatigue Crack Propagation under Christmas Tree Load Pattern
## 3 Results
### Plastic Strain  
Figure 5a presents the evolution of the plastic strain at the node containing the crack tip during some node releases starting at a crack length \(a=15.30\) mm. The plastic strain at the beginning of each new propagation is approximately 0.8 for all loading patterns. This results from the accumulation of damage while the node is positioned some distance ahead of the crack tip. On the other hand, for the high-frequency loading pattern, the deformation starts almost from zero. Therefore, the size of the plastic zone associated with the high-frequency loading pattern is very small in comparison to the other loading patterns studied. Due to the relatively low load range (see Figure 2), only one element ahead the tip is accumulating plastic deformation. Thus, this loading pattern requires more load cycles to reach the critical value of accumulated plastic deformation, yielding a lower FCG rate than the other patterns. The horizontal dashed line defines the critical value of cumulative plastic strain. Node release occurs when the accumulated strain value exceeds this limit. Despite the difference in the loading patterns (see Figure 2), the overall plastic strain growth rate between nodal releases is identical for all load spectrums, as shown in Figure 5a. Figure 5b is a detail of the evolution of the plastic strain evaluated at the crack tip for \(a=15.30\) mm for the different load patterns. In order to compare the behavior of the increase in plastic deformation during the same time interval, 17 load cycles of the high-frequency pattern were used. For the remaining load patterns, only one load block is shown in Figure 5b. Note that the plastic deformation graphs have undergone a translation of both axes, setting both the initial plastic strain and time to zero. Considering the constant-amplitude loading cases, both present exponential growths in plastic deformation  
Figure 4: Comparison of different loading spectrums: (**a**) CTOD curves evaluated for \(a=15.3\) mm. (**b**) Detail of applied load and of CTOD versus load plot. at the crack tip. However, the low-frequency pattern presents more prominent increases, in comparison to the high-frequency one, due to the higher \(\Delta K\) levels. For both Christmas Tree loading patterns, the plastic deformation has several increasing steps. These steps occur due to successive loading and unloading phases. During the global loading phase (0-8.5 s), the magnitude of these jumps increases, following the exponential growth observed for the constant amplitude case. A similar trend is observed during the global unloading phase (8.5-17 s). The biggest \"jump\" in the plastic strain occurs at the maximum load, for a time of 8.5 s. Additionally, the Christmas Tree (15-9) curve is above the results obtained for the CA low frequency and for the 9-3 Christmas Tree, which is particularly evident in the global unloading phase. This is relevant, as it indicates an influence of small cycles on crack tip damage quantified by cumulative plastic strain. This influence is not evident in the da/dN results presented in Figure 3 due to the oscillatory behavior associated with the discrete character of the numerical procedure. In addition, this does not occur for the 9-3 Christmas Tree load pattern, since the corresponding curve overlaps the CA curve in Figure 5b. In order to better understand the effect of the loading patterns on the plastic deformation at the crack tip, the portion of the loading cycle with plastic strain rise was identified. The results are presented, in Figure 6, for the constant-amplitude loading patterns. Considering the low-frequency loading pattern (Figure 6a,c), before the crack opening loading, \(\mathrm{F_{open}}=21.55\) N, there is no plastic deformation. After its opening, it enters in the elastic domain for about half a second, where there is still no plastic deformation. After around 3 s from the beginning of the load cycle, the plastic deformation starts and increases exponentially until it reaches the maximum load.