# Fatigue Crack Propagation under Christmas Tree Load Pattern
## 3 Results
### Without Contact of Crack Flanks  
The numerical elimination of the contact between the crack flanks is interesting to understand the relevance of the crack closure phenomenon. Figure 8a shows the FCG rates for the four load patterns neglecting the contact between the crack flanks. The da/dt curves of these load patterns behave similarly to those obtained with contact (see Figure 3). However, non-contact loading patterns show a more pronounced initial increase, and their stabilization value tends to increase linearly along the crack length. Note that, although the  
Figure 7: Evolution of plastic deformation during loading to a = 15.30 mm: (**a**) Christmas Tree (15–9) pattern; (**b**) Christmas Tree (9–3) pattern; (**c**) plastic strain rise of Christmas Tree (15–9) pattern; (**d**) plastic strain rise of Christmas Tree (9–3) pattern.  
value of da/dt also increases linearly, along the crack length in the simulations with contact, this is less evident, when compared with non-contact results. Indeed, da/dt increases because during propagation, the crack length gets larger and larger, which implies a growth in the magnitude of crack tip fields, and so of the stress intensity factor and propagation rate. Finally, here, the overall values of da/dt are superior to those obtained in the simulations with contact, where the stabilization value is 0.33 \(\upmu\)m/s, as the protective effect provided by crack closure is not present. Similarly to Figure 3, the high-frequency loading is also an exception, presenting a da/dt value of about 0.08 \(\upmu\)m/s; this is around 4 times lower than the other load patterns. The oscillations that the low-frequency pattern present are caused by the relatively high value of \(\Delta\)K. The propagation always occurs at the minimum value of the load cycle; therefore, there may exist some oscillation in the number of load cycles required to reach the critical cumulative plastic strain. On the other hand, for the VA loading patterns, although the \(\Delta\)K is similar, the crack propagation on these patterns can occur for different values of force (any local minimum), which tend to reduce the oscillations in a node release strategy.  
Figure 8b presents the CTOD plots obtained without contact between the crack flanks. Negative values are obtained, which indicate an overlap of the crack flanks at the minimum load, which cannot occur in a physical specimen. The crack tends to open immediately from the minimum load, unlike what happens in Figure 4a, in which the contact between the crack flanks is taken into account. Anyway, the shapes of the CTOD curves are identical to the curves obtained with contact. Note that the CTOD curves of the variable amplitude loadings were obtained after a node release at very low loading. Since the crack propagation did not occur at the global minimum load of the load block, a translation of the CTOD curves of the VA loading patterns occurred.  
Figure 9 shows the evolution of the plastic deformation evaluated at the crack tip for \(a=15.30\) mm, comparing different load patterns from the non-contact simulations. The plastic strain behavior of each simulation has several similarities with the results obtained considering the contact of the crack flanks (Figure 5b). Compared with the situation of constant-amplitude loading, the effect of the Christmas Tree (15-9) loading case is evident, particularly in the unloading part of the loading block, being more evident than that observed in Figure 5b for the situation with contact. The Christmas Tree (9-3) loading case does not produce an increase in damage, once again.  
Figure 8: Comparison of different loading spectrums neglecting the contact between the crack flanks: (**a**) da/dt versus crack length; (**b**) CTOD curves for \(a=15.30\) mm.