# Fatigue Crack Growth from Notches:
## 4 Results
### Insertion of Contact at the Crack Flanks in the Unnotched-Cracked Specim  
The overlap of the crack flanks was then numerically removed, which enabled the simulation of crack closure. In fact, the phenomenon of crack closure reduces the effective load range, leading to a reduction of elastic and plastic deformations. As previously stated, this phenomenon is expected to have a major effect, particularly in plane stress simulations. Figure 5a-f compare the evolution of \(\delta_{p}\) against \(\Delta a\) with and without contact at the crack flanks for the three materials in study, and plane strain and plane stress conditions. As can be seen, in plane strain conditions (Figure 5a,c,e), the contact simulations have a slight reduction of \(\delta_{p}\), but maintaining the trends observed for the _no contact_ simulations. For the plane stress cases (Figure 5b,d,f), the insertion of contact at the crack flanks causes a major reduction of \(\delta_{p}\) with \(\Delta a\). The reduction of \(\delta_{p}\) is more pronounced for AA7050-T6 with pure isotropic behavior, and the decrease rate is higher for pure kinematic behavior. On the other hand, the AA6082-T6 shows a relatively slow reduction of \(\delta_{p}\) and the stabilization occurs for higher values of \(\Delta a\). The increase of \(\delta_{p}\) with \(\Delta a\) observed for the _no contact_ simulations is almost eliminated with the contact of crack flanks. Anyway, there is a small increase of \(\Delta a\) with \(\delta_{p}\), which is evident in Figure 5d. The minimum values, indicated by small arrows in Figure 5d,f, indicate the end of a transient regime associated with the formation of residual plastic wake. This transient regime is much more extensive than the transient behavior associated with the stabilization of cyclic plastic deformation, observed in Figure 4a. The variation of crack closure level, quantified by \(U^{*}\), is presented in Figure 6. \(U^{*}\) is given by:  
\[U^{*}\ =\ \frac{F_{open}-F_{min}}{F_{max}-F_{min}}\times 100, \tag{3}\]  
where \(F_{open}\) is the crack opening load and represents the percentage of load cycle during which the crack is closed. The variations of \(\delta_{p}\) observed in Figure 5a,d,f for the simulations with contact, are perfectly symmetric to those observed in Figure 6. This means that the decreases seen in Figure 5 have exactly the same upward trends seen in Figure 6. \(U^{*}\) increases with \(\Delta a\) and stabilizes for higher values of \(\Delta a\), which explains the reduction of \(\delta_{p}\) with \(\Delta a\) and the consequent stabilization. The growth rate of \(U^{*}\) with \(\Delta a\) is higher for AA7050-T6 with pure kinematic behavior, and higher values of \(U^{*}\) are achieved for AA7050 with pure isotropic behavior, which are in accordance with Figure 5d,f. There is a progressive increase of \(U^{*}\) that is linked to the formation of the residual plastic wake, as the crack propagates. The level of crack closure depends on the plastic elongation of the wedges behind crack tip and on crack tip blunting, and these two opposite mechanisms depend on crack tip deformation [35]. The crack propagation required for the stabilization of \(U^{*}\) is greatly dependent on material, being significantly higher for the AA6082-T6. In any case, this stabilization distance is alwayssignificantly higher than that needed to stabilize the cyclic plastic deformation. The stabilized values of \(U^{*}\) are in the range 38-50%, which are typical values for plane stress state. After stabilization there is a small but progressive increase of \(U^{*}\), which explains the lower variation of \(\delta_{p}\) observed in Figure 5d,f, compared to the _no contact_ simulations. Under plane strain conditions, the level of crack closure is significantly lower, less than 10%, which explains the small variation observed in Figure 5a,c,e. Figure 6b represents the CTOD versus F curves. The reduction of CTOD and therefore of \(\delta_{p}\) is evident when the contact at the crack flanks is introduced.