## 3 Results
Note that the damage \(d\) did not reach one yet. In other words, the results show the state of the rings before any crack initiation. Notice that, as expected, the different damaged zones are almost equi-distributed for the perfectly circular ring (Figure 6a) due to the equal opportunity of damage and crack initiation. As the ratio \(\dfrac{a}{b}\) increases, the damaged zones are more and more located in the major axis vicinity (Figure 6b), and their position converges towards the singularity when \(\dfrac{a}{b}\rightarrow\infty\) (Figure 6c). In addition, a comparison between the numerical results issued from the TLS model and the experimental ones collected from [17; 27] is presented in Figure 7. The latter comparison not only shows a good agreement between the numerical and experimental campaigns, but also confirms that both damage and crack positions are affected by the oblateness severity. For the slightly elliptical ring (\(\dfrac{a}{b}=1.2\)), one crack passing through the wall thickness is visible lightly close to the minor axis (Figure 7a,b). As \(b\) decreases, the position of the crack moves closer towards the major axis (Figure 7c,e). Figure 5: View from the top: von Mises stress (in MPa) contour of the concrete elliptical rings. Note that the color-map limits are not unified for all cases to better represent the stress concentration. (**a**) \(\dfrac{a}{b}=1.2\); (**b**) \(\dfrac{a}{b}=1.5\); (**c**) \(\dfrac{a}{b}=2\); (**d**) \(\dfrac{a}{b}=2.5\). As mentioned earlier, the current version of the TLS model adopts a time-independent evolution law. In other words, the algorithm is governed by damage front advance rather than a time-step control. This is why the exact day on which a crack first appears cannot be determined automatically. Nevertheless, it is still possible to determine a relationship between the ratio \(\frac{a}{b}\) and the needed shrinkage to initiate a damage. Additional post-processing is needed to retrieve the response pattern of the TLS model. For the sake of clarity, the post-processing is explained on the circular ring case (\(\frac{a}{b}=1\)) in Figure 8 before presenting the results for the whole range of test cases. The evolution of the load factor \(|\mu e^{shr}|\) as a function of the simulation increments is plotted. The ring state is presented for four different points, denoted A, B, C and D on the curve, respectively in Figures 8a-d. Point A corresponds to the first damage introduced for a load factor \(\mu_{0}=209.94\)  
Figure 8: Critical shrinkage strain evolution as a function of increments and a top view of the damage field for four different moments of the simulation illustrated for \(\frac{a}{b}=1\). (**a**) First damage initiation; (**b**) crack initiation; (**c**) crack propagates on the top side; (**d**) crack breaks the ring. (i.e., shrinkage strain equal to \(-209.94\frac{\mu m}{m}\)). Point B corresponds to the increment for which the damage reaches one and a crack is initiated for the first time. Point C corresponds to the increment for which the crack propagated completely on the top side of the ring. Finally, Point D corresponds to the increment for which the crack goes through the whole thickness and wall, breaking the ring altogether. To superpose the curves for the different test cases and due to the lack of time notion, the increment for which the crack is first initiated (i.e., Point B) is manually determined for each case and is called \(incr_{a/b}^{d-1}\). The latter is equal to 60 for the circular ring. Relative increments are then determined as \(\dfrac{incr}{incr_{a/b}^{d-1}}\), and all curves are consequently plotted on \([0,1]\) (i.e., between Points A and B) in Figure 9.  
All curves are bound between two limits, the upper one for \(\dfrac{a}{b}=1\). and a lower one for \(\dfrac{a}{b}\rightarrow\infty\). Increasing the ratio \(\dfrac{a}{b}\) yields smaller critical shrinkage loads capable of initiating a damage (i.e., Point A), then a crack (i.e., Point B). The overall behavior is consistent and promising.