## 3 Results
We focus next on determining some kind of time interval for when the damage first appears. Table 2 and Figure 10 summarize the different shrinkage values \(|\mu_{0}e^{shr}|\) for which an initial partial damage (not yet a visible macroscopic crack) is initiated (i.e., Point A). The higher the oblateness a ring has, the less shrinkage it tolerates before a damage appears. The trend line fit of \((|\mu_{0}e^{shr}|)_{a/b\in[1,150]}\) shows that the needed shrinkage to introduce damage is a decreasing function of \(\dfrac{a}{b}\) with a decreasing rate. The authors of [17; 27] determined that \"Cracking ages of the circular ring and elliptical rings with \(\dfrac{a}{b}<2\), i.e., in this case \(b=125\) or \(100\) mm, are all close to 15 days. However, elliptical rings with \(\dfrac{a}{b}\geq 2\), i.e., in this case \(b=75\) or \(60\) mm, cracked much earlier at around 11 and 13 days, respectively, on average\". Aiming to determine a time interval, values summarized in Table 2 are placed on the free shrinkage experimental measurements obtained by [17; 27]. Figure 11 illustrates two different scenarios where numerical shrinkage values \(|\mu_{0}e^{shr}|\) are linked to free shrinkage data for respectively sealed and unsealed specimens. If linked with the experimental curve under sealing conditions, the damage first starts around Day 12 for the circular and slightly elliptical rings (i.e., \(\dfrac{a}{b}<2\)) and around Day 10 for more elliptical rings (i.e., \(\dfrac{a}{b}\geq 2\)). Figure 9: Shrinkage strain evolution for the different \(\dfrac{a}{b}\) ratios limited between the first damage and the first crack initiation (i.e., between Points A and B). manner. The overall behavior of the model is tested. Despite the simplicity considered herein, the first results are promising. 1. Rings' oblateness induces a higher degree of auto-restraint and therefore has a great impact on stress distribution, damage zone initiation and crack positions: * Increasing the ratio \(a/b\) and the ellipticity of the rings showed that stress concentration (i.e., critical zones) is more and more located near the major axis. Different damaged zones were consequently detected and initiated. * A comparison between critical shrinkage strains needed to initiate damage and eventually cracks proved to be dependent on the oblateness severity. Less shrinkage is needed to initiate damage for highly elliptical rings suggesting that damage and cracking times are shorter for elliptical rings. 2. The crack position and trajectory matches the experimental observations presented in [17; 27]. A single crack going through the ring wall was detected. The TLS model offers a more complete insight in particular into the transition between damage and fracture. Contrary to other numerical models and experimental tests, the damage evolution is tracked, and crack initiation is observed when damage reaches exactly one. For illustration purposes, let us take the case using \(\dfrac{a}{b}=1.2\). Figure 12 presents the state of the ring at different increments just after initial cracking. Notice that a small crack appears on the top outer side of the ring in Figure 12a, followed by similar ones on the inner upper and outer bottom sides of the ring, respectively illustrated in Figure 12a,b. The cracks grow closer and then merge to form one single crack going through the ring wall (see Figure 7a). This information is very important for industrial cases where the position of the initial crack and the interaction between neighbor cracks is crucial. To summarize, the numerical results are encouraging and agree fairly well with the experimental findings. The capacity of the TLS model was tested and judged solid. Moving forward, there is plenty of room for improvements. The model can be enhanced by introducing:  
1.