## 3 Results  
Inspired by the study presented in [17; 27], highlighting the effects of the ring geometry on the stress development and consequently crack initiation, similar test cases are proposed. Figure 3: Possible damage profile choices with respect to the level set function \(\Phi\). Figure 4 sketches the ring geometry. To switch from a circular ring to slightly and extremely elliptical rings, we play with the ratio \(\frac{a}{b}\); the ring is circular if \(\frac{a}{b}=1\) and elliptical otherwise. Five different ratios are proposed herein. The inner major radius \(a\) is always kept equal to 150 mm, while the minor inner radius \(b\in\{150;125;100;75;60\}\) mm with a wall thickness \(\Delta R\) of 37.5 mm. All rings share the same height \(H\), equal to 75 mm. The TLS model treats damage and crack growth at the macroscopic scale. Therefore, only the homogenized properties are needed. To achieve a normal strength concrete, the authors of [17; 27] used the following mixture proportions (summarized in Table 1) with commercially available CEM II Portland cement and a maximum aggregate size \(agg_{max}\) of 10 mm. Based on their regression fit to measurements, [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\) are respectively equal to 15.915 GPa and 0.2. Note that Day 15 was chosen to retrieve the latter properties since it is approximately the day for which the concrete rings cracked. When preparing the different rings for the experimental study, they were restricted by inner steel rings, and a coat of a release agent was applied to eliminate friction between concrete and steel. For now, these conditions are imposed by applying directly appropriate conditions on the inner surface of the concrete ring (\(u_{x}=0\), \(u_{y}=0\) and \(u_{z}\) free). This point will be revisited in future works, but remains for now the simplest way to treat the restricted shrinkage without treating contact and friction for multi-material problems. We start by choosing \(l_{c}=0.03\) m \(\simeq 3.agg_{max}\). For a fracture energy \(G_{c}\) of approximately 8.4956 N/m, the value \(Y_{c}=283.18\) Pa is chosen to respect Equation (16) as explained earlier. To bring to light the effects of shrinkage in absence of any external loading, no external load is applied (\(f=0\)), and we impose a uniform shrinkage strain all over the domain \(\mu e^{shr}=\left( \begin{array}{ccc}-\mu&0&0\\ 0&-\mu&0\\ 0&0&-\mu\end{array} \right)\), where \(\mu\) is the load factor to be computed as explained in Section 2.4. Figure 5 illustrates the von Mises stress contour of the different ring configurations ranging from slightly to extremely elliptical before any damage detection. Clearly the geometry (the only variable in this computation series) has a great impact on the stress distribution. When the ring is perfectly circular, the stress is axisymmetric, resulting in an equal opportunity of crack initiation along the circumference. The more elliptical the ring gets, the more the stress field loses its symmetry (with respect to the origin). This suggests that if any damage or cracks are detected, the zone of interest will grow smaller as the ratio \(\frac{a}{b}\) increases and will be more and more located near the major axis \(a\). One could imagine that when \(b\) tends towards zero, the ring will be completely flattened with singulari \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mixture Identification** & **Cement** & **Sand** & **Gravel** & **Water** \\ \hline w/c = 0.5 & 1 & 1.5 & 1.5 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mixture proportions by weight. Figure 4: A schematic representation of the elliptical ring geometry. major axis is supposed to be. It is only natural that the position of any detected damage or crack will converge toward this singularity. To reinforce this reasoning, the study is completed by a test case with \(b=1\) mm (\(\dfrac{a}{b}=150\)) and is considered as the borderline case. Figure 6 illustrates the different damage zones detected after 15 increments for \(\dfrac{a}{b}\in\{1,\,2.5,\,150\}\).