## 2 Model Development
### Stress Downscaling to Hydrates via ITZs
Therefore, we introduce the dimensionless radial coordinate \(\bar{r}\), which amounts to zero at the covered aggregate-old paste interface and to one at the old cement paste cover-new cement paste interface (see Appendix B for more details). This way, the sought ITZ stress in the immediate vicinity of the old covered aggregate reads as  
\[\mathbf{\sigma}_{\mathcal{I}_{oa}^{ocp}}=\mathbf{\sigma}_{ocp}(\bar{r}=0)\,. \tag{8}\]  
Finally, the stress state of the ITZ between the old cement paste cover and the new cement paste matrix, labeled as \(\mathbf{\sigma}_{\mathcal{I}_{ocpc}^{ncp}}\) can be obtained by applying the perfect bond-related interface stress concentration rule (5) to the stresses of the old cement paste cover evaluated at \(\bar{r}=1\) as  
\[\mathbf{\sigma}_{\mathcal{I}_{ocpc}^{ncp}}=\mathrm{B}_{ocpc}^{ncp}:\mathbf{\sigma}_{ ocp}(\bar{r}=1)\,. \tag{9}\]  
So far, we dealt with stress concentration relation at the millimeter-to-centimeter-sized concrete and mortar scales. Following earlier micromechanical approaches [18; 19], the ITZs are considered as perfectly bonded 2D interface phases, rather than 3D phases at these scales. However, at the micrometer-sized cement paste scale, the ITZ's microstructure is resolved according to Figure 1c. This way, the stress states according to Equations (5) and (7)-(9) can be considered as homogeneous macroscopic stress states imposed on an ITZ-related RVE of cement paste. Notably, the depicted microstructure is relevant for both the new cement paste as encountered in four of the six ITZs, namely in \(\mathcal{I}_{na}^{ncp}\), \(\mathcal{I}_{opa}^{ncp}\), \(\mathcal{I}_{oma}^{ncp}\), and \(\mathcal{I}_{ocpc}^{ncp}\), and for the old cement paste encountered in the two remaining ITZs, namely in \(\mathcal{I}_{oa}^{ocp}\) and \(\mathcal{I}_{oa}^{ocp}\). Concentrating the cement paste-related ITZ stresses further down to failure-relevant hydrate stress measures \(\overline{\overline{\sigma}}_{hyd}^{vol}\) and \(\overline{\overline{\sigma}}_{hyd}^{dev}\), which enter the Drucker-Prager failure criterion (1), is discussed next. It has been shown by extensive model validation against independent experimental results that the second-order moments of microstresses of hydrates qualify as relevant stress measures for quantifying hydrate failure [21; 29; 30]. To get access to the second-order stress invariants, we follow the pioneering approach of [PERSON] [38] and consider that the elastic energy stored in the RVE can be expressed from a macroscopic and a microscopic standpoint. This allows for the derivation of closed-form expressions of the scalar stress invariants \(\overline{\overline{\sigma}}_{hyd}^{vol}\) and \(\overline{\overline{\sigma}}_{hyd}^{dev}\), as summarized next and discussed in full detail in Refs. [29; 39; 40]. First, we use the inverse form of the isotropic elasticity law, associated to each ITZ, to obtain ITZ-specific strains \(\mathbf{\varepsilon}_{\mathcal{I}}\) as  
\[\mathbf{\varepsilon}_{\mathcal{I}}=\left(\mathbb{C}_{\mathcal{I}}\right)^{-1}:\mathbf{ \sigma}_{\mathcal{I}}\qquad\forall\mathcal{I}\in\left\{\mathcal{I}_{na}^{ncp},\mathcal{I}_{opa}^{ncp},\mathcal{I}_{oma}^{ncp},\mathcal{I}_{oa}^{ocp}, \mathcal{I}_{ocpc}^{ncp}\right\}, \tag{10}\]  
where \(\mathbf{\sigma}_{\mathcal{I}}\) represents the ITZ-specific stresses according to Equations (5) and (7)-(9), and \(\mathbb{C}_{\mathcal{I}}\) is the ITZ-specific elastic stiffness tensor. The latter follows from elastic homogenization based on the RVE microstructures depicted in Figure 1c,d (see Appendix A for analytical expressions).