## 2 Model Development
### Material Phase Properties
This way, the cement-paste related phase volume fractions, related to Figure 1c, for both old and new cement paste are expressed as functions of the (effective) water-to-cement mass ratios, denoted \((w/c)_{ocp}\) and \((w/c)_{ncp}\), and of the hydration degrees, \(\xi_{ocp}\) and \(\xi_{ncp}\), by means of [PERSON] hydration model [41] as [29]  
\[f^{i}_{clin}=\frac{20\left(1-\xi_{i}\right)}{20+63\left(w/c\right)_{i}}\geq 0\,, \qquad f^{i}_{hf}=1-f^{i}_{clin}\,,\qquad\forall i\in\left\{ocp,ncp\right\}, \tag{18}\]  
whereby \(f^{i}_{clin}\) denotes the (old or new) cement paste-related volume fraction of unhydrated clinker and \(f^{i}_{hf}\) the one of the hydrate foam matrix. At the scale of the (old or new) hydrate foam within ordinary cement paste (see Figure 1d), the volume fractions of capillary pores and of hydrates, \(f^{i}_{por}\) and \(f^{i}_{hyd}\), read as [29]  
\[f^{i}_{por}=\frac{63\left[(w/c)_{i}-0.367\,\xi_{i}\right]}{20\,\xi_{j}+63\left( w/c\right)_{i}}\geq 0\,,\qquad f^{i}_{hyd}=1-f^{i}_{por}\,,\qquad\forall i\in \left\{ohf,nhf\right\}. \tag{19}\]  
The hydration reaction stops at ultimate hydration degrees \(\xi^{ult}_{ocp}\) and \(\xi^{ult}_{ncp}\), which read as [29]  
\[\xi^{ult}_{i}=\min\left\{\frac{\left(w/c\right)_{i}}{0.42},1\right\}\,,\qquad \forall i\in\left\{ocp,ncp\right\}. \tag{20}\]  
Next, elastic phase properties of all phases are discussed. The elastic behavior of old and new hydrate foam phases, of old and new cement paste phases, and of the old mortar phase, respectively, result from homogenization over the corresponding RVEs, as detailed in Appendix A. All remaining phases--aggregates (subscript \(a\)), unhydrated clinker (\(clin\)), hydrates (\(hyd\)), and capillary pores (\(por\))--are considered to exhibit _invariant_ isotropic elastic behavior according to  
\[f_{na}^{con} =(1-f_{ra}^{a})\left(1-f_{ncpm}^{con}\right)\,, \tag{16}\]0  
where \(\mathds{C}_{i}\), \(k_{i}\), and \(\mu_{i}\), respectively, denote the elastic stiffness tensor, the bulk moduli, and the shear moduli of phase \(i\) (see Table 1 for the constants considered herein); where \(\mathds{I}^{dev}=1/3\,\mathbf{1}\otimes\mathbf{1}\) and \(\mathds{I}^{dev}=\mathds{I}-\mathds{I}^{vol}\), respectively, stand for the volumetric and the deviatoric part of the fourth-order unity tensor \(\mathds{I}\) with components \(I_{ijkl}=1/2\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{kj}\right)\), and \(\mathbf{1}\) denotes the second-order unity tensor with components equal to Kronecker \(\delta_{ij}\), which is equal to 1 for \(i=j\), and 0 otherwise. Notably, both old and new aggregates are considered to exhibit the same elastic moduli, denoted as \(k_{a}\) and \(\mu_{a}\), respectively. We are left with discussing the modeling of the mechanical properties of the ITZs. Microhardness testing around recycled aggregates at 28 days revealed that the Vickers microhardness is found to be 30% larger around natural aggregates than around the old mortar from recycled aggregates [42]. These observations were further corroborated by scanning electron microscopic studies, showing that the ITZs around old mortar appeared to be more porous than around natural aggregates [11; 12; 43]. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**Bulk Modulus**} & \multicolumn{2}{c}{**Shear Modulus**} \\  & \multicolumn{1}{c}{\(k\) [GPa]} & \multicolumn{2}{c}{\(\mu\) [GPa]} \\ \hline clinker & \(k_{clin}\) & 116.58 & \(\mu_{clin}\) & 53.81 \\ hydrates & \(k_{hyd}\) & 18.69 & \(\mu_{hyd}\) & 11.76 \\ aggregates (old and new) & \(k_{agg}\) & 35.35 & \(\mu_{agg}\) & 29.91 \\ hydrates & \(k_{por}\) & 0.00 & \(\mu_{por}\) & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Non-zero isotropic elasticity constants of material phases according to [21]. Notably, the ITZs might be improved if recycled aggregates are saturated [12], if a two-stage mixing approach is applied [11], or if the aggregate surface is treated chemically [43].