## Appendix A Mathematical Expressions for Stiffness Homogenization and Stress Concentration  
Herein, we provide analytical expressions for stiffness homogenization (upscaling) to hydrate foam, cement paste, mortar, and concrete, respectively, and for stress concentration (downscaling) to ITZ-related cement paste scales. The self-consistent stiffness estimates for the hydrate foam (Figure 1d) for the three encountered realizations of the hydrate foam microstructure--i.e., for the old hydrate foam (with homogenized stiffness tensor denoted as \(C_{\mathit{off}}\)), for the new hydrate foam (\(C_{\mathit{nhf}}\)), and for the weak ITZ-related hydrate foam (\(C_{\mathcal{I}^{*}\mathit{nhf}}\)) read as [40]  
\[ \begin{split} C_{\mathit{nhf}}=& f_{\mathit{nhf}}^{ \mathit{nhf}}C_{\mathit{nhf}}\cdot\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi} \left[\mathbb{I}+\mathbb{F}_{\mathit{cyl}}^{\mathit{nhf}}(\varphi,\theta): \left(C_{\mathit{nbd}}-C_{\mathit{nhf}}\right)\right]^{-1}\frac{\sin\theta}{4 \pi}\mathrm{d}\theta\,\mathrm{d}\varphi:\\ &\left\{f_{\mathit{pcr}}^{\mathit{nhf}}\left(\mathbb{I}-\mathbb{ F}_{\mathit{sph}}^{\mathit{nhf}}:C_{\mathit{nhf}}\right)^{-1}+f_{\mathit{nhd}}^{ \mathit{nhf}}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\left[\mathbb{I}+ \mathbb{F}_{\mathit{cyl}}^{\mathit{nhf}}(\varphi,\theta):\left(C_{\mathit{nbd }}-C_{\mathit{nhf}}\right)\right]^{-1}\frac{\sin\theta}{4\pi}\mathrm{d} \theta\,\mathrm{d}\varphi\right\}^{-1},\\ &\forall\mathit{nhf}\in\{\mathit{acp},\mathit{ncp},\mathit{ \mathcal{I}^{*}\mathit{nhf}}\}\end{split}  \tag{10}\]  
whereby \(f_{j}^{\mathit{nhf}}\) denotes the hydrate foam-related phase volume fractions (see Equations (19) and (22)), and \(C_{\mathit{nwd}}\) denotes the hydrate stiffness tensor (see Equation (21) and Table 1). The spherical and cylindrical phase shapes, respectively, manifest in the corresponding Hill tensors \(\mathbb{F}_{\mathit{sph}}^{\mathit{nhf}}\) and \(\mathbb{F}_{\mathit{cyl}}^{\mathit{nhf}}\)(see [40] for analytical expressions of their components). Notably, orientation angles \(\varphi\) and \(\vartheta\) refer to the azimuth and zenith angles of the hydrate needle (see Figure 5b). Next, the cement paste RVE (Figure 1c) is dealt with. The Mori-Tanaka estimates for the homogenized stiffness tensors of old cement paste \(\mathbb{C}_{ocp}\), of the new cement paste (\(\mathbb{C}_{ncp}\)), and of the weak ITZ-related cement paste (\(\mathbb{C}_{\mathcal{I}^{*}cp}\)), respectively, read as [40]  
\[ \begin{split}\mathbb{C}_{icp}=&\left\{\left(1-f_{ clin}^{icp}\right)\mathbb{C}_{lift}+f_{clin}^{icp}\mathbb{C}_{clin}:\left[ \mathbb{I}+\mathbb{P}_{sph}^{lhf}:\left(\mathbb{C}_{clin}-\mathbb{C}_{lift} \right)\right]^{-1}\right\}:\\ &\left\{\left(1-f_{clin}^{icp}\right)\mathbb{I}+f_{clin}^{icp} \left[\mathbb{I}+\mathbb{P}_{sph}^{lhf}:\left(\mathbb{C}_{clin}-\mathbb{C}_{ lift}\right)\right]^{-1}\right\}^{-1},\\ &\forall icp\in\{ocp,ncp,\mathcal{I}^{*}cp\}\quad\text{and}\quad \forall ihf\in\{ohf,nhf,\mathcal{I}^{*}hf\},\end{split}  \tag{10}\]  
with cement paste-related clinker volume fraction \(f_{clin}^{icp}\) according to (18) or (22). The Mori-Tanaka estimate of the homogenized stiffness tensor of old mortar (Figure 1b), in turn, reads as  
\[ \begin{split}\mathbb{C}_{oma}=&\left\{\left(1-f_{ oeu}^{oma}\right)\mathbb{C}_{ocp}+f_{oeu}^{oma}\mathbb{C}_{a}:\left[\mathbb{I}+ \mathbb{P}_{sph}^{ocp}:\left(\mathbb{C}_{a}-\mathbb{C}_{ocp}\right)\right]^{-1 }\right\}:\\ &\left\{\left(1-f_{oeu}^{oma}\right)\mathbb{I}+f_{oeu}^{oma}\left[ \mathbb{I}+\mathbb{P}_{sph}^{ocp}:\left(\mathbb{C}_{a}-\mathbb{C}_{ocp}\right) \right]^{-1}\right\}^{-1}\,,\end{split}  \tag{11}\]  
with old mortar-related embedded aggregate volume fraction \(f_{oeu}^{oma}\), old cement paste-related Hill tensor \(\mathbb{P}_{sph}^{ocp}\), and with aggregate stiffness \(\mathbb{C}_{a}\) (see Equation (21) and Table 1). Finally, we discuss the stiffness homogenization of the RVE of concrete, as depicted in Figure 1a.