## Appendix B Covered Aggregate Inclusion  
Herein, we deal with the matrix-covered inclusion problem, encountered during homogenization of the stiffness of the concrete RVE as well as for stress downscaling to the old cement paste cover around the covered aggregate. The corresponding matrix-inclusion problem consists of a spherical inclusion representing the old covered aggregate (with radius \(R_{1}\) and stiffness \(\mathbb{C}_{a}\)), a layer of old cement paste (radius \(R_{2}\) and stiffness \(\mathbb{C}_{ocp}\)), and an infinite matrix of new cement paste (with stiffness \(\mathbb{C}_{ncp}\)) (see Figure 11). The radius \(R_{1}\) is arbitrary and \(R_{2}\) follows from the phase volume fractions of old covered aggregates and of old cement paste cover as \(R_{2}=R_{1}\sqrt[3]{1+f_{ocp}^{\mathit{con}}/f_{oca}^{\mathit{con}}}\). Figure 11: Matrix-covered inclusion problem related to the covered recycled aggregate (Class III). The following calculations are based on the pioneer work of [PERSON] and [PERSON] [52], which provided analytical expressions for the 3D displacement, stress and strain fields inside a covered inclusion embedded in an infinite matrix. Thereby, two loading cases in terms of two homogeneous strain states \(E_{\infty}\) applied on the infinite boundary \(r\to\infty\), respectively, are considered: isotropic strains and a simple shear strains. We here follow [PERSON] and [PERSON] [53], and express the corresponding displacement field \(\underline{u}_{\infty}\) at the boundary as  
\[\underline{u}_{\infty}= \begin{cases}r^{\theta}_{3}\,\varepsilon_{r}&\ldots \text{isotropic},\\ \gamma\left(x\varepsilon_{x}-y\varepsilon_{y}\right)&\ldots\text{simple shear}, \end{cases}  \tag{10}\]  
with \(\theta\) and \(\gamma\) as constants related to the applied strain, \(r,\theta,\varphi\) as spherical coordinates with base vectors \(\underline{e}_{r},\underline{e}_{\theta},\underline{e}_{\varphi}\) and \(x,y,z\) as Cartesian coordinates with base vectors \(\underline{e}_{x},\underline{e}_{y},\underline{e}_{z}\), defined by analogy to Figure 5. The resulting displacement fields of the inclusion (subscript \(i=1\) in the following equation), of the cover (subscript \(i=2\)), and of the matrix (subscript \(i=3\)) read as [53]  
\[\underline{u}_{i}= \begin{cases}\left(E_{i}\,r+\frac{G}{r^{2}}\right)\, \varepsilon_{r}&\ldots\text{isotropic},\\ U^{r}_{i}(r)\sin^{2}\theta\cos^{2}2\varphi\,\varepsilon_{r}+U^{\theta}_{i}(r) \sin\theta\cos\vartheta\cos 2\varphi\,\underline{e}_{\varphi}+U^{\theta}_{i}(r) \sin\vartheta\sin 2\varphi\,\underline{e}_{\varphi}&\ldots\text{simple shear}, \end{cases}  \tag{11}\]  
whereby \(\vartheta\) denotes the polar angle and \(\varphi\) the azimuth angle (see Figure 5), and whereby the \(U^{r}_{i}\), \(U^{\theta}_{i}\) and \(U^{\varphi}_{i}\) read as  
\[U^{r}_{i}(r) =A_{i}r-6\frac{v_{i}}{1-2v_{i}}B_{i}r^{3}+3\frac{C_{i}}{r^{4}}+ \frac{5-4v_{i}}{1-2v_{i}}\frac{D_{i}}{r^{2}}, \tag{12}\] \[U^{\theta}_{i}(r) =-U^{\theta}_{i}(r)=A_{i}r-\frac{7-4v_{i}}{1-2v_{i}}B_{i}r^{3}+2 \frac{C_{i}}{r^{4}}+2\frac{D_{i}}{r^{2}}.\]  
In Equations (12), \(F_{i}\) and \(G_{i}\) as well as \(A_{i}\), \(B_{i}\), \(C_{i}\), and \(D_{i}\) are constants which result from traction and displacement continuity conditions at the interfaces between inclusion, cover, and infinite matrix, by considering that \(C_{1}\), \(D_{1}\) and \(G_{1}\) vanish to avoid singularities at \(r=0\) and that \(B_{3}\) vanishes to avoid a singularity at \(r\to\infty\), and by considering the boundary displacements according to Equation (10) (see [PERSON] and [PERSON] [53] for full analytical expressions). The strain fields within the three domains (inclusion, cover, and matrix) follow by considering the linearized strain tensor \(\epsilon_{i}=1/2\left[\partial\underline{u}_{i}/\partial\underline{x}+\left( \partial\underline{u}_{i}/\partial\underline{x}\right)^{T}\right]\). Stress fields, as required to evaluate ITZ-driven failure, finally result from the inverse form of [PERSON]'s law \(\sigma_{i}=\left(C_{i}\right)^{-1}:\epsilon_{i}\).