## Appendix B Covered Aggregate Inclusion
As for stiffness homogenization of the RVE related to concrete (see Figure 1a and Equation (10)), we need access to the strain concentration tensors of the old covered aggregates \(\mathbb{A}_{\infty,\infty c}\) and of the old cement paste cover \(\mathbb{A}_{\infty,\infty cpc}\), respectively, which are related to the matrix-inclusion problem depicted in Figure 11. The tensors are isotropic,  
\[\mathbb{A}_{\infty,i}=A_{\infty,i}^{\text{vol}}\mathbb{I}^{\text{vol}}+A_{ \infty,i}^{\text{dev}}\mathbb{I}^{\text{dev}}\qquad i\in\{oca,ocpc\} \tag{13}\]  
and the volumetric and deviatoric components read as [53]  
\[A_{\infty,\infty c}^{vol} =\frac{F_{1}}{F_{3}}\,, A_{\infty,\infty c}^{dev} =\frac{1}{A_{3}}\left[A_{1}-\frac{21}{5}\frac{R_{1}^{5}}{\left(1-2v _{\alpha}\right)R_{1}^{3}}B_{1}\right], \tag{14}\] \[A_{\infty,\infty cpc}^{vol} =\frac{F_{2}}{F_{3}}\,, A_{\infty,\infty cpc}^{dev} =\frac{1}{A_{3}}\left[A_{1}-\frac{21}{5}\frac{R_{2}^{5}-R_{1}^{5}} {\left(1-2v_{\alpha cp}\right)\left(R_{2}^{3}-R_{1}^{3}\right)}B_{2}\right].\]  
We are left with providing access to stress concentration tensor \(\mathbb{B}_{ocpc}^{con}\) as a function of the dimensionless radial coordinate \(\bar{r}\). The latter coordinate is defined as  
\[\bar{r}=\frac{r-R_{1}}{R_{2}-R_{1}}\,,\qquad 0\leq\bar{r}\,. \tag{15}\]The stress fields under the isotropic load case and the shear load case, respectively, were developed before (see the analytical expressions for the displacements according to (A12) and the paragraph which follows that equation). Notably, the such obtained stress fields refer to loading in terms of homogeneous strains \(\mathbf{E}_{\infty}\) applied at the infinite boundary. In order to relate the RVE of recycled concrete (Figure 1a) to the matrix-inclusion problem (A1), we need to identify the matrix-inclusion problem-related strain state \(\mathbf{E}_{\infty}\) based on an RVE strain state. We follow [PERSON] (1992) and consider that the strain average rule in the RVE has to be fulfilled, requiring that the concrete-related strain state \(\mathbf{E}\) results from average phase strains in all six phases of which the RVE is built:  
\[\mathbf{E}=\sum_{i}f_{i}^{con}\mathbf{\varepsilon}_{i}=\sum_{i}f_{i}^{con}\mathds{A}_{ \infty,i}:\mathbf{E}_{\infty}.\] (A17)  
Rewriting (A17) and considering the macroscopic Hooke's law yields \(\mathbf{E}_{\infty}\) as a function of the macrostresses \(\mathbf{\Sigma}\) that are applied to the RVE of recycled concrete:  
\[\mathbf{E}_{\infty}=\left(\sum_{i}f_{i}^{con}\mathds{A}_{\infty,i}\right)^{-1}:( \mathds{C}_{con})^{-1}:\mathbf{\Sigma}.\] (A18)  
For a general loading \(\mathbf{\Sigma}\), this results in six different non-zero components for \(\mathbf{E}_{\infty}\). Since the solutions for the displacement and the corresponding stress fields are available for isotropic and shear loading only, we need to superimpose these solutions in order to account for the actual strain state \(\mathbf{E}_{\infty}\). First, the solution under isotropic loading is built, whereby the constant \(\theta\) in Equation (A11) is specified for strain states (A18) as \(\theta=\text{tr}\mathbf{E}_{\infty}\). Next, five simple shear loading cases are considered, whereby the constant \(\gamma\) follows as  
\[\gamma\in\{e_{11},e_{12},e_{13},e_{11},e_{33}\}\quad\text{where}\quad\mathbf{e}= \mathbf{E}_{\infty}-\text{tr}\mathbf{E}_{\infty}\mathds{1}.\] (A19)  
The five displacement fields that correspond to the five shear loading cases and the displacement field that corresponds to the isotropic loading are then superimposed (under consideration of the case-specific orientation of the shear loading). This results in an RVE-related solution for the displacement field of the old cement paste cover. In the corresponding stress field \(\mathbf{\sigma}_{ocpc}(\underline{x})\), we finally substitute the radial coordinate \(r\) by its dimensionless counterpart \(\bar{r}\) according to Equation (A16), which finally allows for establishing the sought stress concentration tensor \(\mathds{B}_{ocpc}^{con}\).