## 2 Materials and Methods
### Implementation of Missing Grains: Porosity (Voids/Pores)
The volume average of stresses over an RVE volume, when utilizing the divergence theorem, can expressed with the following equation:  
\[\sigma_{ij}=\frac{1}{V}\int_{V}\sigma_{ij}(\mathbf{x})\;dV(\mathbf{x})=\frac{ 1}{V}\int_{\Sigma}x_{i}t_{j}\;d\Sigma, \tag{14}\]where \(x_{i}\) is the \(i-th\) component of the position vector of the point lying on the RVE boundary \(\Sigma\), i.e., \(i\in\Sigma\). The strains could be evaluated instead as follows, where, again, the divergence theorem has been applied:  
\[\bar{\varepsilon}_{ij}=\frac{1}{2V}\int_{V}\big{(}u_{i,j}(\mathbf{x})+u_{j,i}( \mathbf{x})\big{)}dV(\mathbf{x})=\frac{1}{2V}\int_{\Sigma}\big{(}u_{i}n_{j}+u_ {j}n_{i}\big{)}\ d\Sigma. \tag{15}\]  
It is interesting to highlight here that, in the case of cohesive cracks or internal discontinuities, the average stress was not affected because they were either in equilibrium or traction free, and the stress integral was evaluated only along the external boundary of the domain (RVE). The same simplification did not apply, however, to the strains; therefore, internal discontinuities or decohesions had to be considered explicitly when computing the integrals due to the local displacement jumps \(u_{i}^{I}\). Hence, the following more general relationship applied:  
\[\bar{\varepsilon}_{ij}=\frac{1}{2V}\bigg{\{}\int_{\Sigma}\big{(}u_{i}n_{j}+u_ {j}n_{i}\big{)}\ d\Sigma-\int_{\Sigma^{I}}\big{(}\delta u_{i}^{I}n_{j}^{I}+ \delta u_{j}^{I}n_{i}^{I}\big{)}\ d\Sigma^{I}\bigg{\}}, \tag{16}\]  
where \(\Sigma^{I}\) are the surfaces of the discontinuities and \(n_{i}^{I}\) the components of their unit normal vectors. Once the stress \(\Omega\) and strain \(\Gamma\) in the macro-scale were known, the apparent macro-properties could be defined by the following:  
\[\Omega=\hat{C}\Gamma, \tag{17}\]  
where (\(\hat{\cdot}\)) denotes macroscopic properties, which are used to distinguish them from the microscopic ones. Equation (17) is the homogenized stress-strain equation, and it forms the basis for the homogenization procedure. The macroscopic properties were computed for each considered micro-structural realization. Hence, under the plane strain assumption, the constitutive equation in Voigt notation was as follows:  
\[ \begin{bmatrix}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{bmatrix} =\frac{E}{(1+v)(1-2v)} \begin{bmatrix}1-v&v&0\\ v&1-v&0\\ 0&0&1-2v\end{bmatrix}\begin{bmatrix}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \varepsilon_{xy}\end{bmatrix} , \tag{18}\]  
where \(E\) and \(v\) are the [PERSON]'s modulus and [PERSON]'s ratio, respectively. The volume fraction of porosity \(V_{f}\) (per unit length) is as follows:  
\[V_{f}=\left(\frac{\Big{(}\frac{l_{x}\cdot l_{y}}{N_{g}}\Big{)}\cdot N_{mg}}{l _{x}\cdot l_{y}}\right)\cdot 100\%, \tag{19}\]  
where \(l_{x}\) and \(l_{y}\) are the \(x\) and \(y\) dimension of the micro-structure, respectively. \(N_{g}\) is the number of grains and \(N_{mg}\) is the number of missing grains. Considering that the stress-strain field (\(\Omega\) and \(\Gamma\)) was obtained by simulations, the only unknown to be evaluated from the system of Equation (17) was the mechanical stiffness matrix (i.e., \(\hat{C}\) where, if needed, the Young's and Shear moduli can be evaluated).