## 2 Overview of the LDEM Approach
Since the area under the curve was proportional to the energy accumulated  
Figure 1: LDEM discretization strategy: (**a**) Basic cubic module in the solid discretization, and (**b**) Constitutive law adopted for LDEM uni-axial elements. by the bar under strain, for a given point P, the area of the triangle OAP (striped sector) would be proportional to the energy dissipated by the damage, whereas the OCD (gray area) would be related to the elastic strain energy stored in the bar. If the external load was relaxed at point P, the return slope would be lower than the initial slope because the bar would then be partially damaged. When \(\varepsilon=\varepsilon_{r}\), the damage energy density would equal the fracture energy, and then the element would fail and lose its load-carrying capacity. Each element's equivalent fracture area \(A_{i}^{*}\) was chosen to ensure that the energy dissipated by continuum fracture matched its discrete representation. For this purpose, the fracture of a cubic sample of dimensions \(L_{n}\times L_{n}\times L_{n}\) was considered. The energy dissipated by a cubic LDEM module and its distribution in the various elements were examined by [47], with an alternative constitutive relation proposed in [48]. Under compression, as the material was assumed to remain linearly elastic with an initial slope \(EA\), failure would be induced by indirect tension. In a quasi-brittle material, tensile collapse stress is commonly reached at around \(1/10\) of its compressive strength. Therefore, the model could capture compression effects indirectly without considering compression-induced damage because tensile failures are the most common. Shear effects had to be considered in order to capture the post-failure behavior, as discussed extensively in [27]. Further discussion on the representation of compression effects on concrete using LDEM were also found in [49]. As described comprehensively by [50], there were two approaches for representing the material's random micro-structural features, either separately or simultaneously:  
* Introducing small disturbances throughout the mesh with the node coordinates \((x,y,z)\) defined by: \[(x_{n}+r_{x}L_{n},y_{n}+r_{y}L_{n},z_{n}+r_{z}L_{n})\] (5) where \(x_{n}\), \(y_{n}\), and \(z_{n}\) are the node coordinates for a perfect cubic array, whereas \(r_{x}\), \(r_{y}\), and \(r_{z}\) are normally distributed random numbers with zero mean and variation coefficient \(CV_{p}\). * Defining the material's specific fracture energy \(G_{f}\) as a random 3D-field, according to a Type-III (Weibull) distribution, where the mean and the variation coefficient would appear as input parameters. This option also considered a spatial correlation (\(l_{c}\)) for \(G_{f}\) when \(l_{c}>L_{n}\). In addition to the aforementioned approaches, the strains \(\varepsilon_{p}\) and \(\varepsilon_{r}\) (see Figure 1b) were related the material's characteristic length \(d_{\alpha q}\), as follows: \[\varepsilon_{p}=\sqrt{\frac{G_{f}}{E\,d_{\alpha q}}},\] (6) \[\varepsilon_{r}=\varepsilon_{p}d_{\alpha q}\bigg{(}\frac{A_{i}^{*}}{A_{i}} \bigg{)}\bigg{(}\frac{2}{L_{i}}\bigg{)}.\] (7)  
In Equation (7), \(A^{*}\) was each element's equivalent fracture area, defined to ensure that the energy dissipated by the fracture of the continuum was equal to that of its discrete counterpart. The sub-index indicated the type of element referenced (diagonal or normal). Finally, the characteristic length was especially relevant because it represented the minimum fissure dimension for an unstable fracture to start propagating. According to [51], this length also related to other significant parameters, such as fragility \(s\):  
\[s=\frac{\sqrt{\frac{G_{f}E}{d}}}{\sigma_{p}}, \tag{8}\]  
where \(d\) represents the characteristic length of the structure and \(\sigma_{p}\) is the failure stress.