## 2 The Lattice Discrete Element Method (LDEM)  
This work's numerical analysis uses the Lattice Discrete Element Method (LDEM), a version of the Discrete Element Method where solid bodies are modeled as nodal lumped masses interconnected by massless uniaxial elements, where such elements are only able to carry axial loads. Each node has three degrees of freedom, corresponding to the displacements in the three orthogonal coordinate directions. [PERSON] and [PERSON] [39] determined the conditions for representing an orthotropic elastic continuum medium through this model, leading to a cubic cell with nine nodes (Figure 1a) that was adopted by [PERSON] [40] to solve dynamic structural problems. For this geometric arrangement used herein, the lengths of longitudinal and diagonal elements are \(L_{\pi}=L\) and \(L_{d}=\sqrt{3L/2}\), respectively, with equivalent stiffnesses given by  
\[EA_{n} = E\phi L^{2},\] \[EA_{d} = \frac{A_{n}2\sqrt{3}}{3}, \tag{1}\]  
where [PERSON]'s modulus is \(E\), \(L\) is the length of longitudinal elements, and \(\phi=(9+8\delta)/(18+24\delta)\) and \(\delta=9\
u/(4-8\
u)\) are coefficients that relate those parameters to their corresponding linearly elastic solid properties, with \(\
u\) representing [PERSON]'s coefficient. The LDEM model used herein is completely equivalent to an isotropic elastic solid when \(\
u=0.25\), as shown in the extensive examination by [41]. Assuming the internal material damping as a linear function of the nodal masses' velocities, the solid's spatial discretization yields \(N\) equations of motion in the well-known form  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{F}_{r}-\mathbf{P}=\mathbf{0}, \tag{2}\]  
in which \(\mathbf{u}\) represents the vector of generalized nodal displacements, where an upper dot denotes a temporal derivative; \(\mathbf{M}\) is the diagonal mass matrix; \(\mathbf{C}\) is the the damping matrix, assumed diagonal to generate an uncoupled system; \(\mathbf{F}_{r}\) comprises the internal forces acting on the nodal masses; and \(\mathbf{P}\) is the vector of external forces. The fracture analysis performed here adopts the softening law for quasi-fragile materials proposed by [42], which assumes the force-strain relation presented in Figure 1b for elements subjected to tension. In such a relation, the area under the force vs. strain curve (triangle OPB) is proportional to the energy density necessary to fracture the element's area of influence, which allows one to account for the irreversible effects of crack nucleation and propagation throughout the modeled structure. Once the damage energy density reaches fracture level, the element fails and loses its load-carrying capacity. The material is assumed to be linearly elastic under compression, inducing failure to occur through indirect tension in such circumstances. The constitutive parameters and symbols appearing in the figure are defined as follows. The element's axial force \(F\) depends on the axial strain \(\varepsilon\), whereas \(A_{i}^{*}\) is the element's equivalent fracture area, defined to ensure that the energy dissipated by a fracture in the discrete arrangement matches that of the continuum medium it emulates. For a continuous cubic sample of dimensions \(L\times L\times L\), the fracture energy due to a crack parallel to one of its faces is \(\Gamma=G_{f}\Lambda=G_{f}L^{2}\), in which \(\Lambda\) is the actual fractured area, i.e., \(L^{2}\). In the corresponding LDEM module with the same dimensions, fracture occurs in two parts, consisting of the contributions from five longitudinal elemen  
Figure 1: View of basic LDEM model: (**a**) the basic cubic module within a LDEM array; (**b**) classical uniaxial constitutive law used by the LDEM elements. the module edges and one internal element) and four diagonal ones.