## 1 Introduction
This method was used by [PERSON] [7] to solve structural dynamics problems in an orthotropic continuum, with stiffness coefficients derived for a linearly elastic solid by [PERSON] and [PERSON] [37], and by using the explicit direct numerical integration of the equations of motion. The LDEM was also employed to represent the behavior of quasi-brittle specimens in [10; 38], showing a good correlation with numerical methods and experimental results, not only in terms of final configuration and global parameters, but also in the simulation of acoustic emission. The method consists of representing a solid by an arrangement of masses with three degrees of freedom, interconnected by massless elements carrying only axial loads. Each nodal mass's three degrees of freedom correspond to the nodal displacements in the coordinate directions. For the adopted cubic array presented in Figure 2(a), the lengths of longitudinal and diagonal elements are \(L_{n}\) and \(L_{d}=\left(\sqrt{3}/2\right)L\), respectively. The effective areas of normal and diagonal elements, for an isotropic medium, are given by \(A_{n}=\alpha{L_{n}}^{2}\) and \(A_{d}=\left(2/\sqrt{3}\right)\delta\alpha{\left(L_{n}\right)}^{2}\) where \(\delta=9\
u/\left(4-8\
u\right)\) and \(\alpha=\left(9+8\delta\right)/\left(18-12\delta\right)\). The initial axial stiffness (Figure 2(b)) of normal and diagonal elements are \(EA_{n}/L_{n}\) and \(EA_{d}/L_{d}\) respectively, where \(E\) is the Young's modulus of the material. The cubic array adopted herein results in the exact representation of an isotropic continuum for \(\
u=0.25\). On the other hand, shear differences arise for \(\
u\
eq 0.25\), as detailed in [39]. Such differences are very small and can be neglected if \(0.20\leq\
u\leq 0.30\). A system of equations of motion is obtained by applying [PERSON]'s second law to each node. The material's internal damping is assumed to depend on nodal masses and corresponding velocities. The equations of motion are thus uncoupled and may be integrated in the time domain using an explicit finite difference scheme. The convergence of the solution using LDEM for both linear elastic problems and elastic stability problems was verified by different authors, as reported in [40]. The uniaxial constitutive law adopted for the LDEM bars is based on [PERSON]'s [41] proposal for quasi-brittle materials, which adopts a triangular-element constitutive relationship, Figure 2(b). Note that failure in compression occurs by indirect tension. The element axial force \(F\) depends on the axial strain \(\varepsilon\). An equivalent fracture area \(A_{i}^{*}\) of the \(i\)-th element is defined to ensure that the energies dissipated by fracture both in the continuum and in its discrete representation are equivalent, considering a cubic sample of dimensions \(L\times L\times L\)[10]. A basic parameter of the bilinear constitutive relationship shown in Figure 2(b) is given by:  
\[\varepsilon_{p}=\sqrt{\frac{G_{f}}{d_{eq}E}} \tag{5}\]  
Figure 3: LDEM: (**a**) Basic cubic module employed in the discretization and (**b**) bilinear constitutive law attributed to the elements. where \(\varepsilon_{p}\) denotes the strain at the peak of the bilinear law, \(G_{f}\) is the fracture energy and \(d_{eq}\) is a material characteristic length, also considered a material property, discussed in detail by [PERSON] [42]. The \(i\)-th element's ultimate strain \(\varepsilon_{r}\), at which an element subjected to tension has no residual carrying capacity, is defined by:  
Figure 2.0  
It is observed that \(\varepsilon_{r}\) depends both on the material properties and the discretization level. More details about the LDEM formulation employed here and how to calibrate it for quasi-brittle structures appear in previous contributions by the authors, such as [43]. The non-homogeneous nature of engineering materials such as concrete is accounted for in the LDEM using two approaches.