# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 3 Numerical Simulations
### Uniaxial Tension Test  
Uniaxial tension tests were conducted on specimens with identical geometric and mechanical characteristics (Table 2), employing two different meshes, as illustrated in Figure 10. Symmetrical constraints were applied to the bottom edge of the specimen, as depicted in Figure 10, to ensure a symmetric response in terms of deformations in the lateral direction. The test involved applying an imposed displacement on the top edge of the specimen. The primary objective of this test was to examine the discrete model's capacity to accurately simulate local deformations in the elastic regime and post-peak softening behavior. Simulations are conducted using imposed displacement control. Figure 11 presents the complete macroscopic reaction-displacement curve, where the specimen initially behaves like a linear elastic continuum. Subsequently, cracks initiate, marking the transition from a continuous to a discontinuous state. The observed drop in the macroscopic curve and subsequent softening signify the progressive degradation of the specimen until it is entirely softened, losing its carrying capacity. \begin{table}
\begin{tabular}{c} \hline
**Uniaxial Tension and Compression Test** \\ \hline \(E=10\) GPa, \(v=0.2\) \\ \(\sigma_{u,c}=12\) MPa, \(\sigma_{u,t}=2\ MPa\), \(\tau_{u}=2.5\) MPa \\ \(G_{f,l}^{(c)}=200\) N/m, \(G_{f,l}^{(l)}=2\) N/m, \(G_{f,l1}=30\) N/m \\ Dimensions of the specimen \(10\times 10\times 1\) cm \\ \hline \end{tabular}
\end{table} 
Table 2: The mechanical characteristics and geometry of the specimen. Figure 11: Comparison of results for tension test for two different meshes. Figure 10: Uniaxial tension test meshes: (**a**) Mesh1-1210; (**b**) Mesh2-2570. Figure 12 illustrates displacement plots on the discrete lattice network of elements at the imposed displacement level of 0.001 cm. At this stage, the specimen remains in the linear elastic regime, and no cracking has occurred. These plots demonstrate that the proposed algorithm of fictitious forces ensures correct displacements and deformations of the specimen. The lateral displacements correspond to a Poisson ratio of 0.2, as specified in the input parameters of the simulation. The algorithm achieves accurate deformation of the discrete lattice network by computing fictitious forces based on the stress tensor, correctly determining the principal stress in the nodes aligned with the global \(y\)-axis. Uniform results in linear elastic regime deformations are attained for both lattice meshes when the input parameters are defined on the global level. Conversely, failure in the discrete lattice network initiates with the initiation of the first cracks, with the failure criteria and parameters defined on the local element level. The final state of the cracks at the end of the simulation is depicted in Figure 13, where the failed elements are highlighted in red. On a global scale, a dominant macro-crack propagates through the entire domain, extending perpendicular to the imposed displacement direction. However, the cracks leading to total failure exhibit slight variations in length and local directions between the two meshes, even with the use of identical fracture parameters. This discrepancy arises due to the inherent randomness in stiffness within each lattice element, impacting the failure regime. As the failure criteria are applied to each lattice beam element, this internal heterogeneity determines the weakest link, highlighting the effectiveness of the lattice model in terms of robust crack propagation and path determination. Thus, the proposed lattice model effectively simulates deformations related to a continuum response in the linear elastic regime, while the fracture behavior is governed by the internal heterogeneity within the network, rendering it robust in terms of crack propagation.