# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 2 Novel Discrete Lattice Element Model  
The discrete lattice element model proposed in this work is based on the foundation of various existing lattice models commonly employed for the representation of material failure in engineering, particularly in concrete and rocks. Discrete lattice models generally use one-dimensional lattice elements to transmit forces between material particles or distinct portions of the material (grains), which is efficient in simulations of crack and fracture growth. The lattice element models can be represented by truss bars, Euler-Bernoulli beams, or Timoshenko beams [1], or, alternatively, the models may use springs, giving rise to a rigid-body spring model [2]. The unstructured lattice configuration possesses structural randomness, resulting in efficiency in capturing cracks and fractures, which is very beneficial in simulations of failure at lower scales. However, their response at macroscale causes a non-uniform response in the elastic regime, which may significantly reduce their efficiency in capturing crack initiation, especially in case of macroscopic homogenized models.  
In the model presented here, we employ a discretization approach based on the Delaunay triangulation of the domain. The Delaunay triangulation is a dual form of Voronoi tessellation, where Voronoi cells represent material particles, such as grains or portions of the material. The Delaunay edges within the triangulation serve as lattice elements, symbolizing cohesive links between Voronoi cells (Figure 1). This structural arrangement enables the representation of cracks through the progressive failure of adjacent lattice elements, providing a comprehensive framework for simulating the complex failure mechanisms observed in concrete and rock materials [1]. In our previous studies, we effectively employed enhanced Timoshenko beams as lattice elements equipped with embedded strong discontinuities to simulate failure in concrete and rocks [13; 14; 32; 33; 34; 35; 36]. In the present study, instead of standard straight Timoshenko beam elements, we employ generalized beam lattice elements based on the kinematics of rigid-body spring models, ensuring the proper elastic displacements and deformations of the discrete model. Previous findings demonstrated the capacity of such discrete models to achieve a correct elastic response [8]. On the other hand, lattice models employing standard beam elements are unable to produce homogeneous strain and stress fields even when Poisson's ratio is equal to zero, although in certain cases an improved response can be achieved if the element parameters are extracted directly from the Voronoi diagram [10; 14]. In our generalized beam element model, we introduce the embedded strong discontinuity to ensure captureof the fracture behavior, leading to a new lattice model based on generalized beam elements with embedded discontinuities.  
Given that in our work the nodes are placed along the edges of the designated domain and Voronoi cells intersect the boundaries, we employ a mirroring technique to obtain complete cells. This allows us to extract the comprehensive geometrical characteristics of the elements.