# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 1 Introduction  
The study of fracture propagation plays an important role in engineering and materials science, providing insights into the structural integrity of materials and the behavior of complex systems under various conditions. In this context, discrete models have proved to be powerful tools for accurate representation of internal processes associated with fracture propagation [1]. These models offer the capability not only to simulate macroscale fractures and cracks but also to capture lower-scale crack propagation, including mesoscale and microscale phenomena. Among the diverse range of discrete models, classical particle discrete models such as the discrete element method [2], discrete lattice element method [1], and finite discrete element method [3] have proven particularly effective in various fields of engineering failure analysis of concrete, rocks, and other materials [4; 5; 6; 7]. While discrete models properly capture the discontinuous nature of fracture propagation problems, they have limitations in accurately capturing the global and/or local elastic response of materials and structures [8; 9; 10; 11; 12]. Namely, discrete element models commonly employ one-dimensional elements such as bars, beams, or springs to simulate particle interactions, restricting their ability to capture a wide range of Poisson ratios. This leads to a mismatch between local and global elastic constants, where the elastic input parameters of the models fail to align with the overall global elastic response of the material. Consequently, these discrepancies can lead the models to generally fail in the prediction of material behavior. In this paper, we focus on improving the existing discrete lattice elementmodels based on Timoshenko beams with embedded strong discontinuities [13; 14] for the objective simulation of fracture processes by enhancing the representation of the elastic response. More precisely, our focus is on bridging the gap between local and global elastic constants, while still allowing the full failure representation of the material. In order to achieve an improved elastic response and accurate representation of the fracture propagation mechanisms, we adopt the kinematics typical of rigid-body spring models [10; 11]. More precisely, such connections are considered here as generalized beam lattice elements, closely related to Timoshenko beam elements, which enable the application of a procedure for correcting the elastic response of undamaged material, as in [8; 9; 15; 16]. Such a generalized beam lattice element is equipped with embedded strong discontinuities [17; 18] for the propagation of fractures in the domain. In the proposed approach, the stress tensor at the lattice element network nodes is computed by considering all stress resultants from the elements connected to a specific node. Subsequently, fictitious forces [8; 9; 19] are computed by extracting information from the transverse strains that need to be achieved. By introducing these fictitious forces in the lattice elements of the model, we ensure that the local element stiffness aligns with the global material properties, correcting the elastic response. Moreover, a wide range of Poisson ratios is possible to be achieved with this approach, which was not the case in classical discrete beam lattice models. Following this correct computation of the elastic response, fracture criteria pertaining to the maximum tensile strength and maximum shear strength are systematically checked in all elements [1]. The procedure of fictitious forces and correction of the elastic response is terminated once the first elements exhibit failure, signifying the material's progression towards a fully discontinuous state and eventual total failure. This approach aims to provide a more realistic representation of the elastic response in discrete lattice element models, while correctly capturing the complexities of fracture propagation. The softening response represents material failure where classical finite element procedures struggle.