# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 1 Introduction
Namely, the representation of failure with softening in traditional methods is prone to mesh dependency. Recognizing this limitation, various advanced methods have emerged to correct softening behavior. Among these, there are non-local damage and plasticity models [20; 21; 22; 23], various localization limiters like those listed in [18], and, more recently, the very popular phase field models [24]. The extended finite element method (X-FEM) is another type of approach to address this issue, in which discontinuous functions are introduced in the formulation, while treating additional unknowns associated with fractures on a global scale [25; 26]. Alternatively, the embedded discontinuity finite element method (ED-FEM) addresses the additional discontinuity-related unknowns locally at the element level [27; 28; 29; 30; 31]. Within our lattice element method, we adopt a solution for softening regularization by using the embedded strong discontinuity framework proposed in [17]. This involves the implementation of embedded strong discontinuities within the lattice element models, as presented in [32; 33; 34; 35; 36]. Here, the material parameter governing softening is the fracture energy, which enables the mesh-independent softening response. It is worth noting that alternative models utilizing the granular micromechanics damage approach with finite deformation also produce proper modelling of the localization zone [37; 38; 39], which is similar to non-local damage models. The models based on lattice structure can also be used in large-motion dynamics [40] and buckling problems [41]. The proposed novel model integrates the embedded strong discontinuity framework and previously mentioned correction of elastic response within the lattice element model aiming to fully capture the material failure. To that end, the proposed model is implemented in an incremental and iterative fashion, achieving both local and global convergence. The iteration on the local level involves internal model variables and is performed using the Newton iterative method. Global convergence is attained through a Newton iterative solver as well. The mechanical model is implemented in the FEAP (Finite Element Analysis Program) [42]. The stress tensor computations and the computation of fictitious forces are carried out in MATLAB at the end of each increment after extracting the stress resultantsfrom the FEAP model. The exchange of data is facilitated by MATFEAP [43], a framework extension for FEAP computations. To show the capabilities of the proposed model, we conduct several numerical simulations. Initially, a linear elastic analysis is performed to establish the stress tensor in a shear-dominated example. Subsequently, more computations involving failure under the influence of both tensile and compressive load cases, where the material failure is followed by a global softening response, are presented. In Section 2, we present the formulations of the proposed model. Numerical simulations are provided in Section 3, while the Discussion and Conclusion are given in Sections 4 and 5, respectively.