# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 3 Numerical Simulations  
In this work, we conduct several numerical simulations to present the capabilities of the proposed discrete lattice element model with improved elastic response. While discrete lattice models are commonly applied in mesoscale simulations of failure with heterogeneous structures, our simulations involve specimens composed of a single material, designed to first exhibit elastic continuum behavior prior to the appearance of fractures. At each increment of the simulation, the fracture initiation criteria are examined for all lattice elements. Once the initial fractures start to propagate, the specimen undergoes a transition from a continuum material to a fractured state, while at the same time, the algorithm for fictitious forces ensuring continuum-like behavior in the lattice model stops. Quasi-static propagation of fractures is considered here, although fast dynamic simulation can be conducted by adding inertia terms to the discrete lattice model [14].  
The presented damage softening model is suitable for brittle and quasi-brittle failure of engineering materials, usually observed in composite materials and structures [14]. The ductile failure can alternatively be conducted with a plasticity-based softening model [13]. The distinction between these two models is in using internal thermodynamics variables. Although the focus here relies on quasi-brittle failure with a damage softening model, a plasticity-based model can be implemented with the proposed enhanced discrete lattice model.  
To illustrate the model performance, we initiate the simulation with a test where shear stresses dominate. In this example, only stress computations are carried out and compared with those from a standard continuum element model. Subsequently, monotonic uniaxial tension and compression tests are conducted until failure, where the model is providing the continuum-like behavior prior to the initiation of cracks. Post-fracture initiation, the model allows for the further propagation of tensile and shear cracks by activating the formulation of embedded strong discontinuities. The proposed model simulates monotonic load increase until failure, but it can also simulate unloading. Indeed, an unloading mechanism is present in a significant number of elements that initiated cracks but did not suffer significant damage. Moreover, it is also possible to build a model that can capture cyclic loading conditions on the basis of the proposed kinematics.  
The implementation of the model is carried out in both FEAP and MATLAB, with the MATFEAP interface facilitating the exchange of data between the two codes, as elaborated in the preceding section.  
In all simulations, the domain is discretized using the Delaunay algorithm, and this process is accomplished through the utilization of GMSH [44]. The discrete lattice elements are then extracted from the resulting mesh. Concurrently, the geometric properties and characteristics pertaining to the Voronoi network are also obtained from the discretized domain. In our work, we employ rectangular domains. The utilization of more complex geometries involving potential inclusions and multi-phase materials would necessitate an iterative