# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 2 Modeling  
In the LBM used herein, the continuum is first discretized by a network of Timoshenko beam elements. Timoshenko beam was utilized to account for shear deformation, considering the ratio between length and cross-section of the beam element to be low. Specifically, the domain was divided into several cubic cells. Then, a sub-cell was defined within each cell and the lattice nodes were randomly placed in each sub-cell using a random number generator. A parameter, defined as the ratio between the length of sub-cell and cell, was used to represent the randomness of the mesh. In order to avoid large variations in element length and introduce geometry disorder of material texture,randomness of 0.2 was adopted in the following analyses in this paper. The final lattice network was generated through Delaunay triangulation, see Figure 1.  
Then, a series of linear elastic calculations were conducted for the stress distribution among lattice elements under prescribed loading and boundary conditions. Normal force and bending moment were taken into account for the element stress calculation.  
\[\sigma=\alpha_{\text{N}}\frac{F}{A}+\alpha_{\text{M}}\frac{\left(\left|M_{i} \right|,\left|M_{j}\right|\right)_{\text{max}}}{W} \tag{1}\]  
\[W=\pi D^{3}/32 \tag{2}\]  
where \(F\) is the normal force for the lattice beam element, \(A\) is the element cross-sectional area, and \(W\) is the cross-sectional factor for bending resistance, as indicated in Equation (2) (\(D\) is the diameter of the lattice element). The coefficients \(\alpha_{\text{N}}\) and \(\alpha_{\text{M}}\) are the normal force factor and the bending influence factor, determining the final failure mode in which either normal force or bending moment play a dominant role.  
The \(\alpha_{\text{N}}\) is commonly adopted as 1.0 [21; 39; 40]. However, publications on bending influence factor determination are scarce. Several researchers use the aforementioned fracture criterion but the \(\alpha_{\text{M}}\) value varies from the material type and experimental conditions in their publications [19; 32; 38; 39; 41]. Because of the inconsistency of \(\alpha_{\text{M}}\), the value was determined through a systematic calibration approach in this work.  
In LBM, for each step of the analysis, loading was increased until exactly one element reached a stress equivalent to the assigned compressive/tensile strength. This element was defined as the critical element and removed from the system. The removed element determined crack initiation and propagation as well as load-displacement response. Then, the lattice network was updated and relaxed and the loading procedure was repeated until the system reached the pre-determined failure criterion (e.g., load or displacement).  
In terms of computational effort, parallel computing technique, operating on multicore computer or clusters was integrated into the LBM, which significantly decreased computation efforts, making its application to model large-scale structures and/or when a considerable number of elements is needed possible.  
More details on the lattice fracture model are available in the literature [41]. Building upon this literature, the 3D lattice fracture model was revisited here and relevant fracture analyses can then be implemented by using the input parameters proposed in this study.  
Figure 1: (**a**) Schematic view of LBM (lattice beam network model) generation; (**b**) node and mesh generation procedure using Delaunay triangulation. Reproduced from [20]. Copyright 2018, Elsevier.