# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 5 Discussion  
While the LBM is capable of simulating the fracture process of cementitious materials under various loading and boundary conditions, relatively brittle post-peak behavior exists. Several potential approaches for overcoming the post-peak brittleness have been proposed in the past [3]: (1) Considering the material mesostructured explicitly in the simulations, as shown in Figure 14a; (2) using a multi-linear constitutive relationship (instead of the typical elastic-brittle one), essentially considering material softening at the elemental level, as shown in Figure 14b; (3) changing the removal mechanism of \"failed\" lattice beams (i.e., not removing the element altogether in a single step, but first removing its bending stiffness, essentially making it a truss element, and then the axial stiffness), as shown in Figure 14c. Herein, these potential methods quantitively reflect the influence on post-peak behavior through numerical analyses and the detailed input parameters can be found in Table 9. The model No.1 is regarded as the reference model; in the model No.2, one element is removed by two steps as explained before. Accurately, in the first step, the critical element degrading from a Timoshenko beam element into a truss element loses the capacity to bear the torque and bending; in the second step, the truss element is thoroughly removed and a crack path is recorded. The second step can simulate the friction  
Figure 14: (**a**) Mesostructured model, (**b**) multi-linear constitutive relationship, (**c**) schematic view of the element degradation at the first step [53]. Reproduced from [53]. Copyright 2019, Elsevier. of crack faces in a way; the model No.3 investigates the effect of a multi-linear constitutive relation. To be specific, a multi-linear curve (in black) with five segments is utilized standing for the constitutive relation and capturing its characteristics. The points are taken at: (1) Peak load; (2) point for which stress is 70% peak load; (3) point for which strain is 0.5%; (4) point which the strain reaches 1.25%; and (5) point for which the stress is 10% peak load. After the last point, the element is removed from the system, representative for the crack in the system; model No.4 includes the mesostructured procedure, in which coarse aggregate and interface elements are introduced for a realistic sample. The corresponding material properties for different phases are listed in Table 9. The high friction boundary condition is applied to all simulations. Figure 15 showcases the influence of material heterogeneity, multi-linear constitutive relationship, and removal mechanism, clearly showing that their influence can contribute to the ductility and completely reproduce the load-displacement curve of the experiments [3; 54; 55; 56]. However, since these factors significantly complicate the calibration process, the brittle-linear mechanical properties and homogeneous material are utilized during the calibration and validation process. For completely reproducing the load-displacement curves, an experimentally validated multi-scale modeling scheme has recently been proposed by [PERSON] [57]. In his work, the constitutive relation of lattice elements is derived from lower-scale (i.e., micro-scale) simulations or measurements. Based on the experimental measurements of micro-scale sized specimens, micromechanical models for both the bulk cement paste and the interfacial transition zone have been calibrated and validated under various loading conditions [39; 44; 53; 58]. This can offer reliable constitutive relations of local elements for mesoscale models [40; 59]. Together with the current work, it is possible to extend the multi-scale modeling scheme. Figure 15: Normalized stress and normalized strain curve with different influence factors.