# Lattice Fracture Model for Concrete Fracture Revisited: Calibration and Validation
## 6 Conclusions  
In this paper, a systematic calibration procedure that relies on a combination of two uniaxial tensile tests is proposed to determine input parameters of lattice elements--tensile strength, elastic modulus, and \(\alpha_{\text{M}}\). The procedure is then validated by simulating concrete fracture under different complex loading and boundary conditions: Uniaxial compression, three-point bending, tensile splitting, and double-edge-notch (DEN) beam shear. Simulation results are compared to experimental findings in all cases and show the LBM capacity to simulate the fracture process of cementitious materials under various loading and boundary conditions. While the post-peak behavior is brittle due to the simplicity of the model used, potential solutions have been provided for a complete load and displacement curve. Based on this calibration and validation work, parametric analyses including the size effect, slenderness, notch ratio, and boundary condition are performed to give further explanations on the fracture mechanisms in cementitious materials.  
Based on the results obtained in this study, the following conclusions can be formulated.  
* The calibration method with a combination of two uniaxial tensile tests is utilized to get the value of the bending influence factor (\(\alpha_{\text{M}}=0.05\)), which can be used to simulate the fracture process of cementitious materials under different boundary conditions.
* The LBM can simulate the fracture process of concrete under uniaxial compression and tension. The model captures the effect of high friction boundary and low friction boundary conditions on compressive strength well. Also, the LBM can reproduce the differences in crack patterns and the load-displacement curve for different boundary conditions and slenderness.
* The LBM can reproduce the fracture process accurately and analyze the size effect on the load-displacement curve for concrete subjected to a three-point bending test. Furthermore, the LBM is also able to capture the influence of the notch size and the size effect on the load-displacement response.
* The LBM can simulate the fracture process in the Brazilian splitting test. Furthermore, it predicts the size effect on the splitting test and produces the same crack patterns observed in the experiments.
* The LBM can simulate the fracture process during a DEN beam shear experiment. It does not only predict the crack nucleation and propagation accurately but also shows the influence of different boundary conditions.  
At present, the LBM has been extensively developed as an efficient numerical model to simulate the fracture process of cementitious materials. Based on this summary, systematic calibration and validation as guides for the determination of input parameters is proposed for future research. Also, the multiscale research suggestion provides a promising choice for further research about mechanical properties and fracture processes for cementitious materials.  
**Author Contributions:** Formal analysis, Z.C.; investigation, Z.C.; methodology, Z.C. and H.Z.; supervision, E.S. and B.S.; validation, Z.C. and H.Z.; writing--original draft, Z.C.; writing--review and editing, H.Z.; E.S., and B.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** [PERSON] and [PERSON] would like to acknowledge the funding supported by China Scholarship Council under grant number 201806060129 and 201506120067, respectively.  
**Acknowledgments:** The authors gratefully thank [PERSON] for her linguistic assistance during the revision of this manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.