# A Novel Method of Calibrating Micro-Scale Parameters of PFC Model and Experimental Validation
## 6 Conclusions  
In this study, a coupling optimization method incorporating Box-Behnken response surface theory and desirability function optimization approach was proposed to quickly find the optimal micro-scale parameters of PFC model. After that, the calibrated model was verified by comparing the brittle fracture behavior of rock with an opening under axial loading obtained by numerical and experimental methods. Based on the results of the research, a couple of points can be concluded as follows:  
Figure 13: Crack development around openings of different shapes under uniaxial loading obtained by experimental method: (**a**) sample with a circular opening, (**b**) sample with a square opening, (**c**) sample with a D-shaped opening, and (**d**) rock failure pattern after testing.  
1. The macroscopic mechanical properties of the parallel-bond model, including the uniaxial compressive strength, Young's modulus, and Poisson's ratio, are mainly controlled by three micro-scale parameters: \(\sigma_{c}\), \(E_{\rm c}\), and \(k_{\rm n}\)/\(k_{\rm s}\).
2. BB-DFO experimental design method can not only optimize multiple response variables simultaneously but also fit the nonlinear relationships between the influencing factors and the response variables. Analysis of variance indicates that the uniaxial compressive strength is extremely significantly affected by \(\sigma_{c}\) and \(k_{\rm n}\)/\(k_{\rm s}\), and the interactions between \(\sigma_{c}\) and \(k_{\rm n}\)/\(k_{\rm s}\) and between \(E_{\rm c}\) and \(k_{\rm n}\)/\(k_{\rm s}\) also have a significant influence on it. For the Young's modulus, the effects of \(E_{\rm c}\) and \(k_{\rm n}\)/\(k_{\rm s}\) as well as their interaction on it are extremely significant. In regard to the Poisson's ratio, the impacts of \(E_{\rm c}\) and \(k_{\rm n}\)/\(k_{\rm s}\) upon it are significant and extremely significant, respectively, but the effect of the interaction is not significant.
3. The optimal micro-parameters of PFC model are \(\sigma_{c}\) = 82.07 MPa, \(E_{\rm c}\) = 20.00, and \(k_{\rm n}\)/\(k_{\rm s}\) = 2.198, and the overall desirability reaches 0.914. Correspondingly, the matched uniaxial compressive strength, Young's modulus, and Poisson's ratio are 102.61 MPa, 21.77 GPa, and 0.258, respectively, which are basically consistent with their experimental values (102.61 MPa, 20.78 GPa, and 0.258). Besides, the crack developments around the openings in rocks under uniaxial compression obtained by the calibrated model demonstrate good agreement with those from experimental investigation. This further proves that the micro-parameters optimized by the BB-DFO method are feasible.  
This research was jointly performed by [PERSON], B.D., G.Z., Y.C., and Y.T. Conceptualization, H.W. and B.D.; methodology, B.D.; software, Y.C. and Y. T.; validation, G.Z. and Y.T.; formal analysis, H.W.; investigation, B.D.; resources, G.Z.; data curation, Y.T and Y.C.; writing--original draft preparation, H.W.; writing--review and editing, H.W., B.D. G.Z., Y.C., and Y.T.; funding acquisition, B.D. and G.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (51804163 and 51774321), the China Postdoctoral Science Foundation (2018M642678), the Hunan Province Engineering Research Center of Radioactive Control Technology in Uranium Mining and Metallurgy, and Hunan Province Engineering Technology Research Center of Uranium Tailings Treatment Technology (2019 YKZX2003).  
The authors declare no conflict of interest.