# A Novel Method of Calibrating Micro-Scale Parameters of PFC Model and Experimental Validation
## 4 Experimental Design and Analysis
### Optimization of Micro-Scale Parameters  
According to the principle of the desirability function method, first, we need to determine the optimization interval for each factor. The purpose of this study is to find the micro-scale parameters to fit the macroscopic mechanical response. Therefore, Equation (8) was utilized to calculate the individual desirability, and then the overall desirability could be obtained according to Equation (9). Assuming these three factors are equally important, their weights were all set to 5 (\(r_{1}=r_{2}=r_{3}=5\)). The optimization intervals of \(A\), \(B\), and \(C\) were defined as (50,150), (20,60) and (1,3), respectively, and their target values were set to be equal to those of the uniaxial compression test results, i.e., \(y_{1}=102.61\) MPa, \(y_{2}=20.78\) GPa, and \(y_{3}=0.258\). The effect of each factor on the overall desirability is plotted in Figure 7. It is shown that the desirability is the maximum when \(A\) is 75-85 MPa, \(B\) is 20-23 GPa, and \(C\) is 1.6\(\sim\)2.7. In these intervals, the desirability increases first and then decrease with the increasing \(A\) and \(C\), while it declines as \(B\) increases. Moreover, Figure 8 illustrates that the interactions between these factors have a significant effect on the overall desirability. It can be observed clearly that the overall desirability is the largest under the condition of \(A\) is about 82 MPa, \(B\) is about 20 GPa, and \(C\) is about 2.2.  
Figure 6: Influence of micro-scale parameters on Poisson’s ratio: (**a**) one-factor effect and (**b**) interaction effect between \(B\) and \(C\).  
Through optimization, we found that the overall desirability of the solution that \(A=82.07\) MPa, \(B=20.00\) GPa, and \(C=2.198\) is the maximum, namely, \(D=0.914\). In such a case, the simulated uniaxial compressive strength, Young's modulus, and Poisson's ratio are 102.61 MPa, 21.77 GPa, and 0.258, respectively, which are exceedingly agreeable with the laboratory results.  
In addition to the simulated responses being consistent with laboratory test results, the failure mode of the model should also coincide with that of the rock specimen. Thus, minor adjustments are required on these parameters and the standard deviations of normal and shear strength (\(\overline{\sigma}_{c}\) and \(\overline{\tau}_{c}\)). The final micro-scale parameters obtained are shown in Table 6, and the comparison between experimental and numerical stress-strain curves is presented in Figure 9. The peak stresses and slopes of the two curves are basically equal, with relative errors of 1.19% and 5.92%, respectively. This indicates that the uniaxial compressive strength and Young's modulus are consistent. Since the built models compressed by two discs are strictly two dimensional, there is no initial pores and cracks compaction stage during the loading process. Consequently, the peak stain of the model is slightly smaller than that of the real rock specimen.  
Figure 8: Interaction effect of multiple factors on overall desirability: (**a**) A–B, (**b**) A–C, and (**c**) B–C.  
Figure 7: Effect of one factor on overall desirability: (**a**) factor A, (**b**) factor B, and (**c**) factor C.