## 4 Fractional Nonlinear Damage Creep Model
### Determination of Creep Parameters and Verification of Model  
In MATLAB, establish the constitutive equation described in this text and set the initial values for the model parameters. Create a function to calculate the difference between predicted values and creep data obtained from carbonaceous slate with various foliation angles. Utilize MATLAB's optimization tools to adjust the parameters in order to minimize this difference. The flow chart of parameters identification method is illustrated in Figure 16. The final parameter fitting results are presented in Table 3.  
Additionally, plot a comparison between the fitted curve (in red) and the actual data (in black hexagons), as illustrated in Figure 17. The FNC model not only captures the deformation characteristics of carbonaceous slate during the steady-state creep stage but also accurately describes the features observed during the accelerated creep stage. Across different foliation angles, creep stages, and loading levels, the calculated curves closely align with the experimental data (\(R^{2}>0.95\)), demonstrating the model's accuracy and validity. Figure 16: Diagram of parameter identification method and model verification. \begin{table}
\begin{tabular}{|c c c c c c c c|} \hline \(\beta\)/\({}^{\circ}\) & \(E_{1}\)/GPa & \(E_{2}\)/GPa & \(\eta_{1}\)/GPa h & \(\eta_{2}\)/GPa h & \(\eta_{n1}\)/GPa h & \(n\) & \(R^{2}\) \\ \hline
0 & 30.26 & 11.55 & 3.20 & 99.72 & 65.13 & 0.55 & \(9.78\times 10^{-1}\) \\
15 & 15.16 & 2.6 & 1.59 & 31.63 & 39.46 & 0.37 & \(9.76\times 10^{-1}\) \\
45 & 11.27 & 13.05 & 0.59 & 19.67 & 24.15 & 0.29 & \(9.66\times 10^{-1}\) \\
75 & 13.81 & 50.43 & 7.61 & 12.32 & 27.54 & 0.48 & \(9.79\times 10^{-1}\) \\
90 & 11.75 & 9.59 & 8.46 & 100.17 & 49.80 & 0.71 & \(9.81\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Fitted parameters of the fractional nonlinear creep model. To further validate the FNC model, it is compared with the classic Nishihara model (Equation (28)). Using the same method, the fitting data are summarized in Table 4. The comparison results between the traditional Nishihara model and FNC model are illustrated in Figure 18.  
\[ \begin{cases}\varepsilon(t)=\frac{\sigma}{\varepsilon_{a}}+\frac{\sigma}{E_{b}} \Big{(}1-e^{-(E_{b}/\eta_{b})t}\Big{)}&\sigma\leq\sigma_{s}\\ \varepsilon(t)=\frac{\sigma}{E_{a}}+\frac{\sigma}{E_{b}}\Big{(}1-e^{-(E_{b}/ \eta_{b})t}\Big{)}+\frac{\sigma-\sigma_{s}}{\eta_{a}}&\sigma>\sigma_{s}\end{cases}  \tag{28}\]  
Figure 17: Comparison between experimental data and calculation curve. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\beta^{c}\) & \(E_{a}\)/GPa & \(E_{b}\)/GPa & \(\eta_{a}\)/GPa h & \(\eta_{b}\)/GPa h & \(R^{2}\) \\ \hline
0 & 13.26 & 11.55 & 3.20 & 120.42 & \(9.11\times 10^{-1}\) \\
15 & 13.16 & 2.6 & 1.59 & 114.93 & \(9.03\times 10^{-1}\) \\
45 & 14.27 & 13.05 & 0.59 & 26.69 & \(8.34\times 10^{-1}\) \\
75 & 15.81 & 50.43 & 7.61 & 346.17 & \(8.89\times 10^{-1}\) \\
90 & 29.75 & 9.59 & 8.46 & 465.90 & \(8.72\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Fitted parameters of the Nishihara model. Figure 18: Comparison diagram between fractional nonlinear constitutive model and Nishihara model. The \(R^{2}\) of the FNC model is larger than 0.965, which is higher than that of the Nishihara model (\(R^{2}\leq 0.911\)). As can be seen from Figure 18b, the FNC model can well describe the characteristics of the decelerating, steady-state, and accelerating creep stages at any stress level. From Figure 18a,d, it is also noted that the FNC model can effectively cope with the impact of the sudden increase in the strain-time curve. When \(\beta<45^{\circ}\), carbonaceous slate is prone to shear slip along the foliation plane under high stress levels. Therefore, the Nishihara model can only describe the creep process under low stress. When \(\beta>45^{\circ}\), however, the foliation planes are compacted due to the high stress and is not prone to slip. Hence, the Nishihara model can better describe the creep process under high stress.