# A Variable-Order Dynamic Constitutive Model for Clay Based on the Fractional Calculus
## 6 Model Validation and Sensitivity Analysis
### Sensitivity Analysis  
For completeness, a sensitivity analysis should be performed in order to estimate the influence of the parametric error, Equations (46) to (48) show that the accumulative strain of clay depends on the parameters \(E_{0}\), \(\alpha_{1}\), \(\alpha_{2}\), \(\beta\), \(\eta\), \(\beta\) and \(\eta\). To more accurately understand the effects of these parameters on the results of each stage and to check the robustness of the constitutive model, a sensitivity analysis was carried out on the main parameters of \(\alpha_{1}\), \(\alpha_{2}\), \(\beta\) and \(\eta\). The results are depicted in Figure 14, which shows the distribution of accumulative strain in different three-dimensional spaces and the respective influences of \(\alpha_{1}\), \(\alpha_{2}\), \(\beta\) and \(\eta\) on accumulative strain. Taking the experimental data with 50 kPa amplitude as an example, when one of the parameters of \(\alpha_{1}\), \(\alpha_{2}\), \(\beta\), \(\eta\) is adjusted, other parameters in the model remain the same. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\sigma_{d}\)/kPa. & \(E_{0}\)/kPa & \(\alpha_{1}\) & \(\eta_{1}\) & \(\alpha_{2}\) & \(\eta_{2}\) & \(\beta\) & \(\eta\) & \(t_{1}\)/s & \(t_{2}\)/s & \(R^{2}\) \\ \hline \multicolumn{1}{c}{} & \multicolumn{3}{c}{**Initial Stage**} & \multicolumn{3}{c}{**Steady Stage**} & \multicolumn{3}{c}{**Accelerating Stage**} & \\ \hline \multirow{3}{*}{35} & 1642 & 0.418 & 10800 & & & & 225 & & 0.9497 \\  & & & 0.087 & 1013 & & & 225 & & 0.9006 \\  & 20950 & 0.354 & 693 & & & & 210 & & 0.9926 \\  & 184 & 0.706 & 1931 & & & & 210 & & 0.9812 \\  & & & 0.254 & 106 & & & 210 & 640 & 0.9916 \\  & & & & 0.022 & 42 & 210 & 640 & 0.9325 \\  & 28050 & 0.620 & 519 & & & 0.0022 & 42 & 210 & 640 & 0.9926 \\  & 70 & & & & & 160 & & 0.9957 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters determined via fitting analysis based on the dynamic triaxial test of clay. Figure 14: Sensitivity analysis of parameters \(\alpha_{1},\alpha_{2},\beta\) and \(\eta\) of the constitutive model. Figure 14a,b show the accumulative strain distributions under the three-dimensional space \(t-\alpha_{1}-\varepsilon_{d}\), and the effect of parameter increment \(\Delta\alpha_{1}\) on accumulative strain, respectively. The accumulative strain-time curves under parameter increment \(\Delta\alpha_{1}=0.01\) almost coincide in the initial stage. In addition, significant change was no longer observed in dynamic strain once the parameter \(\alpha_{1}\leq 0.4\) (Figure 14a). Therefore, the influence of parameter \(\alpha_{1}\) on the accumulative strain is related not only to the \(\Delta\alpha_{1}\), but also to the initial value \(\alpha_{1}\). On the whole, it was found that there is a certain difference in accumulative strain curves under the parameter increment \(\Delta\alpha_{1}=0.01\), but not an obvious difference. Therefore, the parameter \(\alpha_{1}\) is considered to exhibit low sensitivity in the constitutive model. Figure 14c,d show accumulative strain distributions under the three-dimensional space \(t-\alpha_{2}-\varepsilon_{d}\), and the influence of the parameter increment \(\Delta\alpha_{2}=0.01\) on the accumulative strain, respectively. It is worth noting that the accumulative strain increases as the parameter \(\alpha_{2}\) increases, and the difference in accumulative strain under the parameter increment \(\Delta\alpha_{2}=0.01\) is more distinct compared to \(\Delta\alpha_{1}=0.01\). The reason for this phenomenon is that the magnitude of the cyclic loading duration becomes larger. Meanwhile, it can be understood from Figure 14c that the accumulative strain calculated by the constitutive model exhibits a lower gradual effect when the order is 0\(<\alpha_{2}\leq 0.2\). However, when \(\alpha_{2}>0.2\), the increase in accumulative strain is more obvious with increasing values of the parameter \(\alpha_{2}\).