## 5 Verification and Discussion
### Discussion about the Consolidation Behavior
Figure 7c shows that \(U_{\text{P}}\)-curves under different \(T_{\lambda 1}\) intersect each other with the increase in time factor \(T_{\text{v}}\). Higher \(T_{\lambda 1}\) (i.e., the viscosity coefficient of Kelvin-Voigt body is larger) delivers faster dissipation of excess pore-water pressure in the early stage, but more slowly one at a later stage. For the case of sinusoidal loading, Equation (45) for average degree of consolidation also has seven governing parameters. All of them are the same as those for the case of ramp loading except the loading parameter \(T_{\text{T}}\)(=\(c_{\text{v}}T/H^{2}\)), which reflects the influence of the period (\(T\)) of sinusoidal loading on consolidation behavior. By changing the values of these parameters individually while holding the others constant, the consolidation behavior of fractional viscoelastic soils under sinusoidal loading is investigated in the following discussion. Figures 8-10 illustrate the changes of average degree of consolidation (\(U_{\text{P}}\)) with the increase in time factor (\(T_{\text{v}}\)). It can be observed that the change of \(U_{\text{P}}\) under sinusoidal loading is a vibrant process. In each period, \(U_{\text{P}}\) maximizes at the beginning of unloading stage, whereas minimizes at the beginning of the next loading stage. Moreover, the maximum and minimum of \(U_{\text{P}}\) gradually become stable with the increase in time. Figure 7: Influence of rheological parameters on consolidation behavior under ramp loading: (**a**) influence of \(\kappa\), (**b**) influence of \(\theta\) and (**c**) influence of \(T_{\lambda 1}\). Figure 8: The influence of fractional orders on consolidation behavior under sinusoidal loading: (**a**) influence of \(\alpha_{0}\) and (**b**) influence of \(\alpha_{1}\). Figure 10: Influence of rheological parameters on consolidation behavior under sinusoidal loading: (**a**) influence of \(\kappa\), (**b**) influence of \(\theta\) and (**c**) influence of \(T_{\lambda 1}\). Figure 9: Influence of loading parameters on consolidation behavior under sinusoidal loading: (**a**) influence of \(\zeta\) and (**b**) influence of \(T_{\rm T}\). Figure 8 illustrates the influence of fractional orders on consolidation behavior under sinusoidal loading, where \(\alpha_{0}\) and \(\alpha_{1}\) individually vary from 0 to 1, whereas the other parameters remain constant. As observed in Figure 8a,b, \(U_{\text{P}}\) curves under various fractional orders intersect each other with the increase in time factor \(T_{\text{v}}\). Higher fractional orders result in faster dissipation of excess pore-water pressure in the early stage, but slower one at a later stage. Within each period, the relative variation of \(U_{\text{P}}\) becomes larger with the increase in fractional orders. Moreover, increasing \(\alpha_{0}\) (i.e., fractional order of independent fractional rheological element) gradually decreases the ultimate consolidation degree. However, \(\alpha_{1}\) (i.e., fractional order of Kelvin-Voigt body) has no influence on the ultimate consolidation degree. Figure 9 shows the influence of loading parameters on consolidation behavior under sinusoidal loading. \(\zeta\) varying from \(-0.5\) to \(0.5\) and \(T_{\text{T}}\) varying from \(0.05\) to \(0.20\), respectively, are taken for the investigations in Figure 9a,b. As observed in Figure 9a, the distribution pattern of total vertical stress has a noteworthy effect on the change of \(U_{\text{P}}\) and its relative variation within each period.