## 5 Verification and Discussion
### Discussion about the Consolidation Behavior
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Solution**} & \multicolumn{5}{c}{**Rheological Parameter**} & \multicolumn{2}{c}{**Fractional Order**} \\  & & \(E_{\mathrm{v}}(E_{\mathrm{eff}})\) & \(E_{1}\) & \(\eta_{0}\) & \(\eta_{1}\) & \(\alpha_{0}\) & \(\alpha_{1}\) \\  & & (MPa) & (MPa) & (MPa,d) & (MPa,d) & (MPa,d) & \(\alpha_{0}\) & \(\alpha_{1}\) \\ \hline \multirow{3}{*}{I} & This study & 12 & 12 & 200 & 200 & 1 & 1 \\  & standard four-element & 12 & 12 & 200 & 200 & / & / \\  & model [13] & 12 & 12 & 200 & 200 & 0 & 1 \\  & This study & 12 & 12 & 200 & 200 & 0 & 1 \\ II & standard Merchant & 6* & 12 & / & 200 & / & / \\  & model [20] & 6* & 12 & 200 & 200 & 1 & 0 \\  & This study & 12 & 12 & 200 & / & / & / \\ III & standard Maxwell & 8* & / & 200 & / & / & / \\  & model [13] & 12 & 12 & 200 & 200 & 0 & 0 \\ IV & linear elastic model [9] & 4.8* & / & / & / & / & / \\ \hline \end{tabular}  
* Effective modulus (\(E_{\mathrm{eff}}\)) of independent spring, which is obtained by the expression in Table 1.  
\end{table} 
Table 2: Rheological parameter and fractional order for each viscoelastic model. ultimate degree of consolidation will become smaller, if fluid characteristic is embodied in the independent fractional rheological element. Figure 6 shows the influence of loading parameters on consolidation behavior under ramp loading. In Figure 6a, \(\zeta\) changes from \(-0.5\) to \(0.5\), and the other parameters remain constant. Herein three \(U_{\text{P}}\)-curves, respectively, are corresponding to three types of distribution patterns of total vertical stress; that is, inverse trapezoidal pattern (\(\zeta<0\)), rectangular pattern (\(\zeta=0\)) and trapezoidal pattern (\(\zeta>0\)). It can be clearly observed that the distribution pattern of total vertical stress has an important effect on the consolidation behavior. Inverse trapezoidal pattern brings about the most rapid dissipation of excess pore-water pressure; trapezoidal pattern results in the slowest dissipation of excess pore-water pressure; and under rectangular pattern, the dissipation of excess pore-water pressure falls in between them. During the consolidation process, the differences between three \(U_{\text{P}}\)-curves initially increase and reach a maximum at a given time, and then gradually decrease and finally disappear with the increase in time. Figure 6b illustrates the changes of \(U_{\text{P}}\) with time factor under various construction times. The construction time factor (i.e., \(T_{\text{v}0}\)) varies from \(0\) to \(1\), whereas the other parameters remain constant. It can be obviously observed that larger construction time slow down the dissipation of excess pore-water pressure (i.e., reduce \(U_{\text{P}}\)) in the early stage, and then the difference between \(U_{\text{P}}\)-curves of various construction times decrease gradually as time increases. Figure 7 illustrates the influence of rheological parameters on consolidation behavior under ramp loading. The dimensionless parameters (i.e., \(\kappa\), \(\theta\) and \(T_{\lambda 1}\)) varies from \(0.1\) to \(10\) one by one, whereas the other parameters remain constant. As observed in Figure 7a, the higher the modulus ratio (i.e., modulus of Kelvin-Voigt body is larger), the faster  
Figure 5: Influence of fractional orders on consolidation behavior under ramp loading: (**a**) influence of \(\alpha_{0}\) and (**b**) influence of \(\alpha_{1}\). Figure 6: Influence of loading parameters on consolidation behavior under ramp loading: (**a**) influence of \(\zeta\) and (**b**) influence of \(T_{\text{v}0}\). the dissipation of excess pore-water pressure. Figure 7b indicates that higher viscosity coefficient ratio (i.e., viscosity coefficient of independent fractional-derivative element is smaller) brings about slower dissipation of excess pore-water pressure, and decrease the ultimate consolidation degree as well.