## 4 Verifications and Examples
### Influence of \(k_{a}/k_{\text{w}}\) on Consolidation  
According to Figure 1, mathematical modeling is adopted to explore the consolidation behavior of the viscoelastic unsaturated soils subjected to exponential loading. The initial EPAP \(u_{\text{a}}^{0}=5\) kPa and the initial EPWP \(u_{\text{w}}^{0}=40\) kPa can be determined according to the method given by [PERSON] and [PERSON] [6]. The following are the other parameters displayed: \(H=10\) m, \(S_{\text{r}0}=80\%\), \(k_{\text{w}}=10^{-10}\) m/s, \(n_{0}=50\%\). (The other parameters are as follows in Figure 5; \(k_{\text{a}}/k_{\text{w}}=10\) and other parameters are as follows in Table 1[26; 29], except the analytical parameters in Figures 5-8). Exponential loading \(q(t)=q_{0}+Aq_{0}\left(1-\text{e}^{-bt}\right)\) is exerted on the unsaturated soil layer as shown in Figure 5. Here, \(q_{0}\) represents the incipient surcharge, \(A\) represents the loading constant, and \(b\) represents the nonlinear rate of exponential loading. \(\text{U}_{\text{s}}\) is the degree of consolidation, \(\text{U}_{\text{s}}=w(t)/w(\infty)\). Figure 4: The comparison of the proposed solution with the literature solution [41]: (**a**) EPAP, (**b**) EPWP. In this study, \(q_{0}=100\) kPa, \(A=1\), and \(b=5\times 10^{-4}\) s\({}^{-1}\) were chosen to conduct research on the consolidation behavior of viscoelastic unsaturated soils under exponential loading. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(E_{0\text{n}}^{\text{a}}=E_{1\text{n}}^{\text{a}}\) & \(5\times 10^{3}\) & kPa & \(E_{0\text{n}}^{\text{w}}=E_{1\text{n}}^{\text{w}}\) & \(1\times 10^{3}\) & kPa \\ \(E_{0\text{s}}^{\text{a}}=E_{1\text{s}}^{\text{a}}\) & \(1.25\times 10^{3}\) & kPa & \(E_{0\text{s}}^{\text{w}}=E_{1\text{s}}^{\text{w}}\) & \(2.5\times 10^{3}\) & kPa \\ \(\eta_{\text{n}}^{\text{a}}\) & \(5\times 10\) & kPa-s & \(\eta_{\text{n}}^{\text{w}}\) & \(1\times 10\) & kPa-s \\ \(\eta_{\text{s}}^{\text{a}}\) & \(1.25\times 10\) & kPa-s & \(\eta_{\text{s}}^{\text{w}}\) & \(2.5\times 10\) & kPa-s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example parameters. Figure 5: Exponential loading. Figure 6 shows the variations of EPAP, EPWP, and U\({}_{\rm S}\) at different values of \(k_{\rm a}/k_{\rm w}\). It is shown that the larger \(k_{\rm a}/k_{\rm w}\) is, the faster the excess pore pressures dissipate. There are two periods (i.e., the prior and latter periods) of EPAP and three periods (i.e., the prior, intermediate, and latter periods) of EPWP in the dissipation process. It can be found from Figure 6a that the EPAP starts to dissipate at the end of the prior period when the load reached its maximum except when \(k_{\rm a}/k_{\rm w}=100\). This is because the dissipation rate of EPAP is greater than the increasing rate of the loading when \(k_{\rm a}/k_{\rm w}=100\). Figure 6b shows that the prior and intermediate dissipation periods of EPWP are similar to the whole dissipation process of EPAP. The main reason for this is that the permeability coefficient of EPAP is bigger than that of EPWP in practice; thus, the water phase begins to flow when the air phase flows to termination. In the latter period, the dissipation curves of EPWP at varying values tend to become the same, as the EPAP has been fully dissipated. The dissipation curves of EPWP shown in Figure 6 are roughly double-S-typed, and there is a steady period in these curves. When the EPAP dissipates to zero, it is just the beginning of the steady period. Figure 6c shows the variation in the average U\({}_{\rm S}\) with time at different values of \(k_{\rm a}/k_{\rm w}\). The larger \(k_{\rm a}/k_{\rm w}\), the earlier the soil begins to be consolidated. In the latter period, the dissipation curves of the average U\({}_{\rm S}\) with different values of \(k_{\rm a}/k_{\rm w}\) tend towards the same.