## 5 Verification and Discussion
### Discussion about the Consolidation Behavior  
This part presents two examples to discuss the consolidation behavior of viscoelastic soils incorporating the Caputo-Fabrizio fractional derivative. One example is conducted for the ramp loading, and the other one is performed for the sinusoidal loading. In each example, the average degree of consolidation incorporating various parameters is obtained from the proposed solution, and the results are presented as a function of the time factor. For the case of ramp loading, the proposed solution for average degree of consolidation, which is given in Equation (42), has seven dimensionless parameters governing the consolidation process. These parameters can be classified into three categories, namely fractional orders (i.e., \(\alpha_{0}\) and \(\alpha_{1}\)), loading parameters (i.e., \(\zeta\) and \(T_{\mathrm{v0}}\)), and rheological parameters (i.e., \(\kappa\), \(\vartheta\) and \(T_{\lambda 1}\)). \(\alpha_{0}\) and \(\alpha_{1}\) reflect the influence of fractional orders of independent fractional-derivative element and that in fractional-derivative Kelvin-Voigt body, which further represent the fluid characteristic of soil mass. Indeed, higher fractional orders indicate that fluid characteristic of soil mass is more obvious. \(\zeta\) reflects the influence of distribution pattern of total vertical stress with depth. In practical engineering, total vertical stress is supposed to change linearly with depth according to three types of distribution patterns, i.e., rectangular pattern as \(\zeta=0\), trapezoidal pattern as \(\zeta>0\) and inverse trapezoidal pattern as \(\zeta<0\). \(T_{\mathrm{v0}}\) (=\(c_{\mathrm{v}}t_{0}/H^{2}\)) reflects the influence of construction time \(t_{0}\). When \(T_{\mathrm{v0}}=0\), ramp loading will become instantaneous loading, and the solution for this condition is given in Equation (39). \(\kappa\) (=\(E_{1}/E_{0}\)) reflects the influence of modulus ratio between fractional-derivative Kelvin-Voigt body and the independent spring. \(\vartheta\) (=\(\eta_{1}/\eta_{0}\)) reflects the influence of viscosity coefficient ratio between fractional-derivative Kelvin-Voigt body and the independent fractional-derivative element. \(T_{\lambda 1}\) (=\(c_{\mathrm{v}}\lambda_{1}/H^{2}\)) reflects the influence of viscosity coefficient of fractional-derivative Kelvin-Voigt body. By changing the values of these parameters one by one, consolidation behavior of viscoelastic soil under ramp loading is investigated in the following discussion. Figure 5 illustrates the influence of fractional orders on consolidation behavior under ramp loading. In Figure 5a, \(\alpha_{0}\) changes from 0 to 1 while the other parameters remain constant, and the changes of average degree of consolidation (\(U_{\mathrm{P}}\)) with time factor are presented. It can be clearly seen that \(U_{\mathrm{P}}\)-curves under different \(\alpha_{0}\) intersect each other with the increase in time factor \(T_{\mathrm{v}}\). Higher \(\alpha_{0}\) delivers larger \(U_{\mathrm{P}}\) in the early stage, but smaller one at a later stage. This indicates that for higher \(\alpha_{0}\), excess pore-water pressure dissipates faster in the beginning and then more slowly during consolidation process. Moreover, the ultimate value of \(U_{\mathrm{P}}\) gradually decreases with the increase in \(\alpha_{0}\). In Figure 5b, \(\alpha_{1}\) changes from 0 to 1, and the other parameters remain constant. In comparison with the results obtained under various \(\alpha_{0}\), the results obtained under different \(\alpha_{1}\) also share similar changing features for \(U_{\mathrm{P}}\)-curves. The only difference is that for various \(\alpha_{1}\), the ultimate value of \(U_{\mathrm{P}}\) is the same with each other. These influences demonstrate that fractional orders i.e., fluid characteristic of soil mass has a significant impact on consolidation behavior. More fluid soils consolidate initially rapidly but more slowly at a later stage, and even their