## 4 Verification and Analysis of Solution
### Consolidation Analyses
During the initial phase of consolidation, there is a marginal disparity in the consolidation curves, based on different compression index to penetration index ratios, suggesting minimal influence during this initial stage. However, notable differences emerge in the middle and late stages of consolidation for the various ratios of compression index to penetration index. Furthermore, the consolidation curves for \(c_{1}=0.5\) and \(c_{2}=1.5\) coincide with those for \(c_{1}=1.5\) and \(c_{2}=0.5\), implying that under the conditions of uniform interface parameters at the top and bottom surfaces of the soil and identical permeability and compressibility in the upper and lower layers, the interchange of \(c_{1}\) and \(c_{2}\) exerts no impact on the soil consolidation rate. Figure 6: Influence of the interface parameter value on the \(U_{\text{s}}\) curves. Figure 7: Influence of the ratio \(c_{i}\) of compression index to permeability index on the \(U_{\text{s}}\) curves. Figure 8 analyzes the consolidation characteristics of the foundation when the permeability ratio of the soil layer is equal to the compressibility ratio under the condition that the interface parameters of the top and bottom surfaces of the foundation are equal. From the diagram, it is evident that for \(c_{1}=0.5\) and \(c_{2}=1\), the consolidation degree decreases with increasing \(a\) and \(b\) values. These results underline the great impact the compression index to penetration index has ratio on soil consolidation characteristics when the consolidation coefficients remain consistent across the upper and lower soil layers. The one-dimensional nonlinear consolidation of double-layered soil presents a multifaceted complexity. This intricacy is not solely contingent on the relative permeability and compressibility of soil layers but is also influenced by nonlinear parameters and the form of drainage boundaries. Consequently, a comprehensive analysis of double-layered soil consolidation characteristics necessitates the integration of these multifarious factors. Figure 9 explores the influence of the ratio \(N_{e}\), representing the final effective stress in relation to the initial effective stress, on both the average settlement-defined consolidation degree (\(U_{\text{s}}\)) and the average pore water pressure-defined consolidation degree (\(U_{\text{p}}\)). This exploration is conducted under the circumstance where the compression index to penetration index ratio for both upper and lower layers is 0.5. Figure 8: Influence of the ratio \(c_{i}\) of compression index to penetration index on the \(U_{\text{s}}\) curves (\(c_{v01}=c_{v02}\)). Figure 9: Impact of the final effective stress to initial effective stress ratio (\(N_{e}\)) on consolidation degree curves (\(c_{ci}/c_{ki}=0.5\)). With an increase in the applied external load the ratio \(N_{\sigma}\) of final effective stress to initial effective stress concurrently elevates. As \(N_{\sigma}\) rises, the \(\sigma_{0}^{\prime}/\sigma^{\prime}\) decreases, a trend traceable back to Equations (3) and (4), wherein the parameters \(k_{vi}\) and \(m_{vi}\) decrease with the increasing \(N_{\sigma}\). When \(c_{ci}/c_{ki}<1\), the reduction in \(k_{vi}\) is less pronounced, accentuating the influence of the compression coefficient \(m_{\rm v}\) on pore water pressure dissipation. Reduced \(k_{\rm v}\) implies more challenging pore water drainage, translating into a slower consolidation process. Decreased \(m_{\rm v}\) indicates a lesser capacity for pore compression, correlating with improved soil consolidation. Consequently, the diagram demonstrates an increasing trend in both \(U_{\rm s}\) and \(U_{\rm p}\) as \(N_{\sigma}\) escalates. This outcome is attributed to the reduction in \(m_{\rm v}\) with rising \(N_{\sigma}\), which promotes soil consolidation. Notably, when \(c_{ci}/c_{ki}<1\), an increased external load serves to enhance soil consolidation.