## 2 Derivation of \(e-log\sigma^{\prime}{}_{v}\) Equations for Laboratory Test Results
### Equations for the Hysteresis  
Hysteresis refers to a loop involving unloading followed by reloading in the consolidation test result of all clays due to the requirements in the experimental standard. This is to observe the reversible and irreversible deformation of clays during loading. In some clays, the unloading and reloading curves may overlap or closely align, allowing the hysteresis to be simplified using a straight line with a constant slope \(C_{r}\). In such a situation, calculations using the linear approximation can provide accuracy for overconsolidated clays. However, some clays exhibit differently, such as clays in [22; 23; 24], where it is evident that the unloading and reloading curves deviated significantly. For these clays, the secant slope between two points on the unloading or reloading curves differed to the value of the average value \(C_{r}\). Therefore, the existing linear approximation method cannot provide high accuracy for these clays. To address this, adopting an equation capable of representing both types of hysteresis behavior could improve the accuracy of settlement calculations in overconsolidated clays. This approach enhances the precision of modeling soil compression and helps in predicting deformation under varying loading conditions more reliably.  
Additionally, it was found that some hysteresis can be both symmetric and asymmetric. Symmetric hysteresis refers to cases where the reloading curve returns a point close to the starting point of the unloading curve on VCL. However, certain clays exhibit different characteristics, specifically asymmetric hysteresis, where the reloading curve bends below  
Figure 2: Demonstration of the characteristics of the graphs for Equation (6) with different parameters: (**a**) graphs varying parameter \(\Gamma\); (**b**) graphs varying parameter \(R\); (**c**) graphs varying parameter \(\theta\); (**d**) graphs varying parameter \(\alpha\).  
the starting point of the unloading curve on VCL. This behavior contributes to the increased occurrence of irreversible strain in overconsolidated clays, which tend to experience greater water expulsion than usual.  
\[e=\frac{\left(log\sigma^{\prime}{}_{v}-h\right)^{2}}{4C}+k \tag{8}\]  
It is necessary to consider both accuracy and ease of use in order to create equations that correctly depict the unloading-reloading part. In this study, parabolic equations are employed. Generally, three fitting parameters are used for a parabola, as shown in Equation (8). An upward-facing parabolic equation represents the unloading phase (with parameters \(C\), \(h\) and \(k\)), while a downward-facing parabolic equation represents the reloading phase of the experimental results (with different parameters \(C_{1}\), \(h_{1}\), and \(k_{1}\)). It should be noted that Equation (8) can be valid for the unloading only if \(\sigma^{\prime}{}_{v}\) is less than or equal to the stress at the unloading point \(\sigma^{\prime}{}_{\theta}\) on VCL. Figure 3 illustrates the concept of using parabolas to describe the unloading-reloading graphs.  
It is crucial to first formulate an equation to represent symmetric hysteresis behavior. Once this foundational equation is established, it can be further revised to accommodate asymmetric hysteresis. This approach ensures systematic progression from simpler to more complex representations of hysteresis, enhancing the ability to accurately model and predict soil behavior under varying conditions.