## 2 Derivation of \(e-log\sigma^{\prime}{}_{v}\) Equations for Laboratory Test Results
### Equations for the Hysteresis
#### 2.2.1 Equations for the Symmetric Hysteresis  
For symmetric hysteresis, three parameters (\(C\), \(h\) and \(k\)) should be applicable to both the upward-facing and downward-facing parabolas for simplicity. Therefore, it is necessary to derive the values of \(C_{1}\), \(h_{1}\), and \(k_{1}\) for the downward-facing parabola based on the values of \(C\), \(h\) and \(k\) of the upward-facing parabola. Based on Equation (8), which is the equation for the unloading curve, the change in void ratio \(\Delta e\) and the secant slope \(\overline{C_{r}}(Un)\) between the starting point of unloading (\(\sigma^{\prime}{}_{\theta}\)) and any point on the symmetrical unloading (\(\sigma^{\prime}{}_{v}\)) can be calculated using Equations (9) and (10), respectively. \[\Delta e=e-e_{\theta}=\frac{\left(log\sigma^{\prime}{}_{p}\right)^{2}-\left( log\sigma^{\prime}{}_{\theta}\right)^{2}-2h\cdot log(\sigma^{\prime}{}_{v}/ \sigma^{\prime}{}_{\theta})}{4C} \tag{9}\]  
Figure 3: Schematic of using parabolic equations for unloading and reloading lines. \[\overline{C_{r}}(Un)=-\frac{\Delta e}{\Delta log\sigma^{\prime}{}_{v}}=\frac{2h- log(\sigma^{\prime}{}_{\theta}\cdot\sigma^{\prime}{}_{v})}{4C} \tag{10}\]  
In the case of symmetric hysteresis, the size of the upward parabolas used to represent the unloading and reloading curves is identical. Consequently, the parameter \(C_{1}\) can be expressed in terms of \(C\). Next, to determine the formulas for \(h_{1}\) and \(k_{1}\), it is necessary to set up two additional conditions. From Figure 3, it can be observed that the reloading line intersects two key points; i.e., the point where unloading begins \((\sigma^{\prime}{}_{\theta}\), \(e_{\theta})\) and the point where reloading begins \((\sigma^{\prime}{}_{\omega}\), \(e_{\omega})\). We use the coordinates of these two points to find \(h_{1}\) and \(k_{1}\). First, determine the void ratio \(e\) at both points by substituting the stress \(\sigma^{\prime}{}_{\theta}\) into the VCL equation (Equation (6)) to obtain \(e_{\theta}=f(\Gamma,R,\alpha,\theta)\). Then, substitute the stress value at the reloading start point \(\sigma^{\prime}{}_{\omega}\) into the unloading equation (Equation (8)) to obtain \(e_{\omega}\). Using the condition that \(C_{1}=-C\) for both calculations, by solving the simultaneous equations, we can derive formulas for \(h_{1}\) and \(k_{1}\) that depend on the four parameters of the newly proposed VCL \((\Gamma,R,\alpha,\theta)\), as well as \(C\), \(h\), and \(k\). The equations for \(C_{1}\), \(h_{1}\) and \(k_{1}\) of the downward-facing parabola are shown in Equations (11)-(13), respectively. \[C_{1}=-C \tag{11}\]  
\[h_{1}=\frac{4C_{1}(e_{\omega}-e_{\theta})-(log\sigma^{\prime}{}_{\omega})^{2}+ (log\sigma^{\prime}{}_{\theta})^{2}}{2(log\sigma^{\prime}{}_{\theta}-log \sigma^{\prime}{}_{\omega})} \tag{12}\]  
\[k_{1}=e_{\omega}-\frac{1}{4C_{1}}(log\sigma^{\prime}{}_{\omega}-h_{1})^{2} \tag{13}\]  
where \(e_{\theta}=f(\Gamma,R,\alpha,\theta)=\Gamma-\alpha log\left(\frac{\sigma^{ \prime}{}_{\theta}}{R}\right)-\sqrt{\theta+\left(log\left(\frac{\sigma^{ \prime}{}_{\theta}}{R}\right)\right)^{2}}\) and \(e_{\omega}=\frac{(log\sigma^{\prime}{}_{\omega}-h)^{2}}{4C}+k\). At this step of the optimum parameter technique, a complete symmetric hysteresis (both unloading and reloading phases) can be evaluated using only three parameters (\(C\), \(h\) and \(k\)). Furthermore, the tangent slope \(\kappa\) of the unloading portion in Equation (8) can be expressed in Equation (14). At the starting point of unloading (and on VCL), the tangent slope \(\kappa\) is usually assumed to be a constant \(\kappa_{0}\) for each type of soil. Therefore, \(\kappa_{0}\) can be determined by substituting \(\sigma^{\prime}{}_{v}\) in Equation (14) with \(\sigma^{\prime}{}_{\theta}\) as shown in Equation (15).