# Role of Bimodal Water Retention Curve on the Unsaturated Shear Strength
## 3 Mathematical Equation  
### Applicable Theory  
Various best-fitting methods for the WRC had been developed with great accuracy. In this study, a mathematical equation (Equation (1)) from [PERSON] et al. [17] was used for best fitting the experimental data of the bimodal WRC. This equation was used since the parameter of this equation has a physical definition [12; 23; 33]. The term of physical definition for parameters in the WRC equation refers to the ability of the parameters in the proposed equation to represent the variables of the WRC, such as: air-entry value of soil, inflection point of WRC, and residual suction and residual water content of soil.  
\[\begin{split}\theta_{w}=\left[1-\frac{ln\left(1+\frac{\Psi}{U_{r} }\right)}{ln\left(1+\frac{\Psi}{U_{r}}\right)}\right]\left[\theta_{r}+\left\{ (\theta_{s1}-\theta_{s2})\left(1-(\beta_{1})erfc\left(\frac{ln\left(\frac{ \Psi_{a2}-\Psi}{U_{s1}-\Psi_{s1}}\right)}{s_{1}}\right)\right)\right\}\right. \\ \left.+\left\{(\theta_{s2}-\theta_{r})\left(1-(\beta_{2})erfc \left(\frac{ln\left(\frac{\Psi_{a2}-\Psi}{U_{s2}-\Psi_{s2}}\right)}{s_{2}} \right)\right)\right\}\right.\end{split} \tag{1}\]  
where:  
\(\theta_{s1}=\) saturated volumetric water content  
\(\theta_{s2}=\) volumetric water content related to air-entry value 2  
\(\beta_{1}=0\) when \(\psi\leq\psi_{a1}\); \(\beta_{1}=1\) when \(\psi>\psi_{a1}\)  
\(\beta_{2}=0\) when \(\psi\leq\psi_{a2}\); \(\beta_{2}=1\) when \(\psi>\psi_{a2}\)  
\(\psi_{a1}=\) parameter related to air-entry value 1 (\(AEV_{1}\)) (kPa) (Figure 1)  
\(\psi_{a2}=\) parameter related to air-entry value 2 (\(AEV_{2}\)) (kPa) (Figure 1)  
\(C_{r}=\) input parameter according to Fredlung and Xing [43] (kPa)  
\(erfc=\) the complementary error function, \(erfc(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}exp\left(-\frac{x^{2}}{2} \right)dx\)  
\(\psi_{m1}=\) parameter related to suction at the inflection point 1 (Figure 1)  
\(\psi_{m2}=\) parameter related to suction at the inflection point 2 (Figure 1)  
\(\theta_{r}=\) parameter related to volumetric water content at residual condition (Figure 1)  
\(s_{1}=\) parameter related to standard deviation 1 (Figure 1)  
\(s_{2}=\) parameter related to standard deviation 2 (Figure 1)  
The subscript 1 and 2 in the equation are associated with sub-curve 1 (macro pores) and sub-curve 2 (micro pores) of dual-porosity soils, respectively. The details of explanation of each parameter can be seen in [PERSON] et al. [17].  
Many equations had been developed to model the unsaturated shear strength. In this study, the mathematical equation (Equation (2)) from [PERSON] et al. [44], was used to model the experimental data from the unsaturated shear strength in the initial stage.  
\[\tau=c\prime+(\sigma-u_{a})tan\;\phi\;t+(u_{a}-u_{w})tan\;\phi^{b} \tag{2}\]  
where \(\phi^{\prime}=\phi^{b}\) if \((u_{a}-u_{w})<AEV\)  
\[\tau=c\prime+[(\sigma-u_{a})+(u_{a}-u_{w})_{b}]tan\;\phi\;t+[(u_{a}-u_{w})-AEV] b\Theta^{k}tan\;\phi\;t \tag{3}\]  
\[\text{if}\;(u_{a}-u_{w})\geq AEV\]  
\[y=0.502\;ln\big{(}\;I_{p}+2.7\big{)}-0.387 \tag{4}\]  
\[k=[log(\;u_{a}-u_{w})-log(\;AEV)]^{y} \tag{5}\]  
\[b=-0.245\big{\{}ln\big{[}\;n\big{(}I_{p}+4.4\big{)}\big{]}\big{\}}^{2}+2.114 \big{\{}ln\big{[}\;n\big{(}I_{p}+4.4\big{)}\big{]}\big{\}}-3.522 \tag{6}\]  
where:  
\(AEV=\) air-entry value of soil (kPa)  
\(y\) and \(b=\) fitting parameters.  
\(Ip=\) plasticity index.  
\(n=\) fitting parameter from Fredlund and Xing [43] equation for fitting WRC