## 4 Critical State Model for Silty Sand  
### Impact of Fine Particles on Sand Properties  
Due to the partial participation of fines in the void between sand grains. The equivalent intergranular void ratio (\(e^{*}\)) suggested by [PERSON] and [PERSON] [29] was employed in place of the global void ratio to account for the effect of fines in sand. The equation of \(e^{*}\) is expressed as follows:  
\[e^{*}=\frac{e+(1-\beta)FC}{1-(1-\beta)FC} \tag{18}\]  
where \(\beta\) represents the fines influence factor and FC represents the amount of fines content. The reasonable value of \(\beta\) is \(0\ll\beta\ll 1\). When \(\beta=0\), \(\beta=1\), and \(0<\beta<1\) fines take no action, full action, and partial action in the load bearing mechanism, respectively. The dependence of \(\beta\) on the particle size ratio SR = (D10 of host sand)/(d50 of silt) was discovered using experimental data from [PERSON] et al. [40]. The value of D10 can be evaluated by the grain size distribution test of host sand, while the value of d50 can be determined by a photo of fine grain. The empirical equation for \(\beta\), presented by [PERSON] [9], takes into account the impacts of the particle size ratio SR as well as the roundness of the soil grains and are stated as follows:  
\[\beta=\beta_{0}\times(FC)\times SR^{-0.2} \tag{19}\]  
where  
\[\beta_{0}=\Big{(}1.93+0.04\langle r-1\rangle^{2}\Big{)}\Big{(}1+3.2\langle r- 1\rangle^{2}EXP(-22 FC)\Big{)} \tag{20}\]  
and \(r\) is the ratio of the average roundness of sand (rc) and fines (rf), respectively.  
It is generally accepted that the CSL in the e-p space gradually moves downward when \(FC\) increases, e.g., [41; 42; 43; 44; 45; 41]. However, when drawing in terms of the equivalent intergranular void ratio \(e^{*}\), the CSL are close together in a narrow range. As a result, for different \(FC\), a specific curve of the \(e^{*}\)-\(p\) can be established. In this paper, the \(e^{*}\) in Equation (18) is used to model the silty sand. Therefore, the state parameter for silty sand becomes \(\Psi^{*}=e^{*}-e_{cs}\) and the \(M_{f}\) for silty sand expressed as follows:  
\[M_{f}=M_{c}[1+n_{b}\langle-\Psi^{*}\rangle] \tag{21}\]  
The amount of \(FC\) also affects the stress ratio at the critical state \(M_{c}\). In the work of [PERSON] [9], the value of \(M_{c}\) for the specific \(FC\) can be determined by the concept of threshold fines content (\(FC_{th}\)) expressed in Equation (22). The stress ratio at the critical state for \(FC\) (\(M_{c}(FC)\)) can be calculated by [PERSON] [9] in Equation (23) with two more parameters, i.e., \(M_{c}(0)\) where the \(M_{c}\) at zero fines content and \(\Delta M_{c}=M_{c}(FC_{th})-M_{c}(0)\). The equations for the \(FC_{th}\) by [PERSON] [27], \(M_{c}(FC)\) and by [PERSON] [9] are presented in Equations (22) and (23).  
\[FC_{th}=0.4\bigg{(}\frac{1}{1+EXP(0.50-0.13\times SR)}+\frac{1}{SR}\bigg{)} \tag{22}\]  
\[M_{c}(FC)=M_{c}(0)+\frac{3}{2}(\Delta M_{c})\bigg{(}\frac{FC}{FC_{th}}\bigg{)} ^{2}-\frac{1}{2}(\Delta M_{c})\bigg{(}\frac{FC}{FC_{th}}\bigg{)}^{3} \tag{23}\]  
As described above, five additional parameters are required in the analysis of silty sand, i.e., rc, rf, \(SR\), \(M_{c}(0)\), and \(\Delta M_{c}\). The value of \(M_{c}\), \(M_{c}(0)\), \(M_{c}(FC)\), and \(\Delta M_{c}\) can be determined from the triaxial compression test of sand.