## 3 Effect of Model Constant on Critical State  
### Liquefaction Behaviour in Loose State  
The recommended identification method of model constants a and b is shown in Figure 2[9; 13]. Several critical state points need to be plotted and fitted in the \(\mathrm{e}-\mathrm{p}^{\prime}/\mathrm{p}_{\mathrm{a}}\) space to a straight line where a is the interception value and b (b \(<0\)) is the slope of this line. Figure 1: Initial state parameter for two different initial density indexes \((\mathrm{e}_{0}\#1\
eq\mathrm{e}_{0}\#2)\). The critical state of Hostun RF sand from very loose (\(\rm I_{Di}=-0.10\)) to loose (\(\rm I_{Di}=0.20\)) state is shown in Figure 3. It can be observed that the unification of final state could not be seriously achieved. The simplification of the linear fitting is only a rough hypothesis and the determination of the model parameter a is in general sophisticated. Thus, a parametric study of a was carried out by simulating a specimen of density indices equal to \(0.00\) with different values of a \(=1.05,\ 1.00,\ 0.95\) under an undrained triaxial condition. The material constants according to their functions in separate columns are given in Table 1. One should also note that all parameters listed in Table 1 were dimensionless. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Elastic Parameters**} & **Critical State** & **Dilatancy** & **Hardening** & **Sand Property** \\  & **Parameters** & **Parameters** & **Parameters** & **Sand Property** \\ \hline \(\rm G_{0}=176\) & \(\rm M=1.22\) & \(\rm n=0.42\) & \(\rm\alpha=0.63\) & \(\rm e_{max}=1\) \\ \(\rm\gamma=0.176\) & \(\rm b_{0}=-0.030\) & - & \(\rm\beta=-2.5\) & \(\rm e_{min}=0.656\) \\  & \(\rm a=1.00\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters and physical properties of Hostun RF sand. Figure 3: Critical state of Hostun RF sand from very loose to loose states on (**a**) \(\rm e-log(p^{\prime})\) and (**b**) \(\rm e-log(p^{\prime}/p_{a})\) plan [6]. Figure 2: Parameter calibration of a and b in \(\rm e-p^{\prime}/p_{a}\) plan (\(x\)-axis in log scale) [9, 13]. In Figure 4, it is clear that the drop of deviator stress q and the static liquefaction phenomenon became increasingly important with the decrease in a. Dilatant behaviour was observed for \(\text{a}=1.05\), whereas very contractive behaviour was observed for \(\text{a}=0.95,\ 1.00\), which meant that the result in loose state depended, to a large extent, on the value of a. The critical state line in terms of effective confining pressure is plotted in Figure 5 and the initial state of the simulated case is represented by a black rhombus point. For \(\text{a}=1.05\), the initial state was clearly below the critical state, which caused state parameter \(\psi\) to be greater than 1 during all calculations. However, for the case of \(\text{a}=0.95,\ 1.00\), the initial states were both over the critical state, resulting in the state parameter \(\psi\) being lower than 1. Therefore, a very contractive behaviour could be expected, which is consistent with the generation of excess pore water pressure, as shown in Figure 6. All the above findings suggest that the value of a is the fundamental point to successfully simulate the liquefaction behaviour of very loose sand. Figure 4: Influence of parameter a on effective stress path of Hostun RF sand under a confining pressure of 100 kPa and density index (\(\text{I}_{\text{Di}}\)) of 0.00. Figure 5: Variation of critical state line in \(\text{e}-\log(\text{p}^{\prime})\) plan with model constant a. According to Figure 5, the form of critical state defined by Equation (1) can be separated into two parts. Firstly, it was almost constant before 100 kPa. Then, this line decreased significantly with the increase of \(\mathrm{p^{\prime}}\).