## 4 Constitutive Model and Method
### Generalized Plastic Constitutive Model  
The generalized plastic constitutive model, proposed by Pastor-Zienkiewicz III ([PERSON] et al. (2016)), requires 13 parameters, which can be obtained through a conventional triaxial compression test under monotonic loading and cyclic loading conditions. The elastic relations are  
\[\Delta\epsilon_{\mathrm{v}}^{\mathrm{e}}=\frac{1}{K_{\mathrm{ev}}}\Delta p^{ \prime}, \tag{1}\]  
\[\Delta\epsilon_{\mathrm{s}}^{\mathrm{e}}=\frac{1}{G_{\mathrm{es}}}\Delta q, \tag{2}\]  
with the elastic modulus  
\[K_{\mathrm{ev}}=K_{\mathrm{ev0}}\frac{p^{\prime}}{p_{0}}, \tag{3}\]  
\[G_{\mathrm{es}}=G_{\mathrm{es0}}\frac{p^{\prime}}{p_{0}}, \tag{4}\]  
where \(\Delta\epsilon_{\mathrm{v}}^{\mathrm{e}}\) and \(\Delta\epsilon_{\mathrm{s}}^{\mathrm{e}}\) are the elastic volumetric strain increments and the deviatoric strain increments, respectively; \(\Delta p^{\prime}\) and \(\Delta q\) are the elastic volumetric stress increments and the deviatoric stress increments, respectively; \(K_{\mathrm{ev}}\) and \(G_{\mathrm{es}}\) are the elastic volumetric modulus and the shear modulus, respectively; \(p^{\prime}\) is the mean effective stress; \(K_{\mathrm{ev0}}\), \(G_{\mathrm{es0}}\), and \(p_{0}\) are the parameters of the constitutive model.  
The loading direction \(\{\overline{\mathbf{n}}\}\) is defined as  
\[\{\overline{\mathbf{n}}\}=\frac{1}{\sqrt{1+d_{\mathrm{f}}^{2}}}\big{[}\begin{array} []{cc}d_{\mathrm{f}}&1&-\frac{1}{2}\end{array}qM_{\mathrm{f}}\cos 3\theta \big{]}^{\mathrm{T}}, \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Starting and Ending** & **Length of Compaction** & **Liquidity Index** & **Liquefaction** \\
**Mileage (km)** & **Treatment (m)** & **Grade** \\ \hline \multirow{4}{*}{1416.3–1416.5} & \multirow{4}{*}{200} & 12.6 & Moderate \\  & & 11.1 & Moderate \\  & & 9.6 & Moderate \\  & & 16.3 & Moderate \\
1418.1–1418.48 & 380 & 14.3 & Moderate \\  & & 9.7 & Moderate \\ \hline \hline \end{tabular}
\end{table}
Table 1: Liquidity indices at a depth of 8.0 m after dynamic compaction.  
where \(q\) is the deviatoric stress, \(\theta\) is the Lode angle, \(d_{\rm f}=(1+a_{\rm f})(M_{\rm f}-\eta)\), the stress ratio \(\eta=q/p^{\prime}\), \(a_{\rm f}\) is one of the parameters of the constitutive model, and \(M_{\rm f}\) is dependent on the Lode angle.  
\[M_{\rm f}=\frac{6M_{\rm fc}}{6+M_{\rm fc}(1-\sin 3\theta)}, \tag{6}\]  
where \(M_{\rm fc}\) is one of parameters of the constitutive model.  
The plastic yielding function is defined as  
\[f=q-M_{\rm f}p^{\prime}\bigg{(}\frac{1+a_{\rm f}}{a_{\rm f}}\bigg{)}\bigg{[}1- \bigg{(}\frac{p^{\prime}}{p_{\rm c}}\bigg{)}^{a_{\rm f}}\bigg{]}, \tag{7}\]  
The corresponding plastic flow direction \(\{\overline{\rm{m}}_{\rm g}\}\) is  
\[\{\overline{\rm{m}}_{\rm g}\}=\frac{1}{\sqrt{1+d_{\rm g}^{2}}}\big{[}\begin{array} []{ccc}d_{\rm g}&1&-\frac{1}{2}&qM_{\rm g}\cos 3\theta\end{array}\big{]}^{ \rm T}, \tag{8}\]  
where \(d_{\rm g}\) is the dilatancy law \(d_{\rm g}=\big{(}1+a_{\rm g}\big{)}\big{(}M_{\rm g}-\eta\big{)}\), and \(a_{\rm g}\) is one of the parameters of the constitutive model. \(M_{\rm g}\) is dependent on the Lode angle.  
\[M_{\rm g}=\frac{6M_{\rm gc}}{6+M_{\rm gc}(1-\sin 3\theta)}, \tag{9}\]  
where \(M_{\rm gc}\) is one of the parameters of the constitutive model.  
The plastic potential function is defined as  
\[{\rm g}=q-M_{\rm g}p^{\prime}\bigg{(}\frac{1+a_{\rm g}}{a_{\rm g}}\bigg{)} \bigg{[}1-\bigg{(}\frac{p^{\prime}}{p_{\rm c}}\bigg{)}^{a_{\rm g}}\bigg{]}, \tag{10}\]