# Study on Reinforcement Mechanism and Reinforcement Effect of Saturated Soil with a Weak Layer by DC
## 2 The Fluid-Solid Coupling Model and FEM Implementation  
### u-U-p Formulation  
Based on the ABAQUS/explicit platform of [PERSON] et al. [29; 30], [PERSON] et al. [17; 31] developed a fluid-solid coupling method with a Drucker-Prager-Cap model for the FEM analysis of DC. The u-U-p formulation is briefly introduced as follows.  
The u-U-p formulation is given by:  
\[\mathrm{L}^{\mathrm{T}}\sigma^{\prime}+(\alpha-\mathrm{n})\
abla\mathrm{p}+ \mathrm{c}\left(\dot{\mathrm{U}}-\dot{\mathrm{u}}\right)-(1-\mathrm{n})\rho_{ \mathrm{s}}\ddot{\mathrm{u}}+(1-\mathrm{n})\rho_{\mathrm{s}}\mathrm{b}_{ \mathrm{s}}=0 \tag{1}\]  
\[\mathrm{n}\
abla\mathrm{p}-\mathrm{c}\left(\dot{\mathrm{U}}-\dot{\mathrm{u}} \right)-\mathrm{n}\rho_{\mathrm{f}}\ddot{\mathrm{U}}+\mathrm{n}\rho_{ \mathrm{f}}\mathrm{b}_{\mathrm{f}}=0 \tag{2}\]  
where u is solid displacement; U is pore fluid displacement; p is pore pressure; \(\rho_{\mathrm{s}}\) and \(\rho_{\mathrm{f}}\) are the densities of the soil grains and pore fluid, respectively; \(\mathrm{b}_{\mathrm{s}}\) and \(\mathrm{b}_{\mathrm{f}}\) are the body force vectors on the soil grains and pore fluid, respectively.  
\[\mathrm{p}=\mathrm{K}_{\mathrm{f}}\frac{\left(1-\mathrm{n}-\frac{\mathrm{K}^ {\prime}}{\mathrm{K}_{\mathrm{s}}}\right)\mathrm{m}^{\mathrm{T}}\varepsilon+ \mathrm{n}\
abla^{\mathrm{T}}\mathrm{U}}{\mathrm{n}+(1-\mathrm{n})\frac{ \mathrm{K}_{\mathrm{f}}}{\mathrm{K}_{\mathrm{s}}}-\frac{\mathrm{K}^{\prime} \mathrm{K}_{\mathrm{f}}}{\mathrm{K}_{\mathrm{s}}^{2}}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **Ground Stratum** & **Research Questions** & **Main Conclusion** & **Investigation Method** \\ \hline [PERSON] et al. [22] & Uniform & Model tests of DC on sand with different groundwater tables are performed to investigate the effect of groundwater depth in DC. & Through the comparison of dynamic responses of soils, deviating is used for the treatment of saturated soil with groundwater. & Model tests \\ \hline [PERSON] et al. [23] & Uniform & ABACUS 6.14 software is used to simulate three-dimensional model of lateral DC in the slope. & Impact velocity is the major factor that influences soil different for three different slope. & Numerical method \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(Cont.\)where \(\rm m^{T}=[1\ 1\ 1\ 0\ 0\ 0]\) and \(\varepsilon\) is the strain vector of the soil skeleton (or solid domain), respectively; \(\rm m^{T}\varepsilon=\varepsilon_{v}=v\) volumetric strain; \(\rm K^{\prime}\), \(\rm K_{f}\), and \(\rm K_{s}\) are the bulk moduli of the soil skeleton, pore fluid, and soil grain, respectively; \(\rm n=porosity\).  
\[\alpha=1-\frac{\rm K^{\prime}}{\rm K_{s}} \tag{4}\]  
\[\rm c=\frac{\gamma_{f}n^{2}}{k} \tag{5}\]  
where \(\gamma_{f}\) and \(\rm k\) are the unit weight and coefficient of permeability of the pore fluid, respectively.  
\[\rm L^{T}=\begin{bmatrix}\frac{\partial}{\partial x}&0&0&\frac{\partial}{ \partial y}&0&\frac{\partial}{\partial x}\\ 0&\frac{\partial}{\partial y}&0&\frac{\partial}{\partial x}&\frac{\partial}{ \partial x}&0\\ 0&0&\frac{\partial}{\partial x}&0&\frac{\partial}{\partial y}&\frac{\partial}{ \partial x}\end{bmatrix} \tag{6}\]  
Both Equations (1) and (2) respectively comprise three separate equations, which relate to solid and fluid dynamic equilibrium in the \(\rm x\), \(\rm y\), and \(\rm z\) directions. For the fluid-solid coupling soil assembles, pore fluid is supposed to be incompressible during DC.