# Numerical Simulation of Subdam Settlement in Ash Disposal Based on CGSW Optimization
## 3 Modeling and Test Scheme Design
### Constitutive Model and Boundary Condition  
The Mohr-Coulomb (MC) criterion was chosen for all the modules (subdams, fly ash, CGSW, stone column, and bedrock) in the numerical model. The MC criterion was a simple elastic-plastic criterion that contains five parameters. These material parameters of the five different geological units were estimated from the laboratory geotechnical tests and are presented in Table 1.  
The failure criterion used in this paper was a composite Mohr-Coulomb criterion with a tension cutoff, as illustrated in Figure 5. The magnitude relationship of the three principal stresses was present as \(\sigma_{1}\leq\sigma_{2}\leq\sigma_{3}\). The failure envelope \(f(\sigma_{1},\sigma_{3})=0\) was defined from point A to B by the Mohr-Coulomb failure criterion \(f^{s}=0\) with  
\[f^{s}=-\sigma_{1}+\frac{1+sin\varphi}{1-sin\varphi}\sigma_{3}-2c\sqrt{\frac{1+ sin\varphi}{1-sin\varphi}} \tag{1}\]  
and from point B to C by a tension failure criterion of the form \(f^{t}=0\) with  
\[f^{t}=\sigma_{3}-\sigma^{t} \tag{2}\]  
where \(\varphi\) is the friction angle, \(c\) is the cohesion, \(\sigma^{t}\) is the tensile strength, and its maximum value of \(\sigma^{t}\) is given by  
\[\sigma^{t}_{max}=\frac{c}{tan\varphi} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & **Density** & **Cohesion** & **Friction** & **Bulk Modulus** & **Shear Modulus** & **Tension** \\
**(kg/m3)** & **(KPa)** & **(\({}^{\circ}\))** & **(MPa)** & **(MPa)** & **(MPa)** & **(KPa)** \\ \hline Limestone & 2.69 & 650 & 40 & 11,000 & 11,000 & 6000 \\ Fly ash & 1.41 & 26 & 30 & 30 & 28 & 12 \\ CGSW (saturated) & 2 & 1 & 33 & 8.5 & 6.4 & 0 \\ CGSW (dry) & 1.65 & 26 & 30 & 1200 & 1200 & 0 \\ Stone column & 1.85 & 390 & 42 & 10,000 & 800 & 0 \\ Subdam & 1.53 & 30 & 23 & 32 & 30 & 14 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Static geotechnical properties of materials.  
Figure 5: Mohr–Coulomb failure criterion in Flac3D.  
The incremental expression of [PERSON]'s law in terms of the generalized stress and stress increments has the form  
\[\begin{cases}\Delta\sigma_{1}=\alpha_{1}\Delta e_{1}^{e}+\alpha_{2}\big{(}\Delta e _{2}^{e}+\Delta e_{3}^{e}\big{)}\\ \Delta\sigma_{2}=\alpha_{1}\Delta e_{2}^{e}+\alpha_{2}\big{(}\Delta e_{2}^{e}+ \Delta e_{3}^{e}\big{)}\\ \Delta\sigma_{3}=\alpha_{1}\Delta e_{3}^{e}+\alpha_{2}\big{(}\Delta e_{2}^{e}+ \Delta e_{3}^{e}\big{)}\end{cases} \tag{4}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are material constants defined in terms of the shear modulus, \(G\), and bulk modulus, \(K\), as  
\[\begin{cases}\alpha_{1}=K+\frac{4}{3}G\\ \alpha_{2}=K-\frac{2}{3}G\end{cases} \tag{5}\]  
For the boundary conditions, the free boundaries were applied to the top faces while the roller boundaries were applied to all the other sides of the model. The vertical velocity of the bottom and the horizontal velocity of the four vertical sides were constrained to a minimum to avoid unwanted stress concentrations. In-situ stress distribution was initialized by calculating after the application of gravity in the domain.