# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 8 Numerical Examples  
### Uniaxial Compression  
The GMC yield criterion-based elastoplastic constitutive model's UMAT subroutine is studied and constructed, and the standard soil mechanics experiments conducted under uniaxial compression are analyzed using finite elements. When the strength parameters \(\phi_{0}=\phi_{1}\) and \(c_{0}=c_{1}\), the GMC model degenerates into the Mohr-Coulomb model. The established UMAT subroutine's computational capability, correctness, and effectiveness are confirmed by comparing the calculations of the subroutine and the Mohr-Coulomb model. The specimen's element type is a completely integrated C3D8 solid element, and Figure 11 depicts the calculation model. The specimen is 100 mm \(\times\) 100 mm \(\times\) 100 mm in size. The established calculation model is a cube, and the applied uniform load is 6.0 MPa. Table 3 displays the material properties using the associated flow rule. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Model**} & **Elastic** & **Poisson’s** & \multirow{2}{*}{\(\phi_{0}\) (\({}^{\circ}\))} & \multirow{2}{*}{\(\phi_{1}\) (\({}^{\circ}\))} & \multirow{2}{*}{\(c_{0}\) (Mpa)} & \multirow{2}{*}{\(c_{1}\) (Mpa)} \\  & **Modulus (Mpa)** & & & & & \\ \hline M-C & 1000 & 0.3 & 45 & - & 1.2 & - \\ GMC & 1000 & 0.3 & 45 & 45 & 1.2 & 1.2 \\ \end{tabular}
\end{table} 
Table 3: Material parameters. Figure 10: The flowchart of UMAT. The stress \(\sigma_{33}\) and plastic strain \(\varepsilon_{33}^{p}\) nephogram of the GMC model and M-C model incorporated in ABAQUS under uniaxial compression are shown in Figures 12 and 13, respectively. It is evident from Figures 12 and 13 that the stress nephograms produced using the M-C model integrated into ABAQUS and the UMAT subroutine are essentially identical; the  
Figure 11: 3D finite model. Figure 12: The stress \(\sigma_{33}\) of the two models: (**a**) UMAT; (**b**) M-C.  
Figure 13: The plastic strain \(\varepsilon_{33}^{p}\) of the two models: (**a**) UMAT; (**b**) M-C.  
plastic strain nephogram obtained using the two models also have good consistency, and the plastic strain calculated using the UMAT subroutine is slightly larger. The difference between the calculated results is mainly because the plastic potential function of the M-C model embedded in ABAQUS is a continuous and smooth elliptic function, which is not consistent with the yield function. Nevertheless, the accuracies of both the stress and strain meet the calculation requirements. ### Uniaxial Compression  
To check the UMAT subroutine's accuracy in the multi-element calculation and practical application, an ideal underground cavern under hydrostatic pressure is used as an example. The subroutine's numerical solution is compared with the classical Fenner formula's rigorous solution for analyzing such problems. The tunnel has a 10.0 m radius and is 90.0 m below ground. The model adopts a C3D8 solid element. Due to the structure's symmetry, 1/4 of the model is calculated and a force of 30.0 MPa is applied in the x and y directions. Table 4 displays the surrounding rock-related parameters, and Figure 14 displays the computational model. Under ideal elastoplasticity, the internal stress of the surrounding rock made of M-C material can be calculated using the conventional Fenner formula [56]. The expression is as follows. The plastic zone's stress expression:  
\[\sigma_{r}^{p}=c\cdot\cot\phi\cdot\left[\left(\frac{r}{R_{0}}\right)^{\frac{2 \sin\phi}{1-\sin\phi}}-1\right] \tag{45}\]  
\[\sigma_{\theta}^{p}=c\cdot\cot\phi\cdot\left[\frac{1+\sin\phi}{1-\sin\phi} \bigg{(}\frac{r}{R_{0}}\bigg{)}^{\frac{2\sin\phi}{1-\sin\phi}}-1\right] \tag{46}\]  
\[R_{1}=R_{0}\bigg{[}\left(\frac{(p_{0}+c\cdot\cot\phi)(1-\sin\phi)}{c\cdot\cot \phi}\right)\bigg{]}^{\frac{1-\sin\phi}{2\sin\phi}} \tag{47}\]