# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 8 Numerical Examples
### Modeling of Tunnel Excavation  
Some scholars [57; 58; 59] have carried out corresponding research on tunnel excavation. However, to simulate tunnel excavation, most of them are based on the M-C model. The GMC model, in contrast to the M-C model, considers the impact of \(\sigma_{2}\) and is able to fully utilize the material's strength. Studying the tunnel excavation problem using the GMC model is therefore practical.  
The model has dimensions of 20.0 m in length, width, and height, and the radius of the initial tunnel is 1.0m, which is drilled in a formation with in situ stresses \(\sigma_{V}=\) 15.0 MPa, \(\sigma_{H}=\) 30.0 MPa, and \(\sigma_{h}\) (varying between \(\sigma_{V}\) and \(\sigma_{H}\)). The M-C model is also used to mimic tunnel excavation as a comparison. The elastic mechanical parameters used are \(E=\) 30,000 MPa and \(\
u=\) 0.25, and the plastic parameters are shown in Table 5. A three-dimensional model was created in ABAQUS, as shown in Figure 16, to simulate the analysis as closely as possible. The model adopts the C3D8 solid element and divides into 13,640 units, with a total of 15,666 nodes.  
Figure 15: Comparison of stress distributions computed using analytical and UMAT solution: (a) Radial stress; (b) Circumferential stress.  
Since the M-C model does not reflect the influence of \(\sigma_{h}\), as is shown in Figure 17, the range of the plastic zones computed under various \(\sigma_{h}\) is essentially the same. According to Figure 18, the range of the surrounding rock's plastic zone increasingly shrinks for the GMC model as \(\sigma_{h}\) increases. This shows that the GMC model can accurately represent the intermediate principal stress's effect on the plastic zone, since the surrounding rock's strength continuously increases as the intermediate principal stress rises. Figure 19 shows the maximum equivalent plastic strain corresponding to different \(\sigma_{h}\). The M-C model's maximum equivalent plastic strain is almost unaltered for varied \(\sigma_{h}\) values, proving that it does not account for the intermediate principal stress's influence and that the calculation results are conservative. With an increase in \(\sigma_{h}\), the maximum equivalent plastic strain of the GMC model decreases, thus demonstrating that the GMC model accounts for the intermediate principal stress's effect. Most of the geotechnical materials used in practical engineering are in the three-dimensional stress state, which has a specific enhancing effect on the material's strength. The GMC model may thus fully utilize the material's strength potential while saving on engineering costs, which has some significance for practical engineering issues.  
Figure 19: Maximum equivalent plastic strain under different \(\sigma_{h}\).  
Figure 20 shows surrounding rock's maximum stresses under different \(\sigma_{h}\). The maximum radial stress and maximum circumferential stress of the surrounding rock using the M-C model essentially remain unchanged as \(\sigma_{h}\) increases, whereas the maximum radial stress obtained using the GMC model decreases as \(\sigma_{h}\) increases and the maximum circumferential stress barely changes. Under different \(\sigma_{h}\), the maximum radial stress obtained using the GMC model is lower than the maximum radial stress using the M-C model, while the maximum circumferential stresses obtained using the two models are opposite. As a result, the GMC model can consider how the intermediate principal stress affects the surrounding rock stress. The GMC model's outcomes appear more credible when contrasted with those of the M-C model.