## 3 Plastic Zone Expansion Law
### Theoretical Calculation of the Plastic Zone
#### 3.2.1 Distribution Pattern and Expansion Law of Tunnel Plastic Zone under High Geostress Effect
In this case, the radius of the plastic zone expands from 7.86 m to 14.13 m. Conversely, at a temperature of 20 \({}^{\circ}\)C, the plastic zone assumes a standard vertical butterfly shape with \(\lambda\) at 2.5, distinctly different from the transverse butterfly shape observed under other conditions. 2. Variations in the lateral pressure coefficient induce changes in the fundamental shape of the plastic zone, characterized by both elliptical and butterfly forms, ex  
Figure 3: Morphology evolution of tunnel plastic zone when lateral pressure coefficient changes. hibiting a similar expansion pattern as described in Section 3.2.1. Within the range of \(1.0\leq\lambda\leq 2.0\), the plastic zone assumes an elliptical shape, particularly evident at \(\lambda=1.0\), where it transitions from a circular to an irregularly expanded ellipse. As \(\lambda\) diverges further from 1, influenced by the profound stress environment, the plastic zone exhibits uneven expansion, narrowing horizontally while widening vertically along the longitudinal axis, thereby increasing the extent of the plastic zone. When \(\lambda<1.0\) or \(\lambda>2.0\), the plastic zone adopts an elliptical and butterfly-shaped configuration, mirroring the expansion law outlined in Section 3.2.1. At \(\lambda=1.0\) or \(\lambda>2.0\), the plastic zone extends towards the tunnel's corners, resembling a complete butterfly shape, with a notable expansion in range. As \(\lambda\) increases to 2.8, the plastic zone takes on a double funnel shape, or transverse butterfly shape, with a substantial increase in range; however, its butterfly shape remains unchanged. 2.3 The Impact of Rock Thermal Stress on the Expansion Law of the Plastic Zone in High Geothermal Temperature and High Geostress Tunnels  
The parameters are set as outlined in Table 2 to elucidate the impact of rock thermal stress induced by high geothermal conditions on the expansion pattern of the plastic zone in high geostress tunnels. Considering that 80 \({}^{\circ}\)C represents the typical temperature for the majority of high geothermal rock excavations, we analyzed the plastic zone morphology across various \(\lambda\) values using Equation (4) (Figure 5). Undoubtedly, rock thermal stress moderately retards the morphological development of the plastic zone, resulting in a slightly slower expansion compared to scenarios where rock thermal stress is not considered. Compared to Section 3.2.1, under the same \(\lambda=2\), the plastic zone morphology remains unaltered, retaining its non-butterfly shape with a limited range. For \(\lambda\leq 2\), there is no significant change in the plastic zone morphology, resulting in a notably slower expansion trend. However, for \(\lambda>2\), the morphology and extent of the plastic zone exhibit heightened sensitivity to variations in the lateral pressure coefficient, expanding rapidly outward to form significant lobes. This trend becomes particularly pronounced after \(\lambda\) exceeds 2.8, with the butterfly lobe spreading noticeably at polar angles \(\theta=25\)\(\sim\)90\({}^{\circ}\). Figure 6 illustrates that when considering the thermal stress of the rock mass, the maximum radius of the plastic zone increases with larger values of \(\lambda>1\). The change  
Figure 4: Morphology evolution of tunnel plastic zone when thermal stress of rock mass changes. in the radius of the plastic zone is especially pronounced in the surrounding rock with \(\lambda>2\), whereas the change in the radius is less significant for other values of \(\lambda\). For instance, at \(\lambda=2.8\), the radius of the plastic zone exhibits a curvilinear increase in the polar angle \(\theta=0\)-\(55^{\circ}\), with a rapid initial increase from 5.17 m to 7.86 m, marking a 34.22% rise, followed by a gradual decrease to 7.4 m, reflecting a more stable decline of 5.85%.