## 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  
Assuming that the temperature remains constant both inside and outside the surrounding rock after tunnel excavation, without the presence of support lining, and considering temperatures below 400 degrees, the influence of high-temperature thermal stress on the strength of the surrounding rock is deemed negligible [17]. Consequently, it is posited here that temperature has no impact on the mechanical parameters of the rock. To elucidate the impact of the high geostress environment on the expansion law of the tunnel's plastic zone, parameters identical to those in Table 2 are utilized. Utilizing Equation (4), and given that the surrounding rock of the tunnel is subjected to high geostress, and cannot exhibit plastic behavior before tunnel excavation, the plastic zone's shape and the expansion characteristics of the high geostress tunnel are elucidated by varying the lateral pressure coefficient (\(0.3\leq\lambda\leq 2.8\)) to alter the stress state of the surrounding rock (Figure 3). Figure 3 illustrates the close relationship between the shape of the surrounding rock plastic zone and the lateral pressure coefficient. Under varying ground stress conditions, the plastic zone exhibits three primary shapes: circular, elliptical, and butterfly. As \(\lambda\) gradually increases from 0.3 to 2.8, the plastic zone expands through a sequence: vertical butterfly \(\rightarrow\) circular \(\rightarrow\) vertical ellipse \(\rightarrow\) vertical butterfly \(\rightarrow\) horizontal butterfly. The maximum radius of the plastic zone experiences rapid growth with \(\lambda\). However, when the plastic zone is circular or elliptical, this growth is slow and linear. Conversely, for a butterfly-shaped plastic zone, the maximum radius increases exponentially with \(\lambda\). These observations suggest that the expansion rate of the plastic zone varies depending on its shape and the lateral pressure coefficient. The morphology and expansion pattern of the plastic zone in high-stress tunnels, as determined in this study, align with findings reported in the literature [18,19]. Both studies suggest that the lateral pressure coefficient increases, and the plastic zone transitions from a circular shape to an ellipsoid before ultimately adopting a butterfly configuration. The discrepancy in the extent of the plastic zone between the two studies is relatively minor, indicating strong consistency and supporting the applicability of the derived theoretical formulas. 2.2 Impact of Rock Thermal Stress on the Distribution Pattern of Plastic Zones in Tunnels Subject to High Geothermal Temperatures and High Geostresses  
To explore the impact of thermal stress on the plastic zone, we generated plots depicting the morphology of the plastic zone at various temperatures and lateral pressure coefficients using Equation (4) (Figure 4). The findings indicate that in conditions of high geothermal temperature and high geostress, the plastic zone typically takes two primary shapes: elliptical and butterfly. The formation of a uniformly circular plastic zone is hindered by the influence of thermal stress on the rock mass. To succinctly explain the expansion patterns of the plastic zone in tunnels subjected to high geothermal temperature and high geostress, we conducted separate analyses based on the lateral pressure coefficient and temperature. 1. As the temperature gradually rises, so does the thermal stress within the rock body. Consequently, the geometry of the plastic zone undergoes significant changes from an ellipse to a butterfly shape. During this process, both the radius and extent of the plastic zone increase, with the rate of expansion progressively accelerating. Particularly noteworthy is the scenario where T reaches 80 \({}^{\circ}\)C and \(\lambda\) equals 2.8, resulting in a noticeable expansion of the plastic zone.