## 2 Description of the Problem
### Strain-Softening Model  
The elastoplastic deformation of surrounding rock can be deduced in accordance with the theory of plastic mechanics by calculating the weakening behavior of the strength parameters [28]. If the D-P criterion satisfies the yield function in the strain-softening zone, the expression of the yield function has the following form:  
\[F(\sigma_{\theta},\sigma_{r},\eta^{p})=\sigma_{\theta}-\sigma_{r}-H(\sigma_{r },\eta^{p}) \tag{11}\]  
In Equation (11), \(H(\sigma_{r},\eta^{p})=(\mathrm{N}(\eta)-1)\sigma_{r}+Y(\eta)\), whereas the most extensively accepted deviatoric plastic strain \(\eta^{p}\), regarded as the strain-softening parameter, can be determined by the difference between the circumferential plastic strain \(\varepsilon_{\theta}^{p}\) and radial plastic strain \(\varepsilon_{r}^{p}\). The expression can be defined as [24]:  
\[\eta^{p}=\varepsilon_{\theta}^{p}-\varepsilon_{r}^{p} \tag{12}\]In Equation (12), \(\varepsilon_{\theta}^{p}\) and \(\varepsilon_{r}^{p}\) represent the major principal plastic strain and minor principal plastic strain of the surrounding rock, respectively. In accordance with the theory of plastic mechanics, the widely used Mohr-Coulomb plastic potential function \(G\) could be written as:  
\[G=\sigma_{\theta}-K_{\psi}\sigma_{r} \tag{13}\]  
A correlation exists between the plastic strain of surrounding rock and the plastic potential function \(G\). Following the theory of plastic potential function, the following relationship can be deduced:  
\[\left\{ \begin{array}{c}d\varepsilon_{r}^{p}=d\lambda\frac{\partial G}{ \partial\sigma_{r}}=-K_{\psi}d\lambda\\ d\varepsilon_{\theta}^{p}=d\lambda\frac{\partial G}{\partial\sigma_{\theta}}= d\lambda\end{array} \right. \tag{14}\]  
In Equation (14), \(K_{\psi}\) is the dilatancy coefficient and \(\psi\) is the angle of expansion of the rock mass, which varies with the surrounding pressure and type of rock mass. The dilatancy behavior of rock surrounding a deep circular tunnel is affected by plastic shear deformation and confining pressure of the rock mass [9; 10; 13]. [PERSON] et al. [3] proposed that the dilatancy coefficient \(K_{\psi}\) and the deviatoric plastic strain \(\eta\) show an exponential decay relationship. Subsequently, [PERSON] [29] analyzed the variation in the dilatancy coefficient with deviatoric plastic strain \(\eta\) under different confining pressures through triaxial experiments and obtained the following fitting formula:  
\[K_{\psi}=1+\left(K_{\psi^{p}}-1\right)e^{-\frac{\eta}{\sigma^{2}}} \tag{15}\]  
where:  
\[K_{\psi^{p}}=\frac{1+sin\psi^{peak}}{1-sin\psi^{peak}}\]  
Through an analysis of previous studies and triaxial tests, [PERSON] [10] identified the relationship between the peak dilatancy angle \(\psi^{p}\), the confining pressures \(\sigma_{r}\), and the peak internal friction angle \(\varphi^{p}\) by fitting the test data as:  
\[\psi^{p}=\frac{\varphi^{p}}{1+lg\sigma_{c}}lg\frac{\sigma_{c}}{\sigma_{r}+0.1} \tag{16}\]  
In Equation (16), \(\sigma_{c}\) is uniaxial compressive strength and \(\sigma_{r}\) is the confining stress. The results of previous studies [23; 30; 31] indicated that the tunnel excavation procedure results in a progressive decline in confining stress from \(\sigma_{0}\) to \(p_{i}\). In Figure 2, point A indicates that the rock mass falls within the initial hydrostatic pressure field. The radial stress \(\sigma_{r}\) of surrounding rock decreases progressively from point A to point B during the tunnel excavation process. The critical radial stress \(p_{ir}\) of surrounding rock at point B of the elastic-plastic boundary can be obtained by theoretical calculation. The surrounding rock enters the plastic yield stage beyond the elasto-plastic boundary point B and the surrounding rock parameters experience approximately linear attenuation with increasing strain-softening parameter \(\eta\). The distance between point C and point D demonstrates that the surrounding rock enters the residual stage.