### Calculation Method of Tunnel Wall Displacement Based on the H-B Criterion  
For the tunnel with an excavation radius \(R_{0}\), the simplified mechanical model after tunnel excavation is presented in Figure 1. A micro-element ABCD is taken in the plastic zone, as shown in Figure 2. Assume that the radial stress in the rock mass is \(\sigma_{\rm r}\) and the hoop stress is \(\sigma_{\rm\Theta}\). Based on the continuum elastoplastic theory and the H-B strength criterion, at the elastoplastic interface, according to the continuity assumption of equal stress, that is, \(\sigma_{\rm r}^{\rm e}=\sigma_{\rm r}^{\rm p}\) and \(\sigma_{\rm\theta}^{\rm e}=\sigma_{\rm\theta}^{\rm p}\), the following results [25; 26] are obtained:  
\[\left\{ \begin{array}{l}\sigma_{\rm r}^{\rm p}|_{r=R_{\rm p}}=\frac{\sigma_{ \rm d}}{m_{\rm b}}[m_{\rm b}(1-a)\cdot\ln\frac{R_{\rm p}}{R_{0}}+(\frac{m_{\rm b }p_{\rm i}}{\sigma_{\rm d}}+s)^{1-a}]^{\frac{1}{1-a}}-\frac{\sigma_{\rm d}s}{m _{\rm b}}\\ =\sigma_{\rm r}^{\rm e}|_{r=R_{\rm p}}=\sigma_{R_{\rm p}}\\ \sigma_{\rm\theta}^{\rm p}|_{r=R_{\rm p}}=\frac{\sigma_{\rm d}}{m_{\rm b}}[m_{ \rm b}(1-a)\cdot\ln\frac{R_{\rm p}}{R_{0}}+(\frac{m_{\rm b}p_{\rm i}}{\sigma_{ \rm d}}+s)^{1-a}]^{\frac{1}{1-a}}-\frac{\sigma_{\rm d}s}{m_{\rm b}}\\ +\sigma_{\rm ci}[m_{\rm b}(1-a)\cdot\ln\frac{R_{\rm p}}{R_{0}}+(\frac{m_{\rm b }p_{\rm i}}{\sigma_{\rm ci}}+s)^{1-a}]^{\frac{1}{1-a}}=\sigma_{\rm\theta}^{\rm e }|_{r=R_{\rm p}}=2p_{0}-\sigma_{R_{\rm p}}\end{array} \right. \tag{10}\]  
where \(p_{0}\) is the initial stress of surrounding rock; \(\sigma_{\rm R_{\rm p}}\) is the radial stress of the elastic-plastic boundary; and \(R_{\rm p}\) is the radius of the plastic zone. Figure 1: Excavation model of a circular cavern. Figure 2: Micro-unit in plastic zone. The analytical equation of plastic zone radius can be obtained by simplifying the above equations:  
\[q^{a}+\frac{2}{m_{\rm b}}q-\frac{2p_{0}}{\sigma_{\rm ci}}-\frac{2s}{m_{\rm b}}=0, \tag{11}\]  
where  
\[q=w^{\frac{1}{1-a}}, \tag{12}\]  
\[w=m_{\rm b}(1-a)\cdot\ln\frac{R_{\rm P}}{R_{0}}+(\frac{m_{\rm b}p_{\rm i}}{ \sigma_{\rm ci}}+s)^{1-a}. \tag{13}\]  
The radius \(R_{\rm P}\) of the plastic zone can be determined by solving Equations (11)-(13) simultaneously. It can be seen that the plastic zone range is not only related to the internal support pressure \(p_{\rm i}\), but also related to the H-B strength criterion parameters. Equation (11) is a transcendental equation, so it is very difficult to obtain the analytical solution of the plastic zone radius. The numerical solution of the plastic zone radius \(R_{\rm P}\) is obtained by substituting engineering data into Equation (11) through the univariate solution block in MS Office Excel 2016 software. The ideal elastoplastic theory assumes that the displacement on the interface is continuous. Therefore, the displacement \(u_{\rm P}\) on the elastoplastic interface can be determined according to the radial displacement formula of the boundary in the elastic zone:  
\[u_{\rm P}=\frac{(1+\mu)R_{\rm P}}{E}(p_{0}-\sigma_{\rm R}). \tag{14}\]  
Substituting the two equations from Equation (10) into Equation (14), the displacement equation on the elastic-plastic interface can be derived:  
\[u_{\rm P}=\frac{(1+\mu)R_{\rm P}\sigma_{\rm ci}}{2E}[m_{\rm b}(1-a)\cdot\ln \frac{R_{\rm P}}{R_{0}}+(\frac{m_{\rm b}p_{\rm i}}{\sigma_{\rm ci}}+s)^{1-a}] ^{\frac{\mu}{1-a}}. \tag{15}\]  
During the deformation process of the plastic zone (i.e., the fractured zone) of the surrounding rock of the cavern, there is usually a dilatancy phenomenon.