## 4 Closed-Form Solutions and Convergence Deformation of the Weilai Tunnel
\begin{array}{c}\sigma_{1}=\sigma_{\theta}\\ \sigma_{3}=\sigma_{r}\end{array} \right. \tag{47}\]  
By replacing Equation (42) into Equation (22), and by overlooking the viscoplastic part, and by taking into consideration Equations (46) and (47), the viscoelastic deformation of the host rocks can written as:  
\[u_{r}^{\text{\tiny{pc}}}=\frac{2R_{0}^{2}N_{1}e^{2M}}{3G_{1}r}+\frac{2R_{0}^{2 }N_{1}e^{2M}}{3G_{2}r}\Big{[}1-exp\Big{(}-\frac{G_{2}}{\eta_{2}\alpha}t^{\alpha }\Big{)}\Big{]} \tag{48}\]  
Here \(u_{r}^{\text{\tiny{pc}}}\) represents the creep deformation of the viscoelastic zone. Concerning the visco-plastic zones of the tunnel, a nonassociated plastic flow law is considered, since the dilation undoubtedly occurs in the plastic zone. As stated by [PERSON] et al. [49], viscoplastic strains and displacements can be therefore linked as below:  
\[\frac{du_{r}^{vp}}{dr}+k_{\phi}\frac{u_{r}^{vp}}{r}=\varepsilon_{r}+k_{\phi} \varepsilon_{\theta} \tag{49}\]  
Here the visco-plastic deformation of the surrounding rocks of a tunnel is noted by \(u_{r}^{vp}\); \(\varepsilon_{r}\) is radial strain; \(\varepsilon_{\theta}\) is hoop strain; the coefficient of dilation is indicated by \(k_{\phi}\), and can be given as follows:  
\[k_{\phi}=\frac{1+sin\phi}{1-sin\phi} \tag{50}\]  
A combination of Equations (22), (43), (47) and (49) leads to the following:  
\[\frac{du_{r}^{vp}}{dr}+k_{\phi}\frac{u_{r}^{vp}}{r}=\left[\frac{1}{3G_{1}}+ \frac{1}{G_{p}}+\frac{1}{3G_{2}}\bigg{(}1-exp\bigg{(}-\frac{G_{2}}{\eta_{2} \alpha}t^{\alpha}\bigg{)}\bigg{)}+k\bigg{(}\frac{t}{\eta_{3}}\bigg{)}^{n}(1-D )^{(1-n)}\right]\bigg{(}S_{r}^{p}+k_{\phi}S_{\theta}^{p}\bigg{)}-\big{(}1+k_{ \phi}\big{)}\frac{S_{s}}{G_{p}} \tag{51}\]  
The three-dimensional forms of \(\sigma_{r}^{p}\) and \(\sigma_{\theta}^{p}\), which are, respectively, \(S_{r}^{p}\) and \(S_{\theta}^{p}\), can be written as follows:  
\[\left\{ \begin{aligned} S_{r}^{p}&=-\frac{(3+sin\phi) }{12}m\sigma_{c}ln\frac{r}{K_{0}}-\frac{(3+sin\phi)}{6}\sigma_{c}\sqrt{s}\\ S_{\theta}^{p}&=\frac{(3-sin\phi)}{12}m\sigma_{c}ln \frac{r}{K_{0}}+\frac{(3-sin\phi)}{6}\sigma_{c}\sqrt{s}\end{aligned} \right. \tag{52}\]  
From Equation (51), a function \(g(t)\) can be defined as below:  
\[g(t)=\left[\frac{1}{3G_{1}}+\frac{1}{G_{p}}+\frac{1}{3G_{2}}\bigg{(}1-exp \bigg{(}-\frac{G_{2}}{\eta_{2}\alpha}t^{\alpha}\bigg{)}\bigg{)}+k\bigg{(} \frac{t}{\eta_{3}}\bigg{)}^{n}(1-D)^{(1-n)}\right] \tag{53}\]  
There is compatibility of deformations at the existing interface between the visco-plastic and visco-elastic zones. This condition is taken into account to evaluate the global visco-plastic deformation of the host rocks. At the aforesaid interface, this condition can be written as follows:  
\[\left\{ \begin{aligned} r=R_{P}\\ u_{r}^{vp}(R_{P},\ t)=u_{r}^{vp}(R_{P},\ t)\end{aligned} \right. \tag{54}\]  
Subsequently, we can have the following:  
\[u_{r}^{vp}(R_{P},\ t)=u_{r}^{ve}(R_{P},\ t)=\frac{2R_{0}^{2}N_{1}e^{2M}}{3G_{1 }R_{P}}+\frac{2R_{0}^{2}N_{1}e^{2M}}{3G_{2}R_{P}}\bigg{[}1-exp\bigg{(}-\frac{G _{2}}{\eta_{2}\alpha}t^{\alpha}\bigg{)}\bigg{]} \tag{55}\]  
The global visco-plastic deformation of the surrounding rocks (dry states) of the tunnel is assessed by combining Equations (51) and (55). Its expression is given by:  
\[ \begin{aligned} u_{r}^{vp}(t)=\frac{m\sigma_{r}}{12}g(t)\big{[}k_{ \phi}(3-sin\phi)-(3+sin\phi)\big{]}g_{1}(r)+\frac{\sigma_{r}\sqrt{s}}{6}g(t) \big{[}k_{\phi}(3-sin\phi)-(3+sin\phi)\big{]}g_{2}(r)\\ +\big{(}1+k_{\phi}\big{)}\frac{S_{s}}{G_{p}}g_{2}(r)+\frac{2R_{0}^{2 }N_{1}e^{2M}}{3G_{1}R_{P}}+\frac{2R_{0}^{2}N_{1}e^{2M}}{3G_{2}R_{P}}\big{[}1-exp \Big{(}-\frac{G_{2}}{\eta_{2}\alpha}t^{\alpha}\Big{)}\Big{]}g_{3}(r)\end{aligned}  \tag{56}\]  
The subfunctions \(g_{1}(r)\), \(g_{2}(r)\), and \(g_{3}(r)\) are defined as follows:  
\[\left\{