## 4 Closed-Form Solutions and Convergence Deformation of the Weilai Tunnel  
Reliable closed-form solutions are necessary to assess the time-dependent deformation of deep-buried tunnels, particularly those built in complex rocky media. The proposed creep constitutive with its two variants has to be converted into two closed-form solutions. To do this, a suitable tunnel mechanical model is adopted. As shown in Figure 20, the tunnel is subjected to a virgin in situ stress field (\(P_{0}\)). It is assimilated to a circular tunnel where the radius is \(R_{0}\). The rocks surrounding the tunnel deform elastically, viscoelastically, plastically, and viscoplastically. The distance between the center of the tunnel and the limit of the viscoplastic zone is denoted by \(R_{P}\). The actual cross section geometry has two parts, which comprise a semi-circular part (the upper part) and a rectangular part (the lower part). For calculation and analysis convenience, and referring to [PERSON] et al. [45], it is assumed that a circular section can represent the two sections. In the postexcavation phase, the actual state of the tunnel can be described by two main zones, namely, the internal and the external zones. The visco-plastic conditions are depicted by the internal zones, and the visco-elastic states by the external zones. Typically, the nonlinear failure characteristics of rocks can be described by the Hoek-Brown criterion,  
Figure 19: Influence of damage index on the surrounding rocks in wet states. Figure 20: Adopted mechanical model of the studied tunnel. where the concerned elastic and plastic parameters can be defined. Thereby, parameters in the elastic and plastic zones can be, respectively, given as follows [46; 47; 48]:  
\[\left\{ \begin{array}{c}\sigma_{r}^{e}=P_{0}-\frac{R_{0}^{2}}{r^{2}}N_{1}e^{2M} \\ \sigma_{\theta}^{e}=P_{0}+\frac{R_{0}^{2}}{r^{2}}N_{1}e^{2M}\\ \sigma_{z}^{e}=P_{0}\\ M=\frac{\sqrt{m^{2}+\frac{16m}{c_{c}}+16s-4\sqrt{5}-m}}{2m}\\ N_{1}=P_{0}-M\sigma_{c}\sqrt{s}-M^{2}\frac{m\sigma_{c}}{4}\end{array} \right. \tag{42}\]  
\[\left\{ \begin{array}{c}\sigma_{r}^{p}=\frac{m\sigma_{c}}{4}\Big{(}ln\frac{r} {R_{0}}\Big{)}^{2}+\sigma_{c}\sqrt{s}ln\Big{(}\frac{r}{R_{0}}\Big{)}\\ \sigma_{\theta}^{p}=\frac{m\sigma_{c}}{4}\Big{(}ln\frac{r}{R_{0}}\Big{)}^{2}+ \Big{(}\frac{m\sigma_{c}}{2}+\sigma_{c}\sqrt{s}\Big{)}ln\Big{(}\frac{r}{R_{0} }\Big{)}+\sigma_{c}\sqrt{s}\\ \sigma_{z}^{p}=\frac{(1+\sin\theta)\sigma_{\theta}^{p}+(1-sin\theta)\sigma_{ \theta}^{p}}{2}\\ R_{P}=R_{0}e^{M}\end{array} \right. \tag{43}\]  
Here the radial elastic stress, axial elastic stress, and hoop elastic stress are, respectively, noted by \(\sigma_{r}^{e}\), \(\sigma_{z}^{e}\), and \(\sigma_{\theta}^{e}\). For intact rocks, the unconfined compressive strength is represented by \(\sigma_{c}\). Likewise, the radial plastic stress, axial plastic stress, and hoop plastic stress are, respectively, noted by \(\sigma_{r}^{p}\), \(\sigma_{z}^{p}\), and \(\sigma_{\theta}^{p}\). The distance related to the central point of the section of the tunnel is noted by \(r\). The dilation angle is indicated by \(\phi\), and is given by:  
\[\phi=\frac{5 GSI-125}{1000}\varphi \tag{44}\]  
Here \(GSI\) is geological strength index, and \(\varphi\) denotes internal friction angle of the rock. The material parameters are \(m_{b}\) and \(s\), and can be determined as follows:  
\[\left\{ \begin{array}{c}m_{b}=m_{i}\text{exp}\Big{(}\frac{GSI-100}{28-14D} \Big{)}\\ s=exp\Big{(}\frac{GSI-100}{9-3D}\Big{)}\end{array} \right. \tag{45}\]  
where \(m_{i}\) stands for strength constant for geomaterials, and \(D\) represents a parameter related to the degree of rock mass perturbation. As already mentioned, \(D\in[0,\ 1]\) where 0 traduces undisturbed rock mass, and 1 is wholly disturbed rock mass. The strain and displacement are correlated. In the visco-elastic zone, this correlation can be given as below:  
\[\left\{ \begin{array}{c}\varepsilon_{\theta}=\frac{u}{r}\\ \varepsilon_{r}=\frac{du}{dr}\end{array} \right. \tag{46}\]  
Additionally, we can write:  
\[\left\{