## 3 Analytical Model  
### Mechanical Analysis of Surrounding Rock  
1. [label=(0)]
2. Unlined tunnel
3. The expression of the fictitious support pressure [36] can be expressed as follows: \[\left\{ \begin{array}{l}p_{\mathrm{f}}(t)=[1-\lambda(t)]P_{0}\\ \lambda(t)=1-\alpha e^{-\beta t}\end{array} \right.\] (4) where \(\lambda(t)\) is the stress release coefficient; \(\alpha\) is the stress release rate; and \(\beta\) is a parameter positively related to the tunnelling rate. When \(\alpha\) or \(\beta\) are equal to zero or positive infinity, then \(\lambda(t)\) is equal to 1, indicating that all stresses induced by the tunnel excavation are released. This study assumes that the tunnel is excavated at time \(t=0\), and the lining is installed at time \(t=t_{0}\). Therefore, the boundary conditions of the unlined and lined tunnels (Figure 3) can be given as follows:  
\[\sigma_{r}^{\mathrm{R}}(t)\Big{|}_{r=+\infty}=P_{0} \tag{5}\]  
\[\sigma_{r}^{\mathrm{R}}(t)\Big{|}_{r=R_{2}}=\left\{ \begin{array}{l}P_{ \mathrm{f}}(t)t\leq t_{0}\\ P_{\mathrm{f}}(t)+Q(t)t>t_{0}\end{array} \right. \tag{6}\]  
where \(\sigma_{r}^{\mathrm{R}}(t)\) is the radial stress of the surrounding rock; and \(Q(t)\) is the supporting pressure provided by the lining. The stress distribution of the surrounding rock caused by a circular tunnel excavation under hydrostatic pressure has been studied by many researchers. Based on the boundary conditions of the unlined tunnel, the stress components of the surrounding rock can be directly given as follows:  
\[\left\{ \begin{array}{l}\sigma_{r}^{\mathbb{R}}(t)=\left[1-\lambda(t)\frac{R_{ 2}^{2}}{r^{2}}\right]P_{0}\\ \sigma_{\theta}^{\mathbb{R}}(t)=\left[1+\lambda(t)\frac{R_{2}^{2}}{r^{2}} \right]P_{0}\end{array} \right. \tag{7}\]  
where \(\sigma_{\theta}^{\mathbb{R}}(t)\) is the tangential stress of the surrounding rock. Taking into account the generalized [PERSON]'s law, the computational formula of the longitudinal strain of the surrounding rock can be written as follows:  
\[\varepsilon_{z}^{\mathbb{R}}=\frac{\sigma_{z}^{\mathbb{R}}-v^{\mathbb{R}}( \sigma_{r}^{\mathbb{R}}+\sigma_{\theta}^{\mathbb{R}})}{E^{\mathbb{R}}} \tag{8}\]  
where \(\sigma_{z}^{\mathbb{R}}(t)\) and \(\varepsilon_{z}^{\mathbb{R}}(t)\) are the longitudinal stress and strain of the surrounding rock, respectively; and \(E^{\mathbb{R}}\) and \(v^{\mathbb{R}}\) are the elastic modulus and the Poisson's ratio of the surrounding rock, respectively. The longitudinal strain of the surrounding rock is equal to zero (\(\varepsilon_{z}^{\mathbb{R}}(t)=0\)) under the state of plane strain [10]. Therefore, the longitudinal stress of the surrounding rock can be given as follows:  
\[\sigma_{z}^{\mathbb{R}}=v^{\mathbb{R}}(\sigma_{r}^{\mathbb{R}}+\sigma_{\theta} ^{\mathbb{R}})=2v^{\mathbb{R}}P_{0} \tag{9}\]  
Moreover, the mean stress of the surrounding rock can be calculated as follows:  
\[\sigma_{m}^{\mathbb{R}}=\frac{\sigma_{r}^{\mathbb{R}}(t)+\sigma_{\theta}^{ \mathbb{R}}(t)+\sigma_{z}^{\mathbb{R}}(t)}{3}=\frac{2(1+v^{\mathbb{R}})}{3}P_{0} \tag{10}\]  
Figure 3: Plan sketch of the mechanical analysis of surrounding rock for unlined and lined tunnels. Therefore, the increments of the deviatoric stress components of the surrounding rock caused by the tunnel excavation can be calculated as follows:  
\[\left\{ \begin{array}{l}\Delta s_{r}^{\mathrm{R}}(t)=\left[1-\lambda(t)\frac{R_{2 }^{2}}{r^{2}}\right]P_{0}-\sigma_{m}^{R}-(P_{0}-\sigma_{m}^{R})=-\lambda(t) \frac{R_{2}^{2}}{r^{2}}P_{0}\\ \Delta s_{\theta}^{\mathrm{R}}(t)=\left[1+\lambda(t)\frac{R_{2}^{2}}{r^{2}} \right]P_{0}-\sigma_{m}^{R}-(P_{0}-\sigma_{m}^{R})=\lambda(t)\frac{R_{2}^{2}}{ r^{2}}P_{0}\\ \Delta\sigma_{m}^{\mathrm{R}}=\sigma_{m}^{R}-\sigma_{m}^{R}=\frac{2(1+v^{ \mathrm{R}})P}{3}-\frac{2(1+v^{\mathrm{R}})P}{3}=0\end{array} \right.