## 3 Novel Creep Constitutive Model for the Host Rocks of the Weilai Tunnel
### Development of the Proposed Creep Constitutive Model
By solving Equation (10), and by considering that \(t=t_{f}\), where \(t_{f}\) represents the rock time-to-failure, we can obtain the following:  
\[ \begin{cases}(1-D)^{n+1}=1-A\sigma^{n}(1+n)t\\ t_{f}=\frac{1}{A(1+n)\sigma^{n}}\end{cases}  \tag{11}\]  
Afterwards, the combination of Equations (10) and (11) leads to the expression of the damage variable as below:  
\[D=1-\left(1-\frac{t}{t_{f}}\right)^{\frac{1}{1-n}} \tag{12}\]Combining Equations (9) and (12), the visco-plastic creep strain can be expressed as follows:  
\[\varepsilon_{pp}=k\sigma\left(\frac{t}{\eta_{3}}\right)^{n}\left(1-\left[1-\left( 1-\frac{t}{t_{f}}\right)^{\frac{1}{1-n}}\right]\right)^{\left(1-n\right)} \tag{13}\]  
In its one-dimension form, the novel creep model can be written as:  
\[\varepsilon\left(t\right)=\frac{\sigma}{E_{1}}+\frac{\sigma}{E_{2}}\left[1-exp \left(-\frac{E_{2}}{\eta_{2}\alpha}t^{\alpha}\right)\right]\sigma<\sigma_{s} \tag{14}\]  
\[\varepsilon\left(t\right)=\frac{\sigma}{E_{1}}+\frac{\sigma-\sigma_{s}}{E_{p}} +\frac{\sigma}{E_{2}}\left[1-exp\left(-\frac{E_{2}}{\eta_{2}\alpha}t^{\alpha} \right)\right]+k\sigma\left(\frac{t}{\eta_{3}}\right)^{n}\left(1-\left[1- \left(1-\frac{t}{t_{f}}\right)^{\frac{1}{1-n}}\right]\right)^{\left(1-n \right)}\sigma\geq\sigma_{s} \tag{15}\]  
The three-dimensional form of the proposed creep model can be built by considering that rocks are typically in three-dimensional stress states. One assumes that creep is mainly provoked by the deviatoric stress tensor \(\left(S_{ij}\right)\), damage is accounted at the tertiary creep phase, and the Poisson's ratio of the rocks is not affected by time. The stress tensor \(\left(\sigma_{ij}\right)\) is mainly composed of the deviatoric stress tensor \(\left(S_{ij}\right)\) and the spherical tensor \(\left(\sigma_{m}\right)\). On the basis of [PERSON]'s law, the deviatoric stress tensor \(\left(S_{ij}\right)\) and the spherical stress tensor \(\left(\sigma_{m}\right)\) can be expressed as follows:  
\[\left\{ \begin{array}{c}S_{ij}=\sigma_{ij}-\varepsilon_{m}\delta_{ij}\\ \sigma_{m}=\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})\end{array} \right. \tag{16}\]  
The shear or deviatoric strain tensor \(\left(\varepsilon_{ij}\right)\), and the volumetric or spherical strain invariant \(\left(\varepsilon_{m}\right)\) are expressed as follows, where \(\delta_{ij}\) stands for the Kronecker function:  
\[\left\{ \begin{array}{c}e_{ij}=\varepsilon_{ij}-\varepsilon_{m}\delta_{ij} \\ \varepsilon_{m}=\frac{1}{3}(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}) \end{array} \right. \tag{17}\]  
The total strain \(\left(\varepsilon_{ij}\right)\) in three-dimensional form can be given by superimposing its components, by assuming that rocks are continuous media, as follows:  
\[\varepsilon_{ij}=\varepsilon_{ij}^{me}+\varepsilon_{ij}^{mp}+\varepsilon_{ij} ^{ve}+\varepsilon_{ij}^{pp} \tag{18}\]  
where \(\varepsilon_{ij}^{me}\) and \(\varepsilon_{ij}^{ve}\) represent, respectively, the three-dimensional form of the instant elastic strain and the visco-elastic strain; and \(\varepsilon_{ij}^{mp}\) and \(\varepsilon_{ij}^{vp}\) are, respectively, the three-dimensional form of the instant plastic strain and visco-plastic strain. The instant elastic strain \(\left(\varepsilon_{ij}^{me}\right)\) and the visco-elastic strain \(\left(\varepsilon_{ij}^{ve}\right)\) can be three-dimensionally expressed as follows:  
\[\left\{ \begin{array}{c}e_{ij}^{me}=\frac{1}{2\mathcal{C}_{1}}S_{ij}\\ \varepsilon_{ij}^{ve}=\frac{1}{2\mathcal{C}_{2}}\left[1-exp\left(-\frac{G_{2} }{\eta_{2}\alpha}\right)t^{\alpha}\right]S_{ij}\end{array} \right. \tag{19}\]  
Here \(G_{1}\) is the shear modulus of the elastic element; \(S_{ij}\) represents the tensor of stress; \(G_{p}\) is the instant shear modulus; and \(S_{s}\) is the long-term strength of rock in three-dimensional form. Likewise, the instant plastic strain \(\left(\varepsilon_{ij}^{mp}\right)\) and visco-plastic strain \(\left(\varepsilon_{ij}^{vp}\right)\) can be three-dimensionally written as follows:  
\[\left\{