## 3 Novel Creep Constitutive Model for the Host Rocks of the Weilai Tunnel
### Development of the Proposed Creep Constitutive Model  
To simulate the instant elastic strain, the Hooke element is taken into account. Thereby, the axial instantaneous elastic strain can be read as follows [32; 34]:  
\[\varepsilon_{me}=\frac{\sigma}{E_{1}} \tag{2}\]  
Here the applied stress is represented by \(\sigma\), and the elastic modulus by \(E_{1}\). Regarding the axial instant plastic strain, referring to [PERSON] and [PERSON] [35], it can be formulated by considering the modulus of the plastic deformation of the St. Venant element, \(E_{p}\), and the scalar function, \(I(\sigma_{1}-\sigma_{3}-\sigma_{s})\), as follows:  
\[\varepsilon_{mp}= \frac{\sigma-\sigma_{s}}{E_{p}}\ I(\sigma-\sigma_{s}) \tag{3}\]  
The scalar function is defined as below:  
\[I(\sigma-\sigma_{s})=\left\{ \begin{array}{ll}0,\ \sigma<\sigma_{s}\\ 1,\ \sigma\geq\sigma_{s}\end{array} \right. \tag{4}\]  
Here \(\sigma_{s}\) represents the long-term strength of rocks. Note that the long-term strength of rocks is of particular importance. In fact, due to aging and fatigue, rock strength diminishes with the passage of time [36]. In general, the instant rock strength is typically greater than  
Figure 6: Mechanical diagram of the proposed visco-elasto-plastic creep model. the long-term strength. For soft rocks, the long-term strength ranges from 70 to 80% of the instant value [36; 37]. Concerning the visco-elastic element, it can be simulated by the nonlinear Kelvin shape [38]. Assuming a constant \(\alpha\) (material constant) can well reflect the inevitable influence of the stress level on rock creep, and the nonlinear variant of the Kelvin model can be written in its differential constitutive equations as follows:  
\[\sigma=E_{2}\varepsilon_{\mathit{ve}}+\eta_{2}t^{(1-\alpha)}\dot{\varepsilon} _{\mathit{ve}} \tag{5}\]  
Here \(\sigma\) is the applied stress, \(E_{2}\) stands for the elastic modulus of the Kelvin shape, \(\eta_{2}\) represents viscous coefficient of the Kelvin shape, and \(t\) is the creep time. The integration of the previous equation can lead to the following:  
\[\varepsilon_{\mathit{ve}}=\frac{\sigma}{E_{2}}\bigg{[}1-\exp\bigg{(}-\frac{E_ {2}}{\eta_{2}\alpha}t^{\alpha}\bigg{)}\bigg{]} \tag{6}\]  
For the viscoplastic body, it can be described by improving the Ramberg-Osgood equation. Indeed, to improve the Ramberg-Osgood equation [39], in particular its plastic component, one considers the basic definition of rock creep where strain evolves with time, and under constant stress. Thereby, the visco-plastic creep strain model of soft rock can be written as follows:  
\[\varepsilon_{\mathit{vp}}=k\sigma\bigg{(}\frac{t}{\eta_{3}(t)}\bigg{)}^{n} \tag{7}\]  
where \(k\) and \(n\) represent constants related to rock materials; \(\sigma\) denotes the applied stress; \(t\) is creep time; and \(\eta_{3}\) stands for coefficient of the initial viscosity of the viscoplastic body. The variable damage (\(D\)) is introduced, since it is of primary consideration at this stage. It should be noted that, under sustained effects of deviatoric stress, the evolution of damage in rock materials is unavoidable, and one of the parameters which is affected by damage is the viscosity coefficient. Hence, the damaged viscosity coefficient can be expressed as below:  
\[\eta_{3}(t)=\eta_{3}(1-D) \tag{8}\]  
Combining Equations (7) and (8), the visco-plastic creep strain can be expressed as follows:  
\[\varepsilon_{\mathit{vp}}=k\sigma\bigg{(}\frac{t}{\eta_{3}}\bigg{)}^{n}(1-D)^{ (1-n)} \tag{9}\]  
At the tertiary creep stage, as related by [40; 41], the evolution of damage in rock materials can be widely expressed as follows:  
\[\frac{D}{dt}=A\ \bigg{(}\frac{\sigma}{1-D}\bigg{)}^{n} \tag{10}\]  
Here \(A\) and \(n\) are constants related to the materials. Under the evolution of time, critical or full-scope damage generally cause rock failure. Therefore, it is important to take into account the time-to-failure.