## 3 Novel Creep Constitutive Model for the Host Rocks of the Weilai Tunnel
### Development of the Proposed Creep Constitutive Model
#### 3.3.1 Identification of Model Parameters  
Designed to depict the creep behavior of soft rocks, the novel creep model takes into account several parameters such as elastic, plastic, and viscous. The shear modulus \(G_{1}\) can be assessed as below:  
\[G_{1}=\frac{E}{2(1+v)} \tag{26}\]  
where \(v\) denotes poisson's ratio; and E elastic modulus. To evaluate \(G_{p}\), the experimental data related to instant plastic creep strains are employed. Concerning the determination of \(G_{2}\) and \(\alpha\), Equation (32) is used. Note that, under the conditions where time \(t\) is infinitely large, and \(\sigma<\sigma_{s}\), the aforesaid equations can be modified as follows:  
\[\varepsilon=\frac{\sigma_{1}-\sigma_{3}}{3G_{1}}+\frac{\sigma_{1}-\sigma_{3}}{3 G_{2}} \tag{27}\]  
Under the aforementioned conditions, adequate operations are utilized. By denoting by \"h\", the subtraction of Equation (24) into Equation (27), we can have the following:  
\[\left\{\begin{array}{c}h=\text{{Equation (\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq  
Figure 8: Model curve compared to experimental data for sandstone under a deviatoric stress of 14.99 MPa. Figure 9: Model curve compared to experimental data for sandstone under a deviatoric stress of 17.99 MPa. As displayed in Figures 7-11, both in the loading stage and in the unloading stage, the model is in good agreement with the experimental data. The proposed model can thus well simulate the creep behavior of the studied tunnel. It can be remarked that there is a huge difference between the values obtained from the loading stage and those obtained from the unloading stage. In fact, the unloading process has considerable effects on the mechanical properties of soft rocks. Since cracks generally tend to evolve rapidly in unloaded rocks, the relevant rock mechanical properties are affected by the unloading phase, and then altered to some extent. It should be noted that, generally, soft rocks have lesser abilities to withstand deformations generated by the unloading stage than those caused by the loading stage. This is why rock rupture can be faster during the unloading process than during the loading process. Thereby, when studying the long-term safety and stability of deeply buried tunnels, the cyclic loading-unloading process of the host rocks plays a key role. There is a rapid evolution of the creep process of sandstone under the deviatoric stress of 23.98 MPa, as revealed in Figure 11. As the stress level is high, there is a rapid appearance of the loading process. The loading process is not a good approximation of the loading process. Figure 11: Model curve compared to experimental data for sandstone under a deviatoric stress of 23.98 MPa. Figure 10: Model curve compared to experimental data for sandstone under a deviatoric stress of 20.97 MPa. of tertiary creep. In fact, high deviatoric stress is capable of shortening the duration of steady creep, and hastening the triggering of tertiary creep. At a high stress level, the rapid onset of tertiary creep will seriously jeopardize the long-term stability of the tunnel. Appropriate measures are quickly needed to ensure the longest duration of steady creep in deep rock tunnels subject to high geostresses. Aiming at expanding the validation of the proposed model, one analyses the effects of rock damage on viscoplastic strain. The variation of rock damage is also investigated. Here, the viscoplastic stage which represents the tertiary creep stage, is selected for that, as it is new with respect to the more familiar models (nonlinear Kelvin, St. Venant, and Hooke). It is well known that the damage variable in rock varies from 0 to 1. The nondamaged or intact rocks have a damage index of 0, while the damage index of the extremely damaged rocks is 1.