## 3 Novel Creep Constitutive Model for the Host Rocks of the Weilai Tunnel
### Development of the Proposed Creep Constitutive Model
#### 3.3.3 Effect of Pore Water Pressure: Second Variant of the Proposed Model  
The influence of pore water pressure cannot be overlooked, since the predominant rock type of the studied tunnel is broken, and groundwater is abundant in the concerned media. Seepage actions can be severe in this rocky environment. They can generally reduce the bearing capacity of the support structure by modifying the stress allotments around the overall tunnel components [43]. Thereby, it is of tremendous importance to take into account the influence of pore water pressure on the creep process. To this end, damage is particularly accounted for in tertiary creep. It is well known that damage augments as the tertiary creep stage evolves. Therefore, in order to account for the effects of pore water pressure on rock creep, the damage variable is employed. In this situation, the damage variable can be read as below, where \(\beta\left(\sigma,\ p\right)\) is a material constant which is affected by pore water pressure (\(p\)) and stress (\(\sigma\)):  
\[D=1-e^{-\beta\ \left(\sigma,\ p\right)t} \tag{38}\]  
A variant of the mechanical model which is affected by pore water pressure is presented in Figure 14.  
The jointed influence of pore water pressure and confining pressure is taken into account. This influence can be written, for the visco-plastic element, as follows:  
\[\eta_{3}(\ \sigma\,\ p,\ D)=\eta_{3}(\ \sigma\,\ p)(1-D) \tag{39}\]  
Equations (22), (38) and (39) are adequately combined. The three-dimensional form of the novel creep model with effects of pore water pressure can be written as:  
\[\epsilon_{ij}\ (t,\ \sigma,\ p) \tag{40}\] \[=\left\{ \begin{array}{l}\frac{1}{2G_{1}}\left(\sigma,\ p\right) S_{ij}+\frac{1}{2G_{2}\left(\sigma,\ p\right)}\Big{[}1-exp\Big{(}-\frac{G_{2} \left(\sigma,\ p\right)}{\eta_{2}\left(\sigma,\ p\right)\mu}t^{\alpha}\Big{)} \Big{]}S_{ij}\ \big{(}S_{ij}<S_{s}\big{)}\\ \frac{1}{2G_{1}}\left(\sigma,\ p\right)S_{ij}+\frac{S_{ij}-S_{s}}{G_{p}\left( \sigma,\ p\right)}+\frac{1}{2G_{2}\left(\sigma,\ p\right)}\Big{[}1-exp\Big{(} -\frac{G_{2}\left(\sigma,\ p\right)}{\eta_{2}\left(\sigma,\ p\right)\mu}t^{ \alpha}\Big{)}\Big{]}S_{ij}+kS_{ij}\Big{(}\frac{t}{\eta_{3}\left(\sigma,\ p \right)}\Big{)}^{n}\Big{(}e^{-\beta\ \left(\sigma,\ p\right)t}\Big{)}^{\left(1-n\right)}\ \big{(}S_{ij}\geq S_{s}\big{)}\end{array} \right.\]  
For the maximum stress direction, the model affected by pore pressure can be read as follows:  
\[\epsilon_{11}(t,\ \sigma,\ p)\] (41) \[=\left  
Taking into consideration the variation trend in the mechanical properties of the soft rocks subjected to pore water pressure change [44], the parameters of the model are scrupulously evaluated for three levels (0.05 MPa, 0.10 MPa, 0.15 MPa) of pore water pressure. Tables 6 and 7 show the parameters. Under loading and unloading conditions, Figures 15-17 present the effects of pore water pressure on the evolution of rock creep. Figures 15-17 clearly reveal that pore water pressure seriously affects the creep process of the rocks surrounding the studied tunnel. The creep strain increases, and the creep life of the rock decreases when pore water pressure increases. In fact, the higher the pore pressure, the faster the creep process evolves. Pore water pressure shortens the duration of both loading and unloading processes. It is one of the major factors causing considerable instability in deep rock engineering. It can even provoke unexpected failure of deep soft rocks subjected to elevated stress, as can be interpreted in Figure 17.