# Creep Behavior of Rocks and Its Application to the Long-Term Stability of Deep Rock Tunnels
## 6 Mechanical Models and Creep Constitutive Models
### Constitutive Creep Models of Rocks  
Creep constitutive models are usually conceived to predict long-term deformations of rocks. They are roughly classified into two types such as empirical equations and differential creep equations [98]. The differential creep equations can be linear and nonlinear. Nonlinear models are generally established to overcome the limitations of common mechanical models which are characterized by linear components [79]. The empirical creep equations are generally in the forms of power-law, exponential, logarithmic, and are commonly derived from typical creep curves. Empirical equations can also be a mixture of the three mentioned forms. The equation reflecting the empirical law of rock creep is as follows [14; 31]:  
\[\varepsilon(t)=\varepsilon_{e}+\varepsilon_{cp}+\varepsilon_{cs}+\varepsilon_{ca} \tag{10}\]  
where \(\varepsilon(t)\) is the total creep strain; \(\varepsilon_{e}\) is instant elastic strain; \(\varepsilon_{cp}\) is ephemeral creep strain; \(\varepsilon_{cs}\) is stable creep strain; and \(\varepsilon_{ca}\) is creep strain at the final stage. With the need to outrival the basic limits of the empirical creep law [80], many creep constitutive models are developed to better represent the rock properties from combinations of mechanical models. Since rheological problems in rocks are complex and are caused by many factors, most scholars extend the one-dimensional rheological model to three-dimensional (3D) with the intention of better describing rock creep. Indeed, in 3D representations, the deviatoric stress tensor (\(S_{ij}\)) and the spherical stress tensor (\(\sigma_{m}\)) are the constituents of stress tensor (\(\sigma_{ij}\)). They are generally expressed as follows from [PERSON]'s law [32; 144]:  
\[ \begin{cases}S_{ij}=2 Ge_{ij}\\ \sigma_{m}=3K\varepsilon_{m}\end{cases}  \tag{11}\]  
where \(\varepsilon_{ij}\) and \(\varepsilon_{m}\) are respectively deviatoric strain and spherical strain tensors; \(K\) is the elastic bulk modulus. \[ \begin{cases}S_{ij}=\sigma_{ij}-\varepsilon_{m}\delta_{ij}\\ \varepsilon_{ij}=\varepsilon_{ij}-\varepsilon_{m}\delta_{ij}\\ \varepsilon_{m}=\frac{\varepsilon_{kk}}{3}\end{cases}  \tag{12}\]  
where \(\delta_{ij}\) is the Kronecker coefficient; \(\varepsilon_{ij}\) is the strain tensor; \(\varepsilon_{kk}\) is the sum of the principal stresses \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) such as \(\varepsilon_{kk}=\sigma_{1}+\sigma_{2}+\sigma_{3}\). Tables 4 and 5 show relevant information on existing constitutive creep models for various specific and general rock types. They can also illustrate the considerable progress made in this area. Figure 9: A mechanical model integrating thermo-damage effect (after [PERSON] et al. [97], Copyright ©2022, with permission from the publisher, License No. 5333771474409). \begin{table}
\begin{tabular}{p{42.7 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Study Group** & **General Back Type** & **Testing Type and Conditions** & **Densities of Tests** & **Crepo Constitutive Model** & **Crepo Equations or Crep Constitutive Equations** & **Equation Number** & **Cremments** \\ \hline [PERSON] and [PERSON] (2012) & \multirow{2}{*}{All types} & Unital composition & \multirow{2}{*}{2-10} & \multirow{2}{*}{VOFM} & \multirow{2}{*}{\(c(t)=\frac{\alpha_{1}}{\alpha_{2}}+\frac{\alpha_{2}}{\alpha_{3}}+\frac{\alpha_{3}}{ \alpha_{4}}+\frac{\alpha_{3}}{\alpha_{4}}+\frac{\alpha_{3}  
It is important to present existing constitutive creep models for specific rock types and general rock types. In fact, creep constitutive models for specific rock types can be applied in specific situations. That is, the long-term stability of a given tunnel can be studied using an appropriate creep constitutive model of the specific rock type which is the most predominant in the surrounding rocks.