# Design of an Anthracite Creep Model Based on Fractional Order Theory: Experiments and Simulations
## 2 Materials and Methods
### Three-Dimensional Extension of NEG  
In the burial depth conditions of the tunnel, the surrounding rock is in a three-directional stress state, and it is difficult to describe the creep characteristics of the tunnel using the one-dimensional creep equation. Therefore, the one-dimensional creep equation was extended into a three-dimensional creep equation. The rock is in a three-dimensional stress state, and the stress state of any unit body inside it can be expressed by the stress sphere tensor \(\sigma_{m}\) and the stress deflection tensor \(S_{ij}\). Equations (13) and (14) are used to represent the stress and strain expressions. \[\sigma_{ij}=\delta_{ij}\sigma_{m}+S_{ij}, \tag{14}\]  
\[\varepsilon_{ij}=\delta_{ij}\varepsilon_{m}+\varepsilon_{ij} \tag{15}\]  
where \(\delta_{ij}\) denotes the unit tensor, \(\delta_{ij}= \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix} ;\sigma_{m}\) and \(\varepsilon_{m}\) denote the stress and strain spherical tensor; and \(S_{ij}\) and \(e_{ij}\) denote the stress and strain bias tensor. Of which:  
\[\sigma_{m}=\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})=\frac{1}{3}\sigma_{kk}, \tag{16}\]  
\[\varepsilon_{m}=\frac{1}{3}(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})= \frac{1}{3}\varepsilon_{kk}, \tag{17}\]  
where \(\sigma_{kk}\) denotes the volumetric stress and \(\varepsilon_{kk}\) denotes the volumetric strain. Equations (13) and (14) can be expressed as Equation (16). \[\left\{ \begin{aligned} S_{ij}=\sigma_{ij}-\frac{1}{3}\delta_{ij} \sigma_{kk}\\ e_{ij}=\varepsilon_{ij}-\frac{1}{3}\delta_{ij}\varepsilon_{kk}\end{aligned}  \right., \tag{18}\]  
1. Hook body  
In the three-dimensional stress state, based on the one-dimensional Hook's law, Equation (17) can be obtained. \[\left\{ \begin{aligned} \sigma_{m}&=3K\varepsilon_{m}\\ S_{ij}&=2 Ge_{ij}\end{aligned} \right., \tag{19}\]  
where \(K=\frac{E}{3(1-2\mu)}\) and \(G=\frac{E}{2(1+\mu)}\) denote the bulk modulus and elastic modulus, respectively. According to the three-dimensional stress rule, the ball stress tensor \(\sigma_{m}\) remains largely independent of the creep state. The spherical stress tensor produces only a transient deformation. The creep of the coal sample is caused by the stress deflection tensor \(S_{ij}\). Equation (18) is used to represent the Hook body creep equation in its three-dimensional state. \[S_{ij}=2G_{0}e_{ij}, \tag{19}\]  
2. Kelvin-Voigt body  
The three-dimensional Kelvin-Voigt body creep equation is transformed from the one-dimensional creep equation in \(\sigma_{0}\) to a constant bias stress \(S_{(ij)0}\). \(S_{(ij)0}\) denotes the constant bias stress applied in the anthracite creep experiment. Equation (20) is used to represent the Kelvin-Voigt body creep equation in its three-dimensional state. \[\varepsilon_{ij(t)_{a}}=\frac{1}{2G_{a}}\left(1-e^{-\frac{G_{0}t}{9t}}t\right) \left(S_{ij}\right)_{\sigma}, \tag{20}\]  
3. Bingham body  
A three-dimensional Bingham body should consider the plastic flow of the material. Equation (21) is used to represent the creep strain rate of the three-dimensional Bingham body. \[\varepsilon_{ij(t)_{b}}=\frac{1}{\eta_{b}}\langle F\rangle\frac{\phi g}{\theta \sigma_{ij}}t, \tag{21}\]  
where \(g\) denotes the potential function and \(\langle F\rangle\) denotes the Heaviside function. According to the creep equation in the one-dimensional state, the intermediate stress is equal to the minimum principal stress under the same enclosing pressure condition: \(\sigma_{2}=\sigma_{3}\). Equation (22) is used to represent the stress spherical tensor \(\sigma_{m}\) and the stress deflection tensor \(S_{ij}\). \[\left\{ \begin{array}{l}\sigma_{m}=\frac{1}{3}(\sigma_{1}+2\sigma_{3})\\ S_{ij}=\frac{2}{3}(\sigma_{1}-\sigma_{3})\end{array} ,\right. \tag{22}\]  
In summary, the creep equation of NEG in the three-dimensional state can be expressed by Equation (23). \[