## 2 Establishment of a Nonlinear Discrete Creep Model
### Construction of the CVISC Model  
Due to the inherent characteristics of soft coal, such as its low strength, soft and brittle nature, and granular structure, it presents challenges in obtaining representative samples for testing. Therefore, the construction of a nonlinear discrete creep model specific to soft coal becomes necessary in order to understand its creep failure mechanism. To ensure the accuracy of the simulation, the parameters of the creep model are determined using the results obtained from uniaxial creep tests. In the selection of a creep model for soft coal, the classical Burgers model [40; 41; 42; 43] is widely utilized due to its high accuracy in describing the steady-state deformation during coal rock creep. However, it has limitations in capturing the large nonlinear deformations observed in the accelerated creep stage. Based on the aforementioned analysis, it is evident that the deformation of soft coal roadways exhibits three distinct stages of creep. To simulate the creep failure of soft coal particles, the CVISC creep model [44] is employed, as it effectively characterizes the viscoelastic and plastic behaviors of soft coal. The model is illustrated in Figure 3.  
Figure 3: CVISC creep mechanics model. Figure 2: Two-rib drilled peep view: (**a**) Left rib 1.0 m; (**b**) Left rib 1.9 m; (**c**) Right rib 0.5 m; (**d**) Right rib 3 m.  
When \(\sigma<\sigma_{\rm s}\), the plastic element in the CVISC model is composed of the Maxwell model and the Kelvin model. The creep constitutive equation is expressed by Equation (1):  
\[\varepsilon(t)=\frac{\sigma}{E_{\rm m}}+\frac{\sigma}{\eta_{\rm m}}t+\frac{ \sigma}{E_{\rm k}}\bigg{(}1-e^{-\frac{E_{\rm k}}{\eta_{\rm k}}t}\bigg{)} \tag{1}\]  
In the formula, \(E_{\rm m}\) represents the elastic modulus of the Maxwell model, \(\eta_{\rm m}\) represents the viscosity coefficient of the Maxwell model, \(E_{\rm k}\) represents the Kelvin rheological coefficient, and \(\eta_{\rm k}\) represents the viscosity coefficient of the Kelvin model. When \(\sigma\geq\sigma_{\rm s}\) ratio is considered, the Burgers model and the plastic element collaborate in the CVISC model. The total creep strain in the three-dimensional state can be represented as follows:  
\[\dot{\epsilon}_{ij}=\dot{\epsilon}_{ij}{}^{k}+\dot{\epsilon}_{ij}{}^{m}+\dot{ \epsilon}_{ij}{}^{p} \tag{2}\]  
In Equation (2), \(\dot{\epsilon}_{ij}{}^{k}\) and \(\dot{\epsilon}_{ij}{}^{m}\) represent the strain rates of the Kelvin and Maxwell components, respectively, while \(\dot{\epsilon}_{ij}{}^{p}\) represents the strain rate of the plastic element. The strain rates of Kelvin and Maxwell are:  
\[\left\{ \begin{array}{l}S_{ij}=2N_{\rm k}\dot{\epsilon}_{ij}{}^{k}+2G_{\rm k }\epsilon_{ij}^{k}\\ \dot{\epsilon}_{ij}^{m}=\frac{S_{ij}}{2C_{m}}+\frac{S_{ij}}{2N_{m}}\end{array} \right. \tag{3}\]  
For the strain rate of plastic elements, there are:  
\[\left\{ \begin{array}{l}\dot{\epsilon}_{ij}{}^{p}=\lambda\frac{\partial g}{ \partial\sigma_{ij}}-\frac{1}{3}\dot{\epsilon}_{vol}{}^{p}\delta_{ij}\\ \dot{\epsilon}_{vol}{}^{p}=\lambda\Big{[}\frac{\partial g}{\partial\sigma_{1 1}}+\frac{\partial g}{\partial\sigma_{22}}+\frac{\partial g}{\partial\sigma_ {33}}\Big{]}\end{array} \right. \tag{4}\]  
where \(\delta_{ij}\) is the unit ball tensor symbol and \(\dot{\epsilon}_{vol}{}^{p}\) is the volumetric strain rate. \(g\) is the potential function, which is used to determine whether the material is yielding or not. \(\lambda\) is a nonzero parameter that only applies during the process of plastic flow, and its value is determined by the specific plastic yield condition being applied. In the Mohr-Coulomb criterion, for shear yielding, the specific expressions of the yield function \(f\) and its potential function \(g\) are as follows:  
\[\left\{ \begin{array}{l}f=\sigma_{1}-\sigma_{3}N_{\phi}+2C\sqrt{N_{\phi}}\\ g=\sigma_{1}-\sigma_{3}N_{\phi}\end{array} \right. \tag{5}\]  
For tensile yielding, the specific expressions of the yield function \(f\) and its potential function \(g\) are as follows:  
\[\left\{