## 4 Numerical Implementation of the Nonlinear Creep Model of AS  
### Finite Difference Form of the Nonlinear Creep Model  
According to the components of the nonlinear model, the total deviatoric stress and the total deviatoric strain tensors can be expressed as follows:  
\[\left\{ \begin{array}{l}\epsilon_{ij}=e_{ij}^{H}+e_{ij}^{K}+e_{ij}^{B}+e_{ij}^{N }\\ S_{ij}=S_{ij}^{H}+S_{ij}^{K}+S_{ij}^{B}+S_{ij}^{N}\end{array} \right. \tag{29}\]  
where \(e_{ij}^{H}\), \(e_{ij}^{K}\), \(e_{ij}^{B}\) and \(e_{ij}^{N}\) represent the partial strains of Hook, Kelvin, Bingham, and NVPB, respectively; \(S_{ij}^{K}\), \(S_{ij}^{H}\), \(S_{ij}^{B}\) and \(S_{ij}^{N}\) represent the deviatoric (partial) stress of Hook, Kelvin, Bingham, and NVPB, respectively. According to the superposition principle, the total deviatoric strain rate of the nonlinear creep model is as follows:  
\[\dot{e}_{ij}=\dot{e}_{ij}^{H}+\dot{e}_{ij}^{K}+\dot{e}_{ij}^{B}+\dot{e}_{ij}^{N} \tag{30}\]  
Equation (30) is converted to an incremental expression as follows:  
\[\Delta e_{ij}=\Delta e_{ij}^{H}+\Delta e_{ij}^{K}+\Delta e_{ij}^{B}+\Delta e_{ ij}^{N} \tag{31}\]  
The Hook element deviatoric stress is expressed as  
\[S_{ij}^{H}=2G_{0}\Delta e_{ij}^{H} \tag{32}\]  
The stress-strain relationship of the combined model of the Hook element and Kelvin element is as follows:  
\[S_{ij}^{K}=2G_{1}e_{ij}^{K}+2\eta e_{ij}^{K} \tag{33}\]  
The strain rate of the Bingham viscoplastic element in three-dimensional form is as follows:  
\[\dot{e}_{ij}^{B}=\frac{H(g)}{2\eta}\frac{\partial F}{\partial\sigma_{ij}} \tag{34}\]  
Furthermore, the deviatoric strain rate of the Bingham viscoplastic element is as follows:  
\[\dot{e}_{ij}^{B}=\frac{H(g)}{2\eta}\frac{\partial F}{\partial\sigma_{ij}}- \frac{\dot{e}_{ij}^{B}}{3}\delta_{ij} \tag{35}\]  
Figure 10: Comparison between experimental data and fitting values under different confining pressures: (**a**) 3 MPa, (**b**) 12 MPa. Based on the assumption in plastic mechanics, the plastic deformation of rock would not change the volume, and the volumetric stress results only from elastic volumetric strain; thus, the following equation can be obtained:  
\[\dot{\sigma}_{m}=K_{B}\dot{e}_{kk}^{B}+K_{H}\dot{e}_{kk}^{H} \tag{36}\]  
where \(K_{B}\) and \(K_{H}\) both represent the bulk modulus of the rock. Based on the principle of centre difference, Equation (33) can be transformed into  
\[\overline{S}_{ij}^{K}\Delta t=2G_{1}\epsilon_{ij}^{K}+2\eta\epsilon_{ij}^{ \Delta K}\Delta t \tag{37}\]  
where \(\Delta t\) represents a time increment step and \(\overline{S}_{ij}\) and \(\overline{e}_{ij}\) represent the average deviating stress and bias strain in a time increment, respectively. Among these parameters,  
\[\left\{ \begin{array}{l}\overline{S}_{ij}=\frac{(\overline{S}_{ij}^{N}+S_{ij }^{O})}{2}\\ \overline{e}_{ij}=\frac{(\overline{S}_{ij}^{N}+\overline{c}_{ij}^{O})}{2} \end{array} \right. \tag{38}\]  
where the superscript symbols \(N\) and \(O\) represent the new and old values in a computing time step, respectively. Substituting Equation (38) into Equation (37), the following equation can be obtained:  
\[e_{ij}^{B-N}=\frac{1}{1+\frac{G_{2}}{2\eta}\Delta t}[(1-\frac{G_{2}}{2\eta} \Delta t)e_{ij}^{B-O}+\frac{\Delta t}{4\eta}(S_{ij}^{N}+S_{ij}^{O})] \tag{39}\]  
The first part of the spherical strain expression of Equation (39) can be calculated as follows:  
\[e_{m}^{B-N}=\frac{1}{1+\frac{K_{B}}{2\eta}\Delta t}[(1-\frac{K_{B}}{2\eta} \Delta t)e_{m}^{B-O}+\frac{\Delta t}{6K_{B}}(\sigma_{ij}^{N}+\sigma_{ij}^{O})] \tag{40}\]  
By substituting Equations (37) and (40) into Equation (31), the following results can be obtained:  
\[S_{ij}^{N}=\frac{1}{A_{1}}[\Delta e_{ij}-\Delta e_{ij}^{B}-\Delta\epsilon_{ij }^{N}-(\frac{1-\frac{G_{2}}{2\eta}\Delta t}{1+\frac{G_{2}}{2\eta}\Delta t}-1) \epsilon_{ij}^{K-O}+A_{2}S_{ij}^{O}] \tag{41}\]  
\[\left\{ \begin{array}{l}A_{1}=\frac{1}{2G_{0}}+\frac{\Delta t}{4(1+\frac{G_{ 2}}{2\eta}\Delta t)\eta}\\ A_{2}=\frac{1}{2G_{0}}-\frac{\Delta t}{4(1+\frac{G_{2}}{2\eta}\Delta t)\eta} \end{array} \right.