## 3 Nonlinear Creep Model of AS  
### Establishment of Nonlinear Creep Model  
In this paper, the establishment of the nonlinear creep model was conducted by a combination of elemental components. As discussed above, the creep results revealed four stages during the creep tests: instantaneous elastic deformation, primary creep stage, secondary creep stage, and tertiary creep stage. Among them, the instantaneous elastic deformation and primary creep stages could be described by a Hook elastic element (Figure 8I) and Kelvin viscoelastic model (Figure 8 II), respectively. When the stress level reached a certain threshold, the surrounding rock began to enter the secondary creep stage, which could be described by the Bingham model (Figure 8 III) ([PERSON] et al. 2017). When the stress level was higher than a certain threshold, the creep curves showed the characteristics of accelerated deformation. For conventional rheological elements to accurately describe the accelerated creep stage of AS was difficult. In this paper, the nonlinear volume plasticity Bingham (NVPB) model (Figure 8 IV) was introduced to describe the tertiary creep stage. Based on the above analysis, this paper proposes a nonlinear viscoelastic-plastic creep model, as shown in Figure 8, which would be suitable for the description of AS. Figure 7: Creep failure strength of AS samples under different confining pressures. In the above models, \(E_{K_{1}}\) and \(E_{K_{2}}\) represent the elastic moduli of elastic elements in Hook and Kelvin-Voigt bodies, respectively; \(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\) represent the viscous coefficients of viscous elements in Kelvin, Bingham, and NVPB bodies, respectively; N is the rheological index in the NVPB body, indicating the accelerated rheological rate of rock, \(n>0\); \(\sigma_{s_{1}}\) and \(\sigma_{s_{2}}\) represent the crack initiation stress and the expansion stress of the rock, respectively. In Figure 8, the physical meaning and constitutive relation of the first three parts are clear and are not discussed in detail in this study. Interested readers can referred to published papers for more detailed information [26; 27]. For the NVPB model, the viscosity coefficient \(\eta_{3}\) is assumed to satisfy the following equation. \[\eta_{3}(\sigma,t)=\frac{\eta_{c}}{n}(\frac{t}{t_{0}})^{1-n} \tag{1}\]  
where \(t_{0}\) is the relative reference time with a value of 1 and \(\eta_{c}\) is the equivalent viscosity coefficient of the NVPB model, which is closely related to the properties of rock materials. Then, the constitutive equation of the NVPB model is  
\[\sigma_{0}-\sigma_{s_{2}}=\frac{\eta_{c}}{n}(\frac{t}{t_{0}})^{1-n}\dot{\varepsilon} \tag{2}\]  
The corresponding constitutive equation is  
\[\varepsilon(t)=\frac{H(\sigma_{0}-\sigma_{s_{2}})}{\eta_{c}}\frac{t^{n}}{t_{0 }^{n-1}}=\frac{H(\sigma_{0}-\sigma_{s_{2}})}{\eta_{c}}t^{n} \tag{3}\]  
where \(H(\sigma_{0}-\sigma_{s_{2}})\) is a plastic switching function, which can be expressed as follows:  
\[H(\sigma_{0}-\sigma_{s_{2}})=\left\{ \begin{array}{ll}0&(\sigma_{0}\leq\sigma _{s_{2}})\\ \sigma_{0}-\sigma_{s_{2}}&(\sigma_{0}>\sigma_{s_{2}})\end{array} \right. \tag{4}\]  
Equation (4) shows that when \(\sigma_{0}\leq\sigma_{s_{2}}\), the model can reflect the instantaneous deformation, primary creep stage and secondary creep stage of AS; when \(\sigma_{0}>\sigma_{s_{2}}\), the model can fully represent the tertiary creep stage. Figure 9 shows the strain-time curves of the NVPB model under different rheological indices \(n\). As the rheological index \(n\) increased, the deformation rate of the AS samples increased gradually. This property could be used to describe the accelerated creep of AS samples under different confining pressures. According to the series-parallel relationship between components, the following equations can be obtained. Figure 8: Nonlinear creep model of AS: (**I**) Hook elastic element, (**II**) Kelvin viscoelastic model, (**III**) Bingham model, (**IV**) the nonlinear volume plasticity Bingham (NVPB) model.