## 3 Nonlinear Creep Model of AS
### Establishment of Nonlinear Creep Model
When \(\sigma<\sigma_{s_{1}}\), the model degenerates into a Hook body and Kelvin elements in series, and its constitutive relationship is as follows:  
\[\frac{\dot{\sigma}}{E_{K_{1}}}+\frac{E_{K_{1}}+E_{K_{2}}}{E_{K_{1}}\eta_{1}} \sigma=\dot{\varepsilon}+\frac{E_{K_{2}}}{\eta_{1}}\varepsilon \tag{5}\]  
When \(\sigma_{s_{1}}\leq\sigma<\sigma_{s_{2}}\), the model is mainly composed of Hook, Kelvin, and Bingham elements. The constitutive relationship is as follows:  
\[\frac{1}{E_{K_{1}}}\ddot{\sigma}+\frac{E_{K_{1}}\eta_{2}+E_{K_{2}}\eta_{2}+E_{ K_{1}}\eta_{1}}{E_{K_{1}}\eta_{1}\eta_{2}}\dot{\sigma}+\frac{E_{K_{2}}}{\eta_{1} \eta_{2}}\sigma-\frac{E_{K_{2}}\sigma_{s_{1}}}{\eta_{1}\eta_{2}}=\ddot{ \varepsilon}+\frac{E_{K_{2}}}{\eta_{1}}\dot{\varepsilon} \tag{6}\]  
When \(\sigma\geq\sigma_{s_{2}}\), the model contains all four elements, and its constitutive relationship is as follows:  
\[ \begin{split} c\frac{\ddot{\sigma}}{E_{K_{1}}}+(\frac{1}{\eta_{1} }+\frac{1}{\eta_{2}}+\frac{E_{K_{2}}}{E_{K_{1}}\eta_{1}}+\frac{n}{\eta_{c}}t^{ n-1})\dot{\sigma}+[\frac{E_{K_{2}}}{\eta_{1}\eta_{2}}+\frac{E_{K_{2}}n}{ \eta_{1}\eta_{c}}t^{n-1}\\ -\frac{n(1-n)}{\eta_{c}}t^{n-2}]\sigma-\frac{E_{K_{2}}\sigma_{s_{ 1}}}{\eta_{1}\eta_{2}}-\frac{E_{K_{2}}n}{\eta_{1}\eta_{c}}t^{n-1}\sigma_{s_{2} }\\ +\frac{n(1-n)}{\eta_{c}}t^{n-2}\sigma_{s_{2}}=\ddot{\varepsilon}+ \frac{E_{K_{2}}}{\eta_{1}}\dot{\varepsilon}\end{split}  \tag{7}\]  
The simultaneous Equations (5)-(7) are used to obtain the constitutive equation of the nonlinear creep model of AS, as follows:  
Figure 9: Effect of parameter \(n\) on the creep behaviours of NVPB. Assume that rock is a homogeneous material, and that creep deformation is caused only by the stress deviatoric tensor. In this case, the creep equation of a three-dimensional viscoelastic body can be obtained by directly replacing \(\sigma_{0}\) in a one-dimensional creep equation with \(\left(S_{ij}\right)_{0}\), in which \(\left(S_{ij}\right)_{0}\) is the constant deviatoric stress imposed on the rock sample during creep tests. Therefore, the three-dimensional constitutive relationship of the viscoelastic body is as follows:  
\[\varepsilon_{ij}^{p-\varepsilon}=\frac{1}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1} }t})S_{ij} \tag{15}\]  
For a viscoplastic body, the creep behaviors are mainly reflected in the shear deformation. Thus, the constitutive relation under the three-dimensional stress state can be expressed as ([PERSON] et al., 2014):  
\[\varepsilon_{ij}^{p-p}=\frac{1}{\eta_{2}}\bigg{\langle}\phi\left(\frac{g}{g_{0 }}\right)\bigg{\rangle}\frac{\partial F}{\partial\sigma_{ij}}t \tag{16}\]  
where \(g\) and \(g_{0}\) are the initial values of the yield function and yield function, respectively; \(F\) is the plastic potential function; and \(\left\langle\phi(\frac{g}{g_{0}})\right\rangle\) is the model switching function under three-dimensional conditions, which satisfies the conditions:  
\[\left\{ \begin{array}{l}If\ g<0,\ then\left\langle\phi\left(\frac{g}{g_{0}} \right)\right\rangle\geq 0\\ If\ g\geq 0,\ then\left\langle\phi\left(\frac{g}{g_{0}}\right)\right\rangle \geq 1\end{array} \right. \tag{17}\]  
Based on the above analysis, substitute Equations (13)-(15) into Equation (9); thus, the nonlinear viscoelastic-plastic creep equation of AS under a three-dimensional stress state is obtained:  
\[\left\{ \begin{array}{l}\varepsilon(t)=\frac{S_{ij}}{2G_{1}}(1-e^{-\frac{G_{1 }}{\eta_{1}}t})+\frac{S_{ij}}{2G_{0}}+\frac{\sigma_{0}}{3R}\delta_{ij}\\ \varepsilon(t)=\frac{S_{ij}}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}t})+\frac{S_ {ij}}{2G_{0}}+\frac{\sigma_{0}}{3R}\delta_{ij}+\frac{S_{ij}-\alpha_{s_{1}}}{ \eta_{2}}t\\ \varepsilon(t)=\frac{S_{ij}}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}t})+\frac{S_ {ij}}{2G_{0}}+\frac{\sigma_{0}}{3R}\delta_{ij}+\frac{S_{ij}-\alpha_{s_{1}}}{ \eta_{2}}t+\frac{S_{ij}-\alpha_{s_{2}}}{\eta_{c}}t^{n}\end{array} \right. \tag{18}\]  
In the current study, the intermediate and minimum principal stresses were equal, i.e., \(\sigma_{2}=\sigma_{3}\).