# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
## 4 Integer-Order Nonlinear Creep Model
### Three-Dimensional Extension of the Integer-Order Nonlinear Viscoelastic-Plastic Creep Model  
Under three-dimensional stress states, the spherical stress tensor typically induces volumetric deformation in rock elements without altering their shape, while the deviatoric stress tensor causes shape changes without affecting volume. The mathematical expressions for stress and strain states are formulated as follows:  
\[\left\{ \begin{array}{ll}\sigma_{ij}=\sigma_{m}\delta_{ij}+S_{ij}\\ \epsilon_{ij}=\epsilon_{m}\delta_{ij}+e_{ij}\end{array} \right. \tag{22}\]  
where \(\sigma_{m}\) denotes the spherical stress tensor, \(\sigma_{m}=\sigma_{ii}/3\); \(S_{ij}\) represents the deviatoric stress tensor, \(\epsilon_{m}\) is the volumetric strain tensor, and \(e_{ij}\) corresponds to the deviatoric strain tensor. In the elastic state, assuming the volumetric strain tensor and deviatoric strain tensor are respectively related to the spherical stress tensor and deviatoric stress tensor, the three-dimensional form of Hooke's law is formulated as  
\[\left\{ \begin{array}{l}\varepsilon_{m}=\frac{\sigma_{m}}{3K}\\ e_{ij}=\frac{S_{ij}}{2G}\end{array} \right. \tag{23}\]  
where \(K\) is the bulk modulus and \(G\) denotes the shear modulus, with their mathematical expressions defined as  
\[\left\{ \begin{array}{l}K=\frac{E}{3(1-2\mu)}\\ G=\frac{E}{2(1+\mu)}\end{array} \right. \tag{24}\]  
From Equation (24), the strain of the elastic material can be expressed as  
\[\varepsilon_{ij}^{e}=\frac{\sigma_{m}}{3K}\delta_{ij}+\frac{S_{ij}}{2G} \tag{25}\]  
Assuming that volumetric changes are elastic and that rheological properties are predominantly manifested in shear deformation, the constitutive relation for the viscoelastic three-dimensional body is formulated as  
\[\varepsilon_{ij}^{pe}=\frac{1}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}})S_{ij} \tag{26}\]  
The three-dimensional constitutive relation for the viscoplastic body in the rheological model is formulated as  
\[\varepsilon_{ij}^{vp}=\frac{1}{\eta_{2}}\left\langle\phi(\frac{F}{F_{0}}) \right\rangle\frac{\partial Q}{\partial\sigma_{ij}}t \tag{27}\]  
where \(F\) denotes the yield function; \(F_{0}\) represents the initial value of the rock yield function; and \(Q\) corresponds to the plastic potential function. This demonstrates that the three-dimensional integer-order nonlinear viscoelastic-plastic creep model can be obtained by substituting Equation (24), Equation (25), and Equation (26) into the creep Equation (27):  
\[\varepsilon_{ij}=\left\{ \begin{array}{ll}\frac{\sigma_{m}\delta_{ij}}{3K}+ \frac{S_{ij}}{2G_{0}}&S_{11}<\sigma_{s1}\\ \frac{\sigma_{m}\delta_{ij}}{3K}+\frac{S_{ij}}{2G_{0}}+\frac{S_{ij}-\sigma_{s 1}}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}t})&\sigma_{s1}\leq\ S_{11}<\sigma_{s2} \\ \frac{\sigma_{m}\delta_{ij}}{3K}+\frac{S_{ij}}{2G_{0}}+\frac{S_{ij}-\sigma_{s 1}}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}t})+\frac{S_{ij}-\sigma_{s2}}{\eta_{2} }t&\sigma_{s2}\leq S_{11}<\sigma_{s3}\\ \frac{\sigma_{m}\delta_{ij}}{3K}+\frac{S_{ij}}{2G_{0}}+\frac{S_{ij}-\sigma_{s 1}}{2G_{1}}(1-e^{-\frac{G_{1}}{\eta_{1}}t})+\frac{S_{ij}-\sigma_{s2}}{\eta_{2} }t+\frac{S_{ij}-\sigma_{s3}}{\eta_{2}}e^{\mu t}&S_{11}\geq\sigma_{s3}\end{array} \right. \tag{28}\]  
In the triaxial creep test of coal samples with equal confining pressures (\(\sigma_{2}=\sigma_{3}\)), substituting into Equation (28) yields the following:  
\[\left\{ \begin{array}{l}\sigma_{m}=\frac{1}{3}(\sigma_{1}+2\sigma_{3})\\ S_{ij}=\frac{2}{3}(\sigma_{1}-\sigma_{3})\end{array} \right. \tag{29}\]  
By substituting Equation (29) into Equation (28), the integer-order nonlinear viscoelastic-plastic creep model under equal confining pressure is obtained as follows:\[\epsilon(t)=\left\{