# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
## 4 Integer-Order Nonlinear Creep Model
### Integer-Order Nonlinear Viscoelastic-Plastic Creep Model
\begin{array}{l}\sigma_{1}=E_{1}\varepsilon_{1},\sigma_{2}-\sigma_{s 1}=E_{2}\varepsilon_{2}+\eta_{1}\dot{\varepsilon}_{2}\\ \sigma=\sigma_{1}=\sigma_{2},\varepsilon=\varepsilon_{1}+\varepsilon_{2} \end{array} \right. \tag{6}\]  
The constitutive equation is obtained as  
\[(E_{1}+E_{2})\sigma+\eta_{1}\dot{\sigma}-E_{1}\sigma_{s1}=E_{1}E_{2}\varepsilon +E_{1}\eta_{1}\dot{\varepsilon} \tag{7}\]  
Figure 7: Effect of parameter \(a\) on improved Bingham body creep properties. When \(\sigma_{s2}\leq\sigma<\sigma_{s3}\), the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The constituent equations for each element in the model satisfy the following relationships:  
\[\left\{ \begin{array}{l}\sigma_{1}=E_{1}\varepsilon_{1},\sigma_{2}-\sigma_{s1}= E_{2}\varepsilon_{2}+\eta_{1}\dot{\varepsilon}_{2},\sigma_{3}-\sigma_{s2}=\eta_{2} \dot{\varepsilon}_{3}\\ \sigma=\sigma_{1}=\sigma_{2}=\sigma_{3},\varepsilon=\varepsilon_{1}+ \varepsilon_{2}+\varepsilon_{3}\end{array} \right. \tag{8}\]  
From the fifth equation in Equation (8), it can be deduced that  
\[\left\{ \begin{array}{l}\dot{\varepsilon}=\dot{\varepsilon}_{1}+\dot{ \varepsilon}_{2}+\dot{\varepsilon}_{3}\\ \ddot{\varepsilon}=\ddot{\varepsilon}_{1}+\ddot{\varepsilon}_{2}+\ddot{ \varepsilon}_{3}\end{array} \right. \tag{9}\]  
By substituting the first four equations from Equation (8) into Equation (9), the constitutive equation is obtained as  
\[\frac{\ddot{\sigma}}{E_{1}}+(\frac{1}{\eta_{1}}+\frac{1}{\eta_{2}}+\frac{E_{ 2}}{\eta_{1}E_{1}})\dot{\sigma}+\frac{E_{2}}{\eta_{1}\eta_{2}}\sigma-\frac{ \eta_{2}\sigma_{s1}+(E_{2}+\eta_{1})\sigma_{s2}}{\eta_{1}\eta_{2}}=\ddot{ \varepsilon}+\frac{E_{2}}{\eta_{1}}\dot{\varepsilon} \tag{10}\]  
When \(\sigma\geq\sigma_{s3}\), the rheological model is composed of the Hook body, Muruyama body, Bingham body, and modified Bingham body. The constituent equations for each element in the model satisfy the following relationships:  
\[\left\{ \begin{array}{l}\sigma_{1}=E_{1}\varepsilon_{1},\sigma_{2}-\sigma_{s1 }=E_{2}\varepsilon_{2}+\eta_{1}\dot{\varepsilon}_{2},\sigma_{3}-\sigma_{s2}= \eta_{2}\dot{\varepsilon}_{3},\sigma_{4}-\sigma_{s3}=\eta_{c}e^{-at}\dot{ \varepsilon}_{4}\\ \sigma=\sigma_{1}=\sigma_{2}=\sigma_{3}=\sigma_{4},\varepsilon=\varepsilon_{1 }+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4},\dot{\varepsilon}=\dot{ \varepsilon}_{1}+\dot{\varepsilon}_{2}+\dot{\varepsilon}_{3}+\dot{ \varepsilon}_{4},\ddot{\varepsilon}=\ddot{\varepsilon}_{1}+\ddot{\varepsilon} _{2}+\ddot{\varepsilon}_{3}+\ddot{\varepsilon}_{4}\end{array} \right. \tag{11}\]  
From Equation (11), the constitutive equation is obtained as  
\[ \begin{array}{l}\ddot{\varepsilon}+\frac{E_{2}}{\eta_{1}}\dot{\varepsilon}= \frac{\ddot{\sigma}}{E_{1}}+\frac{\dot{\sigma}-\sigma_{s1}}{\eta_{1}}+\frac{ \dot{\sigma}-\sigma_{s2}}{\eta_{2}}+\frac{\dot{\sigma}-\sigma_{s3}}{\eta_{c}} e^{at}\\ +\frac{E_{2}\dot{\sigma}}{\eta_{1}E_{1}}+\frac{E_{2}}{\eta_{1}\eta_{2}}(\sigma- \sigma_{s2})+\frac{\sigma-\sigma_{s3}}{\eta_{c}}ae^{at}+\frac{E_{2}\dot{ \sigma}^{at}}{\eta_{1}\eta_{c}}(\sigma-\sigma_{s4})\end{array}  \tag{12}\]  
(2) Creep equations  
When \(\sigma<\sigma_{s1}\), \(\sigma=\sigma_{c}\), the rheological model contains only the Hook body, and the creep equation is  
\[\varepsilon=\frac{\sigma_{c}}{E_{1}} \tag{13}\]  
By differentiating both sides of Equation (13), the following result is obtained:  
\[\dot{\varepsilon}=0 \tag{14}\]  
From Equation (14), it can be deduced that when \(\sigma<\sigma_{s1}\), the creep rate remains zero, indicating that the creep model exhibits only instantaneous deformation. When \(\sigma_{s1}\leq\sigma<\sigma_{s2}\), \(\sigma=\sigma_{c}\), the rheological model is composed of the Hook body and Muruyama body.