# 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
The creep equation is  
\[\varepsilon=\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{-\frac{E_{2}}{\eta_{1}}t })+\frac{\sigma_{c}}{E_{1}} \tag{15}\]  
By differentiating both sides of Equation (15), the following result is obtained:  
\[\dot{\varepsilon}=\frac{\sigma_{c}-\sigma_{s1}}{\eta_{1}}(1-e^{-\frac{E_{2}}{ \eta_{1}}t}) \tag{16}\]When \(\sigma_{s2}\leq\sigma<\sigma_{s3}\), \(\sigma=\sigma_{c}\), the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The creep equation of the rheological model is  
\[\epsilon=\frac{\sigma_{c}}{E_{1}}+\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{- \frac{E_{2}}{\eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s2}}{\eta_{2}}t \tag{17}\]  
By differentiating both sides of Equation (17), the following result is obtained:  
\[\dot{\epsilon}=\frac{\sigma_{c}-\sigma_{s1}}{\eta_{1}}(1-e^{-\frac{E_{2}}{ \eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s2}}{\eta_{2}} \tag{18}\]  
From Equation (18), it can be deduced that when \(a=9\), the creep rate decreases with increasing time and eventually approaches a constant value. This demonstrates that the model is capable of describing three deformation stages: instantaneous deformation, decelerated creep, and steady creep. When \(\sigma\geq\sigma_{s3}\sigma=\sigma_{c}\), the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The creep equation of the rheological model is  
\[\epsilon=\frac{\sigma_{c}}{E_{1}}+\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{ -\frac{E_{2}}{\eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s2}}{\eta_{2}}t+\frac{ \sigma_{c}-\sigma_{s3}}{a\eta_{c}}e^{at} \tag{19}\]  
By differentiating both sides of Equation (19), the following result is obtained:  
\[\dot{\epsilon}=\frac{\sigma_{c}-\sigma_{s1}}{\eta_{1}}(1-e^{-\frac{E_{2}}{ \eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s3}}{a^{2}\eta_{c}}e^{at}+\frac{\sigma_ {c}-\sigma_{s2}}{\eta_{2}} \tag{20}\]  
From Equation (20), it can be deduced that when \(a=4\), the creep rate increases with time, and the growth rate of creep rate escalates progressively as time advances, resulting in significantly amplified acceleration effects. This demonstrates that the model is capable of characterizing four distinct deformation stages: instantaneous deformation, decelerated creep, steady creep, and accelerated creep. In summary, the one-dimensional integer-order nonlinear viscoelastic-plastic creep model for the free-face section coal pillar is formulated as follows:  
\[\epsilon=\left\{ \begin{array}{ll}\frac{\sigma_{c}}{E_{1}}&\sigma<\sigma_{s1 }\\ \frac{\sigma_{c}}{E_{1}}+\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{-\frac{E_{2 }}{\eta_{1}}t})&\sigma_{s1}\leq\ \sigma<\sigma_{s2}\\ \frac{\sigma_{c}}{E_{1}}+\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{-\frac{E_{ 2}}{\eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s2}}{\eta_{2}}t&\sigma_{s2}\leq \sigma<\sigma_{s3}\\ \frac{\sigma_{c}}{E_{1}}+\frac{\sigma_{c}-\sigma_{s1}}{E_{2}}(1-e^{-\frac{E_{ 2}}{\eta_{1}}t})+\frac{\sigma_{c}-\sigma_{s2}}{\eta_{2}}t+\frac{\sigma_{c}- \sigma_{s3}}{a\eta_{c}}e^{at}&\sigma\geq\sigma_{s3}\end{array} \right. \tag{21}\]