# Study on the Creep of Damage-Containing Anthracite: Theory and Experiment
## 2 Fractal Order Model Construction
### Fractal Order Creep Model Considering the Effect of Damage  
The creep properties of coal are very different from those of rocks, such as sandstone, limestone, etc. [36; 37; 38; 39]. Compared to substances with higher load-bearing strength, coal is more susceptible to deformation by slight stress perturbations due to its physical properties. From a microscopic point of view, anthracite, a higher-order coal, has a more complex arrangement of carbon atoms due to its higher carbon content. In general, the creep properties of anthracite coal are more complex. Existing creep models cannot better describe the accelerated creep phase of damaged anthracite because the model's characterization of the accelerated creep phase is primarily a linear model with constant parameters. Based on the traditional creep model, the instantaneous elastic-elastic-viscoplastic model can better characterize the creep characteristics, and by improving the viscoplastic body on this basis, it is possible to establish a creep model conforming to the creep equation of the damage-containing anthracite coal. Figure 1 shows the diagram of the fractal order creep model considering the influence of damage, where \(\sigma\)(t) denotes the stress, \(\mathrm{E_{1}}\) and \(\mathrm{E_{2}}\) denote the elastic modulus of model parts I and II, respectively, \(\eta_{1}\) and \(\eta_{2}\) denote the viscous coefficients of model parts II and III, and \(\sigma_{\mathrm{s1}}\) denotes the dilation stress of the model up to the accelerated creep stage. Figure 1: Diagram of fractal derivative creep model considering the effect of damage. According to the basic principle of the model series-parallel connection,  
\[\left\{ \begin{array}{l}\sigma=\sigma_{\rm I}=\sigma_{\rm II}=\sigma_{\rm III}\\ \varepsilon=\epsilon_{\rm I}+\varepsilon_{\rm II}+\varepsilon_{\rm III}\end{array} ,\right. \tag{5}\]  
When \(\sigma<\sigma_{\rm s1}\), the model I and II parts come into play to characterize the instantaneous creep deformation, deceleration creep deformation, and isokinetic creep deformation of anthracite creep deformation. The model equation of the state can be expressed as follows:  
\[\left\{ \begin{array}{l}\sigma_{\rm I}={\rm E}_{\rm I}\varepsilon_{\rm I}\\ \sigma_{\rm 2}={\rm E}_{\rm 2}\varepsilon_{\rm 2}+\eta_{\rm 1}\dot{\varepsilon}_{ \rm 2}\end{array} ,\right. \tag{6}\]  
According to the principle of the series-parallel connection of components and Equation (5), the model intrinsic equation in the one-dimensional state can be expressed as follows:  
\[\sigma({\rm t})=\frac{1}{{\rm E}_{\rm I}+{\rm E}_{\rm 2}}\Big{(}{\rm E}_{\rm I }{\rm E}_{\rm 2}\varepsilon_{\rm 1}+{\rm E}_{\rm 1}\eta_{\rm 1}\dot{ \varepsilon}_{\rm 2}-\eta_{\rm 1}\dot{\sigma}_{\rm 2}\Big{)},\sigma<\sigma_{\rm s1}, \tag{7}\]  
When \(\sigma_{\rm s1}\leq\sigma\), models I, II, and III characterize the instantaneous creep deformation, deceleration creep deformation, isokinetic creep deformation, and accelerated creep deformation of anthracite creep deformation. Based on the Newton body containing the fractal derivative of damage, the partial equation of the state of model III at this time can be expressed as follows:  
\[\sigma-\sigma_{\rm s1}=\eta_{\rm 2}{\rm e}^{-A\rm t^{\rm q}}\frac{{\rm d} \varepsilon({\rm t})}{{\rm d}{\rm t}^{\rm q}},\sigma_{\rm s1}\leq\sigma, \tag{8}\]  
In summary, under different stress conditions, each part of the model is subjected to equal stress, and the total model strain is equal to the sum of each model strain. The joint solution of Equations (6)-(8) with the adaptive part of Equation (5), respectively, yields the model creep equation as follows:  
\[