# Design of an Anthracite Creep Model Based on Fractional Order Theory: Experiments and Simulations
## 2 Materials and Methods
### Construction of Fractional Order Model for Anthracite Combinations  
According to the classical creep test curve, the test sample exhibits transient mechanical morphology at the moment of loading; a Hook body was used to characterize this phase. The coal sample deformation exhibits a small, accelerated creep after the transient deformation, and then the coal sample creep rate increases; the fractional-order Bingham model was used to characterize this phase. This stage is called the \"pseudo-accelerated creep stage\". After creep, the creep rate of the coal sample decreases; a Kelvin-Voigt body was used to characterize this stage (\"decay creep stage\"). When the stress level reaches a certain threshold, the coal sample enters the isokinetic creep stage; the Bingham model was used to characterize this stage. When the stress level exceeds a certain constant value and the time t and displacement l meet the requirements, the coal sample deformation enters the accelerated creep phase, and the strain rate of the coal sample increases significantly. In a relatively short period of time, the coal sample undergoes significant damage. A modified fractional-order viscoplastic body was used to characterize this phase. Figure 3 shows the general configuration of the NEG. The specific rheological forms are represented as: viscoplastic, viscoelastic, viscoplastic, and viscoplastic. Figure 2: Improved fractional-order viscosity-plastic body model. 1. Hook body  
The Hook body exhibits transient mechanical morphology when Model I is in effect. The internal damage evolution of anthracite under different stress levels varies, resulting in a partial discounting of mechanical properties compared to intact coal samples. Equation (6) represents the constitutive equation of the Hook body in a one-dimensional state. \[\sigma_{1}=E_{1}(1-D)\varepsilon_{1}, \tag{6}\]  
where \(E\) denotes the modulus of elasticity. 2. Fractional-order Bingham body  
Model II produces an effect when the model enters the pseudo-accelerated creep phase, and the viscosity coefficients in the fractional-order Bingham body are discounted accordingly. Equation (7) is used to represent the fractional-order Bingham body constitutive equation in the one-dimensional state. \[ \begin{cases}\varepsilon_{2}=0,\;\sigma<\sigma_{s1}\\ \sigma(t)-\sigma_{s1}=(1-D)\eta_{1}^{a}D^{a}\varepsilon(t),\;\sigma_{s1}\leq \sigma<\sigma_{s2}\end{cases}  \tag{7}\]  
When \(\sigma(t)=\sigma\), Equation (8) is used to represent the fractional-order Bingham body creep equation. \[ \begin{cases}\varepsilon_{2}=0,\;\sigma<\sigma_{s1}\\ \varepsilon(t)=\frac{\sigma-\sigma_{s1}}{(1-D)\eta_{1}^{a}}\frac{t^{a}}{ \Gamma(\alpha+1)},\;\sigma_{s1}\leq\sigma<\sigma_{s2}\end{cases}  \tag{8}\]  
where \(\Gamma()\) denotes the gamma function. 3. Kelvin-Voigt body  
When model III produces the action, the model enters the deceleration creep phase, and the elastic modulus and viscous coefficient in the Kelvin-Voigt body is discounted accordingly. Equation (9) is used to represent the Kelvin-Voigt body constitutive equation and creep equation in the one-dimensional state. \[ \begin{cases}\sigma=E_{2}(1-D)\varepsilon_{2}+\eta_{2}\dot{\varepsilon_{3}}\\ \varepsilon(t)=\frac{\sigma}{E_{2}}\left(1-e^{-\frac{E_{2}}{\eta_{2}}t}\right) \;,\;\sigma_{s1}\leq\sigma<\sigma_{s2}\end{cases}  \tag{9}\]  
4. Bingham body  
Model IV produces the effect when the model enters the stable creep phase, and the viscosity coefficient in the Bingham body is discounted accordingly. Equation (10) is used to represent the Bingham body instanton equation in the one-dimensional state. Equation (11) is used to represent the Bingham body creep equation in its one-dimensional state. \[ \begin{cases}\varepsilon_{4}=0,\;\sigma<\sigma_{s2}\\ \sigma(t)-\sigma_{s1}=(1-D)\eta_{3}\dot{\varepsilon_{4}},\;\sigma_{s2}\leq \sigma<\sigma_{s3}\end{cases}