# A Fractional-Order Creep Model of Water-Immersed Coal
## 4 Fractional-Order-Based Creep Model  
### One-Dimensional Fractional-Order Creep Model  
Due to the highly nonlinear characteristics presented in the creep process of coal rock, the traditional constitutive relationship based on the integer order (such as the Newtonian dashpot) in the creep element model can no longer accurately describe the characteristics of each stage of the creep process. In this paper, we transformed the integer-order derivatives in the Newtonian dashpot into fractional-order derivatives and the Newtonian kettle in the constitutive model into a fractal dashpot, as shown in Figure 11.  
Figure 10: The curve fitting of accelerated creep rates (**a**) time-stress-creep rate (**b**) stress-creep rate (**c**) time-creep. As shown in Figure 11, the proposed fractional creep model includes an elastic body (H), a nonlinear viscous body (N), and an ideal viscoplastic body (N/Y body) in series, which are used to describe the attenuated creep deformation, isochronous creep deformation, and accelerated creep stages during the graded loading of the inundated coal rock, respectively. The fractional-order derivatives have several different definitions such as the Riemann-Liouville definition, the Caputo definition, and the Riesz potential operator, etc. The fractional-order derivatives used in this paper are the conformable derivatives proposed by [PERSON] in 2014 [45]:  
\[T_{a}f(t)=\lim_{k\to 0}\frac{f(t+kt^{1-\alpha})-f(t)}{k} \tag{10}\]  
where \(T_{a}f(t)\) denotes the \(\alpha\)-order derivative of the function \(f\) with respect to \(t\). In addition, [PERSON] gives the conformable fractional-order derivative with respect to the fractal derivative as  
\[T_{a}f(t)=t^{1-\alpha}\frac{df(t)}{dt} \tag{11}\]  
The relationship between the stress and strain rate in the traditional ideal dashpot (Newtonian dashpot) can be expressed as  
\[\sigma(t)=\eta\frac{d\varepsilon(t)}{dt} \tag{12}\]  
where \(\eta\) is the coefficient of viscosity. By changing the first-order derivatives in the ideal elastic dashpot to fractional-order derivatives based on the conformable derivatives, the fractional-order dashpot based on the conformable derivatives can be obtained:  
\[\sigma(t)=\eta T_{a}\varepsilon(t)=\eta t^{1-\alpha}\frac{d\varepsilon(t)}{dt} \tag{13}\]  
when the fractional-order derivative number equals 1, Equation (14) becomes the stress-strain relation \(\sigma(t)=\eta\frac{d\varepsilon}{dt}\) for an ideal fluid. For the creep process, the stress is a constant \(\sigma\). Substituting \(\sigma(t)=\sigma\) and integrating both sides of Equation (14) yields:  
\[\int d\varepsilon(t)=\int\frac{\sigma}{\eta}t^{\alpha-1}dt\rightarrow \varepsilon(t)=\frac{\sigma}{\eta\alpha}t^{\alpha}+C \tag{14}\]  
The initial strain is 0, where one can obtain the constant \(C=0\) in Equation (15); Equation (15) portrays the power-law relationship between strain and time in a fractional-order viscous dashpot. \[\sigma(t)=\eta_{0}e^{-\beta t}\frac{d\varepsilon(t)}{dt} \tag{15}\]  
where \(\eta_{0}\) is the initial cohesion factor and \(\beta\) is the damage factor. In the accelerated creep deformation stage of the coal-rock body, rupture damage occurs in the coal-rock body and it is necessary to consider the damage on the basis of the element model. According to the damage theory, the damage variable \(\alpha\) is introduced into  
Figure 11: A conceptual creep model of water-immersed coal. the elemental model and the relational equation of the stress-strain in the creep process of the ideal viscous kettle under the consideration of damage is obtained. For creep experiments, \(\sigma\) is a constant, so integrating Equation (16) yields the ideal dashpot strain-time relationship as:  
\[\varepsilon=\frac{\sigma e^{\beta t}}{\eta_{0}\beta} \tag{16}\]  
For the proposed fractional derivative constitutive model, the first part, namely the Constitutive equation of elastic elements, can be expressed as:  
Figure 10:0  
where \(\sigma\) is the creep loading stress and \(E\) is the elastic modulus of the elastic body.