# A Fractional-Order Creep Model of Water-Immersed Coal
## 3 Evolution of Creep Performance  
### Evolution of Elastoplastic Properties  
In order to analyze the elastoplastic properties of the immersed coal samples, their loading stress versus strain data during the creep experiments were assembled as the example of sample C, as displayed in Figure 5. Obviously, in the first loading stage, compaction of the sample appeared, exhibiting a curve growth gradient, despite linear growth of the measured curve data after the second loading stage, and plastic deformation was observed after about the sixth load stage, which was increasingly significant before the peak stress. Figure 5: Curve of stress vs. strain. The nature of the deformation of the immersed specimen at the initial loading stage is in accordance with [PERSON]'s law as a linear elastic stress-strain relationship. The calculated elastic modulus of specimens A, B, C, and D are 5.01 GPa, 4.31 GPa, 3.79 GPa, and 3.52 GPa, respectively. As plotted in Figure 6, the data of the elastic modulus yields the fitted function with a variable of the immersion time, in which \(E_{0}=5.015\) GPa, \(a=0.315\), and \(\beta=11.189\), and the residual sum of squares \(R^{2}=0.98\). The overall tendency of Figure 6 suggests that immersion time decreases the elastic modulus of coal, and with the growth of immersion time, the elastic modulus of a specimen immersed for a long period of time tends to stabilize to a steady level. That behavior is manifested in engineering as a gradual softening of the coal-rock mass with the increasing immersion time, and the immersion time has a significant effect on the mechanical properties of the coal pillar dams in underground reservoirs, which may lead to creep instability of the coal pillars. Compared to the non-immersed sample, as plotted in Figure 6, the elastic modulus of the one immersed for 7 days decreased by about 0.70 GPa (47.37 %); the one immersed for 14 days descended by about 1.22 GPa (82.31 %); and the one immersed for 30 days has approximately a 1.48 GPa deduction in the elastic modulus. As a whole, the elastic modulus has an exponential relationship to the immersion time, which can be expressed by Equation (1), in which \(a\) and \(b\) are the coefficients and \(T_{c}\) means the immersion time. \[E^{*}=E_{0}\bigg{(}1-a\bigg{(}1-\exp\bigg{(}-\frac{T_{c}}{b}\bigg{)}\bigg{)} \bigg{)} \tag{1}\]  
The elastic modulus versus loading stress is displayed in Figure 7, where the compaction of the coal specimen was also observed, exhibiting the initial growth in the elastic modulus. From the second level of loading stress, despite a slight fluctuation of the calculated elastic modulus, it kept stable as the corresponding elastic region, as illustrated in Figure 5. Plastic damage appeared with the promoted loading stress, resulting in the continuous deduction of the elastic modulus in Figure 7.  
Figure 6: Curve of elastic modulus and immersion days. It is not only the elastic modulus that was changed via water immersion but also the peak strength. As plotted in Figure 8, the peak strength of the tested specimen degraded non-linearly with the immersion time. Interestingly, such degradation tendency is similar to that of the elastic modulus in Figure 6. Therefore, a consistent formulation of the evolution of peak strength can be expressed by Equation (2), where \(c\) and \(d\) are the coefficients. \[\sigma_{s}=\sigma_{s}^{0}\bigg{(}1-c\bigg{(}1-\exp\bigg{(}-\frac{T_{c}}{d} \bigg{)}\bigg{)}\bigg{)} \tag{2}\]  
Under uniaxial stress state \(\sigma_{3}=0,\sigma_{1}=\sigma_{s}^{0}\),  
\[\frac{\sigma_{s}^{0}}{z}=C*\cos\varphi+\frac{\sigma_{s}^{0}}{2}\sin\varphi \tag{3}\]  
Figure 8: Peak strength–immersion time curve. Figure 5:0 Elastic modulus–stress relationship curve. Figure 5:1  
Substitute Equation (4) into Equation (2)  
Figure 5:2  
The above analysis convinces us that water immersion has a large impact on the elastic modulus of the coal samples, showing a gradually decreasing tendency during water immersion that is ultimately maintained near a stable value.