## 3 Creep Experimental Results
### Laws of Strain and Decaying Time  
The total strain, \(\varepsilon\), consists of instantaneous axial strain, \(\varepsilon_{i}\), and creep strain, \(\varepsilon_{c}\). The \(\varepsilon_{i}\) is independent of the creep process but is closely correlated with the stress environment. On the other hand, \(\varepsilon_{c}\) is related to the creep time, providing insight into the mechanical properties of carbonaceous slate under varying loading times and ratios. The \(\varepsilon_{c}\) is the sum  
Figure 8: Characteristic strength and strength ratio of carbonaceous slate with different foliation angles.  
Figure 7: The calculation method of long term strength.  
of the strain during the decelerated creep stage (\(\epsilon_{d}\)) and the strain during the steady-state creep stage (\(\epsilon_{s}\)) (Figure 5), and it is expressed as follows:  
\[\epsilon =\epsilon_{i}+\epsilon_{c} \tag{2}\] \[\epsilon_{c} =\epsilon_{d}+\epsilon_{s} \tag{3}\]  
Figure 9 shows the variation in \(\epsilon_{i}\) (Figure 9a) and \(\epsilon_{c}\) (Figure 9b) and the duration of the primary creep stage, \(t_{d}\) (Figure 9c), with the loading ratio, \(k\), for carbonaceous slate samples with varying foliation angles. As the loading ratio, \(k\), rises, all three parameters increase monotonically. This relationship can be mathematically expressed by a power function (Equation (4)):  
\[y=a(\sigma_{a}/\overline{\sigma}_{UCS})^{b} \tag{4}\]  
where \(a\) and \(b\) are constants, \(y\) represents any given creep parameter, and \(k=\sigma_{a}/\overline{\sigma}_{UCS}\) denotes the loading ratio, \(\sigma_{a}\) is the predefined loading stress, and \(\overline{\sigma}_{UCS}\) is average uniaxial strength. The curve fitting parameters are listed in Table 2, with all \(R^{2}\) (Coefficient of Determination) exceeding 0.95.  
In Figure 9a, when \(\beta\geq 45^{\circ}\), the variation in \(\epsilon_{i}\) followed the pattern of \(\epsilon_{75^{\circ}}>\epsilon_{60^{\circ}}>\epsilon_{45^{\circ}}\) as the foliation angle increased. These sets of \(\epsilon_{i}\) values corresponding to \(\beta\geq 45^{\circ}\) were greater than the \(\epsilon_{i}\) values at \(\beta<45^{\circ}\). The instantaneous axial strain at \(\beta=0^{\circ}\) was less than that at \(\beta=15^{\circ}\) when \(k<0.65\), and it surpassed the axial instantaneous strain at \(k=0.75\).  
Figure 9: The power law relationship curves between (**a**) instantaneous axial strain, \(\epsilon_{i}\), (**b**) creep strain, \(\epsilon_{c}\), (**c**) duration of the primary creep stage, \(t_{d}\), and stress ratio, \(k\).  
When \(k=0.65\), the \(\varepsilon_{c}\) value for \(\beta=0^{\circ}\) (0.0354) was greater than that for \(\beta=15^{\circ}\) (0.0193), and \(t_{d}\) reached its maximum of 19.53 h, as shown in Figure 9c. Also, for \(k=0.95\), the \(\varepsilon_{c}\) value at \(\beta=75^{\circ}\) (0.1081) was in close proximity to the value at \(\beta=90^{\circ}\) (0.1105), as illustrated in Figure 9b. These results were due to the fact that for the carbonaceous state samples with \(\beta=0^{\circ}\) and \(\beta=75^{\circ}\), a sudden strain increase had been observed during the deceleration creep stage.