## 3 Creep Experimental Results
### Law of the Steady Creep Strain Rate  
The relationship between steady creep strain rate, \(\dot{\varepsilon}_{s}\), and loading ratio, \(k=\sigma_{a}/\overline{\sigma}_{UCS}\), is illustrated in Figure 10. With an increase in \(k\), the creep rate laws exhibited significant anisotropy. Specifically, \(\dot{\varepsilon}_{s}\) showed a considerable rise when \(\beta<45^{\circ}\) and \(k=0.75\). Similarly, when \(\beta>45^{\circ}\) and \(k\geq 0.95\), \(\dot{\varepsilon}_{s}\) increased. The phenomena can be attributed to several factors. Firstly, the slate experienced a short period of steady creep, during which internal fissures stabilized and propagated. Subsequently, the slate quickly transitioned into the failure stage. When \(\beta=45^{\circ}\), \(\dot{\varepsilon}_{s}\) was at the same order of magnitude. In addition, the \(\dot{\varepsilon}_{s}\) values obtained for \(\beta<45^{\circ}\) (gray ellipse) were found to be less than \(1.5\times 10^{-4}\)/h. When \(\beta\geq 45^{\circ}\), the steady creep rate was greater than \(1.5\times 10^{-4}\)/h (red ellipse).  
The relationship between \(\dot{\varepsilon}_{s}\) and \(k\) can also be described by a power law as follows:  
\[\dot{\varepsilon}_{s}=A(\sigma_{a}/\overline{\sigma}_{UCS})^{N} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\beta\) (\({}^{\circ}\)) & **0** & **15** & **45** & **75** & **90** \\ \hline \(a\) (\(\dot{\varepsilon}_{i}\)) & 0.27 & 0.21 & 0.28 & 0.39 & 0.49 \\ \(b\) (\(\dot{\varepsilon}_{i}\)) & 1.08 & 0.52 & 0.78 & 0.90 & 0.73 \\ \(R^{2}\) (\(\dot{\varepsilon}_{i}\)) & 0.9786 & 0.9753 & 0.9987 & 0.9843 & 0.9984 \\ \(a\) (\(\dot{\varepsilon}_{c}\)) & 0.19 & 0.05 & 0.09 & 0.14 & 0.13 \\ \(b\) (\(\dot{\varepsilon}_{c}\)) & 4.32 & 3.08 & 2.25 & 3.25 & 2.39 \\ \(R^{2}\) (\(\dot{\varepsilon}_{c}\)) & 0.9501 & 0.9570 & 0.9695 & 0.9779 & 0.9863 \\ \(a\) (\(\dot{\varepsilon}_{i}\)) & 326.51 & 69.67 & 25.01 & 41.41 & 32.25 \\ \(b\) (\(\dot{\varepsilon}_{d}\)) & 6.52 & 3.96 & 2.21 & 2.37 & 1.84 \\ \(R^{2}\) (\(\dot{\varepsilon}_{d}\)) & 0.9892 & 0.9726 & 0.9674 & 0.9659 & 0.9894 \\ \(a\)(\(\dot{\varepsilon}_{s}\)) & 3.32 & 0.302 & 0.0004 & 0.06747 & 0.02532 \\ \(N\)(\(\dot{\varepsilon}_{s}\)) & 17.92 & 9.12 & 3.95 & 19.73 & 38.96 \\ \(R^{2}\) (\(\dot{\varepsilon}_{s}\)) & 0.9924 & 0.9729 & 0.9812 & 0.9888 & 0.9809 \\ \hline \end{tabular}
\end{table}
Table 2: The fitting parameters and fitting performance.  
Figure 10: The relationship curve between steady creep rate, \(\dot{\varepsilon}_{s}\), and loading ratio, \(k\).  
where \(\dot{\epsilon}_{s}\) is the steady creep rate; \(\sigma_{a}/\overline{\sigma}_{UCS}\) is the loading ratio; and \(A\) and \(N\) are constant parameters. All \(R^{2}\) exceeded 0.97, as listed in Table 2. The specific parameters are shown in Table 2. It is to be noted that similar phenomena have been observed for other geotechnical materials [31].  
Figure 11 presents a scatter plot showing the correlation between the unknown parameter \(N\) and the foliation angle \(\beta\), wherein the curve exhibits a \"U-shaped\" pattern. The data were fitted using the strength formula proposed by [PERSON] [32]. The \(R^{2}\) reached 0.98, justifying the high correlation. The relationship between \(\beta\) and the unknown parameter \(N\) is expressed as:  
\[N(\beta)=27.93-26.44\cos 2(\beta-35.47) \tag{6}\]  
Thus, Equation (5) can be re-written as follows:  
\[\dot{\epsilon}_{s}=A(\sigma_{a}/\overline{\sigma}_{UCS})^{N(\beta)} \tag{7}\]  
That is, the empirical equation for \(\dot{\epsilon}_{s}\), the loading ratio, \(k\), and foliation angle, \(\beta\), is as follows:  
\[\dot{\epsilon}_{s}(\sigma_{a}/\overline{\sigma}_{UCS},\beta)=A(\sigma_{a}/ \overline{\sigma}_{UCS})^{27.93-26.44\cos 2(\beta-35.47)} \tag{8}\]