## 3 Results and Discussion
### Changes in Acoustic Wave Properties
The moduli plotted in Figure 10 were calculated from Equation (7). \[E_{d}\ =\ \frac{\rho v_{s}^{2}(3v_{p}^{2}-4v_{s}^{2})}{v_{p}^{2}-v_{s}^{2}}, \tag{7}\]  
where \(\rho\) is the bulk density of the sample. In order to reduce the effect of sample inhomogeneity on the calculated dynamic modulus, the modulus is normalised by the static modulus (\(E_{s}\)) for each sample, which are derived in uploading phase (listed in Table 3). As shown in Figure 10, despite accumulated damage from creep stress, the dynamic modulus is still 1.2-3.7 times higher than the static modulus. This discrepancy is larger than that reported by other researchers [45; 46; 47] and can be attributed to the presence of numerous fractures, cracks, and voids in the coal samples; an example is shown in Figure 11. These defects contribute significantly to the plastic deformation involved in uploading phase, resulting in a low static modulus. As is clear from Figure 10, there is not a unified correlation between normalised dynamic modulus and stress level. The moduli of unbroken samples (Figure 10a) are consistently larger than those of failed samples (Figure 10b), and this reflects different responses to long-term and short-term creep. Specifically, the normalised dynamic modulus for short-term creep is negatively related to the stress level, but long-term creep and stress level show no obvious correlation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Time (Minutes)**} & \multicolumn{3}{c}{**Change in P-Wave Velocity (100\%)**} & \multicolumn{3}{c}{**Change in S-Wave Velocity (100\%)**} \\ \cline{2-7}  & **11.02 MPa** & **15.99 MPa** & **17.97 MPa** & **11.02 MPa** & **15.99 MPa** & **17.97 MPa** \\ \hline
3 & 100.0 & - & - & 100.0 & - & - \\
5 & - & - & 100.0 & - & - & 100.0 \\
6 & 101.6 & 100.0 & - & 96.5 & 100 & - \\
7 & - & - & 98.5 & - &  
Figure 11: Photograph of a coal sample showing numerous voids and cracks. Figure 10: Normalised dynamic moduli versus time during creep tests for: (**a**) unbroken samples; (**b**) broken samples. The numbers denote the samples tested at 8.03 MPa, 8.98 MPa, 10.00 MPa, 11.02 MPa, 11.98 MPa, 13.00 MPa, 14.02 MPa, 14.97 MPa, 15.99 MPa, 17.01 MPa, and 17.97 MPa, respectively. Concerning short-term creep, the previous discussion about P-wave velocities declining less than S-wave velocities showed that the underlying mechanism triggering violent failure of the coal samples is the development of vertical cracks. The negative correlation of broken samples and stress level indicates that the magnitude of this structural deterioration is controlled by the stress level. In addition, it is interesting to note in Figure 10b that, although the normalised dynamic modulus consistently decreases with time, the modulus ratio does not decline significantly at failure, as is commonly the case at the end of compression tests [48; 49; 50]. The average percentage decrease for the normalised dynamic modulus in this study is only 12%, thus considerable stiffness must still remain when the sample is near failure. In order to illustrate this phenomenon, a sketch showing the structural deformation inside a sample of coal is included as Figure 12. During uploading phase, vertical cracks develop and transform the sample into a structure consisting of several separated columns. This process does not significantly weaken the bearing capacity of the sample to the vertical pressure, but deteriorates its resistance to the lateral perturbation. When creep begins, the structure of the sample becomes very sensitive to the localized shear and tensile failures. When creep compression continues, the vertical cracks gradually enlarge but have only a slight influence on the axial strain and the stiffness. However, this deterioration increases the possibility of tensile and shear failures of individual columns and this would trigger the sudden release of strain energy. Eventually, the violent collapse turns the already fragile structure into fragments.