# Creep and Hardening Characteristics of Anthracite under Graded Static-Dynamic Coupled Loading
## 4 Discussion
### Evolution Trend of the Instantaneous Elastic Modulus in the Loading Stage
After the respective stresses exceed these values, the instantaneous elastic moduli exhibit a decreasing trend in the loading stage and the ability of the samples to resist deformation weakens. The instantaneous strains of the samples in  
Figure 10: Boxplots of the instantaneous elastic modulus. the loading stage considerably increase, and the damage to the sample's internal structure becomes notable. Herein, the stress corresponding to the peak value of the instantaneous elastic modulus is considered a critical value for the samples when entering the yield failure stage. With increasing graded loading rates, the actual yield stresses of the samples linearly decrease (Figure 11), and the samples become more prone to yield failure. In order to analyze the hardening characteristics of the samples in a comprehensive way, we calculated the average hardening coefficient, the peak hardening coefficient, and the failure stress level hardening coefficient separately. The calculation results for the elastic modulus hardening coefficient \(K\) in the creep test are summarized in Table 6. The fitting relation of \(K\) with the graded loading rate is depicted in Figure 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Hardening** & \multicolumn{4}{c}{**Graded Loading Rate \(V_{i}\) (mm s\({}^{-1}\))**} & \\
**Coefficient \(K\)** & **0.01** & **0.02** & **0.04** & **0.06** & **0.1** & **Fitting Relation** \\ \hline Average value & 1.259 & 1.383 & 1.725 & 2.731 & 3.346 & \(K_{A}=24.858\cdot V_{i}+0.9455,\ R^{2}=0.9404\) \\ Peak value & 1.747 & 1.910 & 2.627 & 4.165 & 4.863 & \(K_{P}=37.6204\cdot V_{i}+1.332,\ R^{2}=0.9203\) \\ Failure stress level & 1.310 & 1.285 & 1.751 & 2.921 & 4.291 & \(K_{F}=35.4575\cdot V_{i}+0.6804,\ R^{2}=0.9551\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relation between the hardening coefficient and graded loading rate. Figure 11: Actual yield stress versus the graded loading rate. Figure 12: Relation between the elastic modulus hardening coefficient and graded loading rate. As can be seen from Figure 12, \(K\) and the graded loading rate exhibit an increasing linear relation, and the growth trends of the peak and failure stress levels are similar. Notably, at the failure stress level (12 MPa), the hardening effect of the sample is temporarily enhanced compared with that in the previous stress level (11 MPa) (Figure 9). In particular, the instantaneous elastic moduli of the H-07 and H-08 samples abruptly increase. To express the mutation of the elastic modulus at the failure stress level, the elastic modulus mutation coefficient \(K_{n}\) can be defined as follows:  
\[K_{n}=E_{n}/E_{n-1}, \tag{6}\]  
where \(E_{n}\) and \(E_{n-1}\) are the elastic moduli at the failure stress level and the stress level before failure, respectively. The values of \(K_{n}\) for each sample are calculated to be 1.007, 0.973, 1.173, 1.174, and 2.236. The fitting relation of the elastic modulus mutation coefficient (\(K_{n}\)) with the graded loading rate (\(V_{l}\)) is depicted in Figure 13.  
It can be clearly observed from Figure 13 that the mutation coefficient of the elastic modulus at the failure stress level linearly increases with the graded loading rate. With increasing graded loading rates, the mutation of the hardening effect becomes more clear before the samples are destroyed. In particular, for the H-07 and H-08 samples, their elastic moduli at the failure stress level show a clear abrupt change; a large amount of energy is rapidly accumulated during the loading stage. The abrupt increase in the elastic moduli well explains the transient failure characteristics of the H-07 and H-08 samples. According to the aforementioned analysis results, the graded loading rate directly affects the instantaneous elastic modulus of the samples during the loading stage. The hardening effect of the samples at medium-high stress levels is most sensitive to the graded loading rate; this is primarily reflected after the graded loading rate exceeds 0.04 mm/s.