# 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  
In this section, the instantaneous elastic modulus evolution law of the samples during the creep test is investigated. Moreover, the hardening effect of the samples under different graded loading rates was analyzed. The variation in the instantaneous elastic modulus with the stress level at different loading stages under the same stress gradient conditions is depicted in Figure 9. The boxplots of the instantaneous elastic modulus under different graded loading rates are presented in Figure 10.  
Figure 9 shows that at the initial stress level, the elastic modulus of each sample is 574.6-1042.98 MPa. The hardening effect is mainly reflected in the medium stress level. The elastic moduli of the samples are the largest at this stage, and the hardening effect is the most clear. Figures 6 and 7 show that the instantaneous elastic modulus at each stress level shows increases with increasing graded loading rate. When the grad  
Figure 9: Variation in the instantaneous elastic modulus with the stress level. from 0.01 to 0.04 mm/s, the instantaneous elastic modulus increases by 0.15-0.63 times (average: 0.36). However, when the graded loading rate increases from 0.04 to 0.1 mm/s, the instantaneous elastic modulus is considerably affected and increases by 0.16-2.32 times (average: 0.95), and the hardening effect is considerably enhanced. [PERSON] et al. [18] analyzed the hardening effect of red sandstone under graded loading creep conditions and showed that the instantaneous elastic modulus first increases and then decreases with increasing stress levels, indicating a good agreement with this study. Figure 10 shows that with increasing graded loading rate, the discreteness of the elastic moduli of the samples in the entire hierarchical creep process increases. The macroscopic performance shows that the heterogeneity of the entire creep process for the samples is clearly enhanced under the graded loading. From the perspective of the entire creep process, we can observe that at a low stress level (0-5 MPa), the graded loading rate has little effect on the instantaneous elastic moduli, mainly because the samples are in the compaction stage. The continuous adjustment of the internal primary fractures and mineral units exerts a clear buffering effect on the external loads. The instantaneous elastic modulus is not sensitive to the graded loading rate but exhibits an overall increasing trend. The growth rates of the elastic moduli of the H-07 and H-08 samples are considerably higher than those of the other samples. Under a medium stress level (5-10 MPa), the samples are theoretically in the linear elasticity stage and the elastic moduli remain unchanged. However, under the influence of the graded loading rate, the instantaneous elastic moduli of the samples gradually increase and the effect of elastic energy accumulation inside the samples gradually becomes notable. At this stage, the sample deformation primarily corresponds to elastic deformation. As per the results presented in Figure 9, the sample strain exhibits a decreasing trend with the stress level, and the hardening effect of the samples is clear. Especially, when the graded loading rate is \(>\)0.04 mm/s, the instantaneous elastic modulus and its growth rate are considerably larger than those of the other samples, and the hardening effect caused by the increase in the graded loading rate becomes clearer. After the stress level exceeds 10 MPa, the samples theoretically enter the yield failure stage. The continuous generation and expansion of internal cracks lead to a rapid decrease in the elastic modulus of the samples, and they gradually suffer from plastic failure. At this time, the sample strain increases and the damage degradation effect dominates. The stress levels corresponding to the peak instantaneous elastic moduli of the various samples are 9, 8, 8, 7, and 6 MPa.