# Creep and Hardening Characteristics of Anthracite under Graded Static-Dynamic Coupled Loading
## 5 Conclusions  
By performing creep tests under different graded loading rates (0.01-0.1 mm/s), the effects of graded loading rates on the creep and hardening characteristics of anthracite were explored. The results revealed the mechanical properties of remaining coal pillars under dynamic pressure, providing a basis for their stability determination. The conclusions obtained from this study can be summarized as follows.  
1. The hardening effect of the sample prepared in this study manifested as increasing instantaneous elastic modulus at the loading stage and decreasing strain rate at the creep stage.
2. The graded loading rate directly affects the instantaneous modulus of elasticity of the samples. With stress level increasing, the instantaneous elastic modulus at each stress level exhibited an increase-decrease-increase trend. The actual yield levels  
Figure 16: Variation of the stable creep rate with the stress level.  
Figure 15: Relation between the time of the decay creep stage and the graded loading rate.  
corresponding to the peak instantaneous elastic moduli of the samples are 9, 8, 8, 7, and 6 MPa, which linearly decreased with increasing graded loading rates.
3. With graded loading rates increasing from 0.01 to 0.1 mm/s, the instantaneous elastic modulus increases by 0.15-2.63 times, the elastic modulus hardening coefficient and the graded loading rate exhibit an increasing linear relation; when the graded loading rate was >0.04 mm/s, the hardening effect was considerably enhanced.
4. The hardening effect of the elastic modulus of the samples abnormally increased at the failure stress level compared with that at the previous stress level. With increasing graded loading rates, the mutation coefficient increased from 1.007 to 2.236, which linearly increased with increasing graded loading rates. When the graded loading rate was >0.04 mm/s, the sudden increase in the instantaneous elastic modulus resulted in a rapid accumulation of elasticity energy, which well explains the instantaneous failure and instability of the samples.
5. The initial strain rate attenuation coefficient and graded loading rate exhibit an exponential relation. With increasing graded loading rates, the attenuation of the initial axial strain rate increases from 46.1% to 75.1%, and that of the initial radial strain rate increases from 17.0% to 85.4%. Moreover, the time of the decay creep stage for the radial strain is longer than that for the axial strain; the radial strain indicates clear hysteresis.  
S.Y.: Writing--Original Draft, Data Curation, Methodology, Formal analysis. K.W.: Methodology, Supervision. X.Z.: Methodology Supervision. T.K.: Writing--Review and Editing. J.Y.: Data Curation. Y.J.: Methodology Supervision. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [National Natural Science Foundation of China] grant number [51974194, 51704204].  
Not applicable.  
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.  
The authors would like to appreciate the financial support from the National Natural Science Foundation of China [Grant No. 51974194, 51704204]. Thanks to all co-authors for their help in the preparation of this paper.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.