# Creep and Hardening Characteristics of Anthracite under Graded Static-Dynamic Coupled Loading
## 2 Methodology
### Creep Principle and Hardening Mechanism  
Researchers have found that not only the damage and softening effects but also the hardening effect occur during the creep process of coal and rock mass [15; 16; 17; 18; 19]. The damage effect and the hardening effect restrict the development of each other and act during the entire creep process. The damage effect primarily manifests as the continuous deterioration of the sample, which can well explain the continuous increase in the strain and the generation of accelerated creep. The hardening effect primarily manifests as a gradual increase in the ability of the sample to resist deformation under an external load, which can well explain the generation of decay creep. The effect of the creep hardening of the sample can be quantified using the elastic modulus [44]. Under graded loading, the damage degree of the samples varies with the stress level, and the elastic modulus exhibits a clear timeliness at different loading levels. The hardening effect is directly reflected in the increase in the instantaneous elastic modulus and strain during the graded loading stage. After the samples are damaged owing to continuous creep and loading, the decay creep characteristics in the creep stage can also reflect the hardening characteristics of the samples. In a subsequent test, the influence of different graded loading rates on the creep-hardening effect of anthracite coal was studied based on three aspects: the strain increment in the loading stage, instantaneous elastic modulus hardening in the loading stage, and strain rate attenuation in the creep stage. During the creep test, axial stress was applied to the sample via graded loading (Figure 3). At the \(i\)th stress, the stress remains constant, and the strain increases with time. The diagram of stress and strain with time is shown in Figure 4.  
The generated strain has two components at each stress level: the instantaneous and creep strains. The relation among the increments of these strains at each stress level can be expressed as follows:  
\[\Delta\varepsilon_{i}=\Delta\varepsilon_{i1}+\Delta\varepsilon_{i2}, \tag{1}\]  
where \(\Delta\varepsilon_{i}\), \(\Delta\varepsilon_{i1}\), and \(\Delta\varepsilon_{i2}\) indicate the increments in the total, instantaneous, and creep strains at the \(i\)th stress level, respectively. During the loading stage, the axial stress increases from \(\sigma_{i-1}\) to \(\sigma_{i}\), and the instantaneous elastic modulus of a sample can be obtained as follows:  
\[E_{i}=\Delta\sigma_{i}/\Delta\varepsilon_{i1}, \tag{2}\]  
where \(E_{i}\), \(\Delta\sigma_{i}\), and \(\Delta\varepsilon_{i1}\) are the instantaneous elasticity modulus of the sample, the stress increment, and the instantaneous strain increment, respectively, at the \(i\)th stress level. Figure 3: Schematic of the graded loading path for creep tests. To quantitatively express the influence degree and trend of the graded loading rate on the hardening effect for a sample, in this experiment the elastic modulus hardening coefficient \(K\) can be defined as follows:  
\[K=E_{i}/E_{c}, \tag{3}\]  
\[E_{c}=\Delta\sigma/\Delta\epsilon, \tag{4}\]  
where \(E_{i}\) is the instantaneous elastic modulus of the sample at the \(i\)th stress level, \(E_{c}\) is the elastic modulus in the conventional compression tests, and \(\Delta\sigma\) and \(\Delta\epsilon\) are the stress and strain increments in the linear elasticity phase in the conventional compression tests. During the creep stage, the creep rate constantly decays and shows a \"decreasing-stabilizing\" trend, which is called the decay creep and steady creep stage, respectively. The evolution of the creep rate and strain of the sample in the creep stage is shown in Figure 5.  
Figure 4: Principle of graded loading creep test. Figure 5: Diagram of creep rate evolution during creep stage.