## 3 Mechanical Parameters and Energy Evolution Characteristics  
### Mechanical Characteristics  
The AniMR-150 nuclear magnetic resonance (NMR) meter was used to measure the porosity of sandstone. By detecting different NMR signals, it is possible to directly reflect the changes in the rock's pore structure. The transverse relaxation rate (also known as the spin-spin or \(T_{2}\) relaxation) in nuclear magnetic resonance can be expressed by the following formula:  
\[\frac{1}{T_{2}}=\frac{\rho_{2}s}{V}=\frac{\rho_{2}F_{2}}{r_{c}} \tag{1}\]  
where \(r_{c}\) is the rock pore size; \(\rho_{2}\) is the surface relaxation strength; \(T_{2}\) is the transverse relaxation time (ms); \(s\) is the pore surface area; \(V\) is the pore volume; and \(F_{2}\) is the core pore shape factor, which is usually constant and related to the pore shape. Let \(\rho_{2}F_{2}=C\). In this paper, \(C=10\), then Equation (1) becomes Equation (2) [30]:  
\[r_{c}=CT_{2} \tag{2}\]  
In conjunction with the relevant literature, the pore size of sandstone can be divided into three intervals: microporous (r \(\leq\) 1 \(\upmu\)m), mesoporous (1 \(\upmu\)m \(\leq\) r \(\leq\) 100 \(\upmu\)m) and macroporous (r \(\geq\) 100 \(\upmu\)m) [30; 31; 32]. Based on Formula (2), it can be deduced that the transverse relaxation time \(T_{2}\) spectrum ranges from 0 to 10 ms for microporous, from 10 to 100 ms for mesoporous, and above 100 ms for macroporous. The data derived from the NMR instrument is shown in Figure 3. Figure 3a shows that the \(T_{2}\) spectrum exhibits a three-peak pattern, where the first peak represents macroporous; the second peak represents mesoporous; and the third peak represents microporous. Between the relaxation times of 0 to 10 ms, the cumulative pore fraction rises slowly in the macroporous region. Between 10 to 100 ms, the curve rises more sharply and at a faster rate. In the relaxation time range of 100 to 10,000 ms, the curve rises slowly and eventually approaches a constant value, which represents the porosity of the sandstone. Figure 3b indicates the porosity of macroporous, mesoporous, and microporous pores, the total porosity, and the proportion of each type of pore. From the figure, it can be seen that the proportion of large pores is 0.451%, that of mesopores is 1.696%, that of micropores is 1.233%, and the total porosity is 3.38%. The mesoporous pores have the highest quantity and account for the largest proportion, representing 50.18% of the total porosity. Therefore, the sandstone contains the most mesoporous pores and the fewest macroporous pores. Figure 2: Schematic diagram of loading method. Figure 4 presents the stress-strain curve of the sandstone. The initial part of the curve is relatively dense, and as the loading progresses, the curve becomes more dispersed, exhibiting an overall pattern of \"dense-sparse\" variation. The research found that in the later stages of loading, the curve significantly deviates from the historical stress-strain curve. This deviation is attributed to the cumulative damage caused by fatigue loading, which leads to an increase in internal structural damage within the sandstone [33; 34]. Under cyclic loading and unloading, the stress-strain curve exhibits hysteresis loops, which reflect the damage and energy dissipation of the rock. To further analyze the mechanical properties of sandstone under cyclic loading and unloading, the hysteresis loop curve is used to calculate the elastic modulus, deformation modulus, and damping ratio, as follows:  
\[\varepsilon_{s}=\varepsilon_{e}+\varepsilon_{cr} \tag{3}\]  
where \(\varepsilon_{s}\) is the total strain in each stage; \(\varepsilon_{e}\) is the elastic strain in each stage; and \(\varepsilon_{cr}\) is the plastic strain in each stage. \[E_{e}=\frac{\sigma_{max}-\sigma_{min}}{\varepsilon_{e}} \tag{4}\]  
Figure 4: Stress–strain curve. Figure 3: Porosity analysis. (**a**) \(T_{2}\) spectrum; (**b**) Pore category.