## 3 Experimental Result Analysis
### Energy Density  
Figure 9 presents a schematic diagram illustrating the calculation of the cyclic energy density of the specimen. In the diagram, the area enclosed by the loading curve (OA) and the strain represents the input energy density, while the area enclosed by the unloading curve (BA) and the strain corresponds to the elastic energy density. The dissipated energy density is represented by the area enclosed by the loading curve (OA), the unloading curve (BA), and the strain. The corresponding formula is given as follows:  
\[U_{I}=U_{E}+U_{D} \tag{1}\]  
\[U_{I}=\int\sigma_{L}\;\mathrm{d}\varepsilon \tag{2}\]  
\[U_{E}=\int\sigma_{U}\;\mathrm{d}\varepsilon \tag{3}\]  
where \(U_{I}\) represents the input energy density (MJ\(\cdot\)m\({}^{-3}\)), \(U_{E}\) denotes the elastic energy density (MJ\(\cdot\)m\({}^{-3}\)), \(U_{D}\) refers to the dissipated energy density (MJ\(\cdot\)m\({}^{-3}\)), \(\sigma_{L}\) corresponds to the loading section curve (OA), \(\sigma_{U}\) represents the unloading section curve (BC), and \(\varepsilon\) denotes the axial strain. Figure 8: Peak strain of specimens at different temperatures and loading modes. Figure 9: Schematic diagram of energy density calculation. As cyclic loading and unloading progress, the ratio of dissipated energy to input energy density provides insight into the damage evolution of rock samples. Therefore, the energy dissipation ratio is introduced, defined as the ratio of the dissipated energy of the specimen to the total input energy during each loading cycle [41]. Figure 10 illustrates the variation in constant lower limit cyclic energy density and dissipation ratio with the number of cycles under real-time temperature conditions. As shown in Figure 11, the input energy density (Figure 10a) and elastic energy density (Figure 10b) at each temperature exhibit a nearly linear increase with the number of cycles. In contrast, the dissipated energy density (Figure 10c) rises sharply near failure. This indicates that, during the early to middle stages of cyclic loading, the internal structure of the rock samples primarily undergoes reversible elastic deformation. Microcrack propagation is limited, and both energy input and storage increase steadily, allowing the samples to remain in a relatively stable load-bearing state. As loading cycles progress, microscopic defects accumulate within the rock samples. The number of microcracks increases and eventually coalesces, resulting in enhanced irreversible deformation. Once crack growth reaches a certain level, the rock samples enter a stage of accelerated damage, marked by a sudden increase in dissipated energy density per cycle. Additionally, temperature significantly influences the mechanical behavior of rocks. At elevated temperatures, the bonding forces between mineral grains weaken, and heat-induced cracks form and grow more readily. This accelerates the accumulation of dissipated energy and hastens the destabilization and failure of the rock samples. However, the variations in input, elastic, and dissipation energies at 25 \({}^{\circ}\)C under constant lower limit-graded cyclic loading and unloading are relatively minor compared to those at 200 \({}^{\circ}\)C, due to three main factors. First, at 25 \({}^{\circ}\)C, energy release from crack propagation in rock samples primarily results from brittle failure, leading to instantaneous energy accumulation and release without significant delay. At 200 \({}^{\circ}\)C, although crack propagation may be slightly accelerated, the overall brittle failure mechanism remains unchanged, leading to a rapid and instantaneous energy release. Consequently, variations in input, elastic, and dissipation energies remain minimal. Second, as rock samples gradually accumulate damage and develop microcracks during constant lower limit cycling, they retain sufficient elastic recovery time due to the fixed lower limit stress.