## 3 Results
### The Evolution Law of Energy Absorption  
In the process of rock failure under loading, there is absorption and a release of energy [31]. Studying the relevant energy characteristics and laws of rock fracture under loading can provide important information for a deeper understanding of rock mechanics [32]. By examining the impact of pre-existing cracks on the energy evolution law during the fracture of sandstone under loading, as well as the relationship between the dip angle of pre-existing cracks, the tensile strength of rock, the acoustic emission energy and the maximum energy absorption of rock, and the mechanical properties of sandstone can be further revealed. The instability failure of the rock during loading is caused by the work performed by the testing machine on the rock, which is stored as energy inside the rock. The rock deforms and eventually fractures. Therefore, the energy absorbed by the rock during the entire process is equal to the work performed by the testing machine on the rock, which can be quantified by calculating the area under the load-displacement curve [33], as shown in Figure 13. This can be expressed using the following equation:  
\[u=\int_{0}^{l_{i}}P_{i}dl, \tag{1}\]  
where \(u\) is the absorbed energy (\(l\)), \(P_{i}\) is the vertical load at a certain moment, and \(l_{i}\) is the vertical displacement increment at that moment. Figure 14 presents the variation pattern of energy absorption in sandstones with different crack angles under various vertical displacement ratios, derived from the analysis of experimental data. It provides insights into the relationship between energy absorption and the extent of vertical displacement relative to failure displacement. According to the observations from Figure 14, it is evident that the relationship between absorbed energy and vertical displacement ratio in all sandstone specimens exhibits nonlinear behavior during the loading and failure process. Initially, in the early loading stage, the absorbed energy increases slowly with the displacement growth rate, and the slope of the curve is small. As the vertical displacement ratio approaches approximately  
Figure 13: The load–displacement curve. 40%, the slope of the curve increases rapidly, and the absorbed energy of the specimen increases with the displacement growth rate. This phenomenon can be attributed to the control displacement mode employed by the universal testing machine, ensuring uniform loading throughout the test. During the initial loading stage, the specimen absorbs less energy and the rate is slower, corresponding to the compaction stage of rock loading and failure initiation. Sandstone specimens are heterogeneous materials with pre-existing cracks, and most of the energy is dissipated due to the compaction of micro-cracks and pre-existing cracks inside the rock. In the later stage of loading, more energy is absorbed at a faster rate, indicating less energy dissipation, and the work performed by the testing machine is absorbed by the rock specimen after the pre-existing cracks and micro-cracks inside the rock are compacted. When the vertical displacement ratio reaches approximately 80%, the absorbed energy evolution curve approaches a linear relationship, and the growth rate stabilizes. This indicates that the absorbed energy and dissipated energy reach a relatively balanced state, and the rock will eventually become unstable and fail. When the dip angle of the pre-existing crack \(\alpha=45^{\circ}\), the absorbed energy of the specimen increases at the slowest rate during the loading and failure process. On the other hand, the energy evolution curve of the specimens with the dip angles \(\alpha=30^{\circ}\) and \(\alpha=60^{\circ}\) exhibit similar trends. When the dip angle \(\alpha=90^{\circ}\), the absorbed energy of the specimen increases at the fastest rate, followed by the specimen with \(\alpha=0^{\circ}\). The rate of absorbed energy growth can reflect the severity of instability and failure during the rock loading process.