# Mechanical and Energy Evolution Characteristics of Sandstone under True Triaxial Cyclic Loading
## 3 Test Results and Analysis
### Analysis of Deformation and Failure Characteristics of Specimens
#### 3.1.3 Strength and Deformation Analysis of Sandstone under Graded Cyclic Loading \(\sigma_{1}\)  
Given that this group of tests refers to the process of studying cyclic load \(\sigma_{1}\), the preloading stage before the load reached the predetermined value was not the focus of this study. To comprehensively analyze the deformation, hysteresis loop, and permanent deformation in each principal stress direction during true triaxial cyclic loading, the predetermined loading point was regarded as the zero point of deformation. Figure 6 shows the stress-strain curve of the effect of step cyclic loading \(\sigma_{1}\) on maximum principal strain \(\varepsilon_{1}\), medium principal strain \(\varepsilon_{2}\), and minimum principal strain \(\varepsilon_{3}\). The results showed that \(\sigma_{1}\) cyclic loading and unloading exerted the greatest influence on sandstone \(\varepsilon_{1}\), followed by \(\varepsilon_{3}\). \(\varepsilon_{2}\) was influenced the least, indicating that the \(\sigma_{1}\) and \(\sigma_{3}\) directions were the dominant deformation directions when true triaxial cyclic load \(\sigma_{1}\) was applied. Figure 5: Failure characteristics of sandstone under different stress paths. (**a**) Uniaxial compression, (**b**) true triaxial compression, (**c**) true triaxial cyclic compression. Under the same \(\sigma_{2}\) condition, the unloading curve of the previous level and the loading curve of the next level were closed, thus forming a hysteresis loop. With graded cyclic loading and unloading \(\sigma_{1}\), the \(\sigma_{1}-\varepsilon_{1}\) and \(\sigma_{1}-\varepsilon_{3}\) curves gradually increased in a spiral shape, and obvious permanent deformation (\(\varepsilon_{1p}\), \(\varepsilon_{3p}\)) characteristics were noted. During each cyclic loading \(\sigma_{1}\), when the loading stress exceeded the unloading point of the previous stage, the deformation curve continued to rise along the loading curve of the previous stage, indicating that the rock deformation had memory characteristics at the elastic stage [25,26]. Under different \(\sigma_{2}\) conditions, the stress-strain curves in the process of graded cyclic load \(\sigma_{1}\) were different. As shown in Figure 6a, when \(\sigma_{2}\) was small, the deformation of sandstone in the three principal stress directions was large. With the increase in \(\sigma_{2}\), cyclic load \(\sigma_{1}\) reduced \(\varepsilon_{2}\) and \(\varepsilon_{3}\) considerably, especially \(\varepsilon_{2}\), and the strength of the rock under the true triaxial state increased, as shown in Figure 6b,c. When \(\sigma_{2}\) was small, the deviatoric stress (\(\sigma_{2}-\sigma_{3}\)) was small, and the cyclic load caused large compression deformation in this direction and large expansion deformation in the \(\sigma_{2}\) and \(\sigma_{3}\) directions. When \(\sigma_{2}\) increased, the deviatoric stress (\(\sigma_{2}-\sigma_{3}\)) was large, and \(\sigma_{2}\) inhibited the deformation in the \(\sigma_{1}\) direction and the development of rock fissures, leading to a decrease in \(\varepsilon_{2}\) and \(\varepsilon_{3}\) and an increase in rock peak stress. However, studies have shown that large deviatoric stress (\(\sigma_{2}-\sigma_{3}\)) promotes the \(\sigma_{3}\) direction to produce large expansion deformation [27]. The deformation trends of sandstone under different loading and unloading rates in the true triaxial state were the same, as shown in Figure 7. With the increase in the loading and unloading rate, the \(\sigma_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\), and \(\varepsilon_{p}\) of sandstone failure decreased. In the process of grading cyclic loading and unloading \(\sigma_{1}\) of sandstone, the cracks between the internal grains continued to open and close. When the load rate increased, the cracks  
Figure 6: Cyclic \(\sigma_{1}\) stress–strain curves under different \(\sigma_{2}\) conditions.