## 3 Experimental Results and Analysis
### Damage Evolution Analysis of Sandstone under Cyclic Loading and Unloading  
The process of rock deformation and damage is essentially a process of rock energy dissipation, which can also be described as a process of rock damage evolution. In the process of rock loading damage, as the input energy increases, the primary microcracks within the rock expand, and new cracks are created, accompanied by the continuous dissipation of energy, thus resulting in the continuous weakening of the mechanical properties of the rock. The dissipation energy reflects the degree of rock damage, so the dissipation energy is used to define the rock damage variable as:  
\[D=\frac{U_{\mathrm{d}}^{\varepsilon}}{U_{\mathrm{t}}} \tag{4}\]  
where \(D\) is the damage variable; \(U_{\mathrm{d}}^{\varepsilon}\) denotes the accumulated dissipation energy at a given stress state of \(\varepsilon\); \(U_{\mathrm{t}}\) is the sum of the dissipated energy generated during the whole deformation process. \(U_{\mathrm{t}}\) is to be 0.154, 0.246, and 0.350 MJ.m\({}^{-3}\) at 0.2, 0.5, and 1 KN/s loading rates, respectively (The peak point dissipation energy is calculated by the equation \(U^{\mathrm{d}}=U^{0}-\sigma^{2}/2E\)). The aforementioned results show that the rock energy evolution law exhibits a significant nonlinearity. The rock system is a complex and dissipative system, which has completely different properties from the closed system. [PERSON] [30] pointed out that there are three interactions in the main types of deformation mechanisms (strain hardening and strain softening) during rock deformation: (1) the interaction between rock elastic energy and dissipative energy refers to the fact that an active accumulation of the former provides more energy for the latter to dissipate, while an active dissipative action also inhibits the accumulation of the former. (2) The interaction between micromechanisms in two major types of deformation mechanisms, i.e., the enhancement of one micromechanism will inhibit the action of other mechanisms. For rock systems, this micromechanism cannot be enhanced indefinitely, and other mechanisms are not yet, which, in turn, will inhibit the enhancement of this mechanism. (3) Spatial interaction of different mechanisms, i.e., the more energy consumed by one mechanism in one region, the more it will inhibit the action of this mechanism in other regions, as well as the action of other mechanisms in this region. In summary, there is a competition among the macroscopic mechanisms of rock energy evolution, as well as competition among microscopic mechanisms and different mechanisms in space. Such an energy evolution law with internal constraints is similar to biological population evolution law. This has been discussed by scholars [31], and it is shown that the variation of any energy evolution process in rocks with stress level can be expressed by a logistic equation; then, the damage variable defined by dissipative energy in this paper can be written in the following form:  
\[D=\frac{k}{1+\mathrm{e}^{a-rc}},0\ <k<1 \tag{5}\]  
\(D\) is the damage variable, \(\sigma\) is the axial stress, \(a\) is the parameter reflecting the initial damage level, \(r\) is the endogenous growth rate of damage evolution, and \(k\) is the maximum damage level of 1. Under the three loading rates, the variation of damage variables with axial stress of the rock in cyclic loading and unloading tests can be calculated, as shown in Figure 12. Under the action of cyclic loading and unloading, the damage variation curve of the rock is the \"weakly concave\" type, with the increase of axial stress, the damage variation gradually increases, and the rate of change also gradually increased. When near failure, the change rate of the damage variable with stress is the largest, but there is no obvious mutation. The data from the test were fitted with Equation (5). Its fitting curve and equation can be acquired.