## 5 Analysis of Energy and Brittleness of Plasticity under Uniaxial Compression  
[PERSON] defines the ratio \((U^{d}/U^{e})\) of the dissipated strain energy to the releasable elastic strain energy at the peak point of the specimen as the brittleness index correction value (BIM). The larger the BIM value is, the smaller the ratio of the pre-peak elastic deformation to the total pre-peak deformation becomes, indicating a greater plastic deformation [32; 33]. Therefore, to determine whether the proportion of plastic deformation among the total deformation in the limestone samples increases with an increase in the moisture content, it is necessary to analyze the energy evolution characteristics of the limestone samples with different water content rates in a uniaxial process. Assuming that the specimen does not engage in heat exchange with the outside world during the uniaxial compression test, the specimen is considered a closed system. The total input strain energy generated by the work performed by the external force is \(U\), which uses the following formula according to the first law of thermodynamics [34]:  
\[U=U^{d}+U^{e} \tag{5}\]  
where \(U\) is the total strain energy of the external force work input (KJ/m\({}^{3}\)), \(U^{d}\) is the dissipative energy (KJ/m\({}^{3}\)), and \(U^{e}\) is the elastic strain energy that can be released (KJ/m\({}^{3}\)). The relationship between the dissipative energy \(U^{d}\) and the releasable elastic strain energy \(U^{e}\) is shown in Figure 8.  
Figure 7: Brittleness index B values of limestone samples with different water content rates. In Figure 8, the stress-strain curve and the unloading elastic modulus \(E_{i}\) are enclosed by \(U^{d}\), and the shaded part \(U^{e}\) represents the release of elastic strain energy. The expression of each instance of strain energy during the uniaxial compression test is as follows:  
\[U=\int_{0}^{\varepsilon_{1}}\sigma_{1}d\varepsilon_{1} \tag{6}\]  
\[U^{e}=\frac{\sigma_{1}^{2}}{2E_{0}} \tag{7}\]  
\[U^{d}=\int_{0}^{\varepsilon_{1}}\sigma_{1}d\varepsilon_{1}-\frac{\sigma_{1}^{ 2}}{2E_{0}} \tag{8}\]  
where \(\sigma_{1}\) is the main stress, \(E_{0}\) is the initial elastic modulus of the specimen, and \(\varepsilon_{1}\) is the strain in the direction of the main stress. Using Equations (6)-(8), the total strain energy U, dissipative energy \(U^{d}\), and releasable elastic strain energy \(U^{e}\) corresponding to the peak points of the different water content rates can be obtained as shown in Table 7. The ratio of the releasable elastic strain energy and the dissipated energy to the total input strain energy of each test piece is then plotted as shown in Figure 9.  
As shown in Table 7, most of the total input strain energy of the different water content rate specimens in the uniaxial compression process is stored as releasable strain energy, and the rest is dissipated by rock damage and plastic deformation. As can be seen from Figure 9, the ratio of the releasable elastic strain energy to the total input strain energy gradually decreases with an increase in water content rate, while the ratio of dissipative energy gradually increases. The energy dissipation will induce the deterioration of the specimen and reduce its strength. Therefore, as the water content rate of the test piece increases, the strength of the test piece decreases. Most of the total strain energy input in the uniaxial compression process of limestone is stored as elastic strain energy that can be released. When the input energy is greater than the elastic energy storage limit of the rock itself, the rock begins to fracture and become unstable and releases energy to the outside world. From the calculation results of Table 7, it can be seen that with an increase in the moisture content, the elastic storage energy limit of the limestone gradually decreases. The elastic strain energy storage of a rock depends on the voids and microfractures inside the rock.