## 5 Discussion
### Poisson's Ratio  
The relationship between Poisson's ratio and the degree of initial damage at different strain rates is shown in Figure 11. It can be seen that:  
1. The Poisson's ratio decreases nonlinearly with respect to strain rate increases. 2. The Poisson's ratio is between 0.26 and 0.3, and the value of the maximum elastic modulus in \(D\) = 0.043 is 68 Gpa, which is 28.3%, 78.9%, and 195.7% higher than that in \(D\) = 0.357, 0.625, and 0.754, respectively. In addition, compared with the degree of the initial damage, the strain rate has little effect on the elastic modulus. In order to quantify the extent of the initial damage in the limestone compression behavior, a nonlinear relationship is applied and expressed as: \[\mu=\ e^{(a+b*D+c*D^{2})}\] (10)  
where \(\mu\) is the Poisson's ratio; \(D\) is the initial damage; and \(a\), \(b\), and \(c\) are the quotieties. Comparison of the Two Initial Damage Determination Methods: Cyclic Loading-Unloading and Longitudinal Wave Speed  
Hamza [53] has taken the strain energy and elastic modulus in the loading and unloading stage as the entry point of the initial damage. The definition of the initial damage is a change law of the elastic modulus, and the expression is shown as:  
\[D=1-\ \frac{E^{\prime}}{E} \tag{11}\]  
However, the definition of the initial damage does not consider the effects of plasticity during the loading-unloading process. The influence of the irreversible plastic strain and the total strain is added to the equation. Moreover, the irreversible plastic deformation of  
Figure 11: Plot of Poisson’s ratio versus initial damage for the axial test from 2 \(\times\) 10\({}^{-6}\) s\({}^{-1}\) to 100 s\({}^{-1}\) levels of strain rate: (a) \(\mu-\overline{z}\); (b) \(\mu-D\). the elastic-plastic material in one-dimensional condition is considered. The expression of the new initial damage variable is shown as:  
\[D=1-\frac{\varepsilon-\varepsilon_{p}}{\varepsilon}\bigg{[}\frac{E^{\prime}}{E} \bigg{]} \tag{12}\]  
where \(\varepsilon\) is the total strain; \(\varepsilon_{p}\) is the irreversible plastic strain after unloading; \(E^{\prime}\) is the injury elastic modulus; and \(E\) is the initial elastic modulus. The stress-strain curves of four initial injuries under loading-unloading conditions are shown in Figure 12. It can be seen from Figure 12 that:  
1. The constant normal three, four, five, and six loading and unloading cycles were developed for four of the initial damaged rock samples. Herein, the difference between starting and ending strain values for the last loading is plastic strain \(\varepsilon_{p}\), and the stress peak point strain in the yield phase is the total strain \(\varepsilon\) which were circled in the drawings 1-4. The final parameters and initial damage values are shown in Table 8. The initial damage calculated by the loading and unloading method is greater than that represented by the longitudinal wave velocity. The reason for this phenomenon is the plastic strain after unloading and is the total value of the original accumulated damage during loading, leading to a greater damage than that of the initial damage in the compression process. 2. The effects of two different initial damages on the compressive strength and destructive strain of limestone are illustrated. For \(\overline{\varepsilon}=2\times 10^{-6}\) s\({}^{-1}\), the compressive strengths are 127.43 Mpa, 97.69 Mpa, 94.89 Mpa, and 82.56 Mpa for the initial damage \(D=0.044\), 0.361, 0.628, and 0.459, respectively. Additionally, the compressive strengths are 130.56 Mpa, 106.37 Mpa, 95.51 Mpa, and 81.23 MPa for the initial damages of 0.044, 0.361, 0.628, and 0.459, respectively. Figure 12: Plot of loading and unloading curves for the compression test from 0.043, 0.357, 0.625, and 0.754 levels of initial damage. (**a**) \(D=0.043\); (**b**) \(D=0.357\); (**c**) \(D=0.625\); (**d**) \(D=0.754\). Figure 13 shows the different definitions of two initial damage methods. It can be seen from Figure 13 that:  
1.