## 3 Results
### Damage Evolution Behavior
#### 3.2.3 Damage Evolution Laws Based on Wave Velocity  
From Figures 4 and 5, it can be seen that the characteristics of the damage variable-strain curve established on the basis of wave velocity can be analyzed in two parts.  
Based on the characteristics of the damage variable-strain curve established using wave velocity, no new damage appears within the rock during the first half of the curve. This phenomenon occurs primarily because the initial damage within the rock material is repaired. The value of the damage variable at the intersection of the curve and the y-axis represents the initial damage of the rock specimen. Under the influence of external load, original defects such as microcracks within the rock begin to compress and close, leading to the repair of pre-existing damage and a subsequent numerical decrease in the damage variable. As the stress gradually increases to the crack closure stress, the rate of damage decrease slows, and the slope of the curve approaches zero. Concurrently, the slope of the stress-strain curve increases. Therefore, the first half of the curve can be effectively represented by an inverse proportional function to describe the damage evolution law of the rock.  
In the second half of the curve, new damage begins to form and develop slowly within the rock. New microfractures start to appear and expand, causing the value of the damage variable to increase from slow to fast. When the external load reaches the peak strength of the specimen, the rock undergoes brittle damage, and the damage variable rapidly rises to 1, approximating an inclined straight line. This portion of the curve is approximately exponential. Consequently, the rock damage evolution process at this stage  
Figure 4: The influence of inclination on the evolutionary process of the damage variable \(D_{V}\).  
Figure 5: Relationship between initial damage of specimen and variation in laminar inclination and confining pressure.  
can be abstracted as the growth process of a single population in biology. Utilizing a simple continuous growth model in biology, such as the logistic model, where the growth rate varies with the population size, we can establish damage evolution equations with wave velocity as the damage variable.  
\[D_{V}=\left\{\begin{array}{l}\frac{1}{a+b\epsilon},0\leq\varepsilon\leq \varepsilon_{\text{cc}}\\ \frac{1}{1+e^{-d}},\varepsilon_{\text{cc}}<\varepsilon\leq\varepsilon_{\text{ f}}\end{array}\right. \tag{8}\]  
In the equation, \(a\), \(b\), \(c\), and \(d\) are coefficients; \(\varepsilon_{\text{cc}}\) is the crack closure strain, which can be determined by \(V_{\text{SC}}\); \(\varepsilon_{\text{f}}\) is the peak strain, which can be determined by \(\sigma_{\text{c}}\).  
To check the rationality and reliability of the equations, by analyzing the fitting coefficients and correlation coefficients of the \(D_{\text{V}}\)-\(\varepsilon\) data from various specimens, it was found that in the first segmented function fitting results, the correlation coefficient R ranged from a minimum of 0.839 to a maximum of 0.979, with an average value of 0.924. In contrast, the second segmented function fitting results showed a minimum correlation coefficient R of 0.964, a maximum of 0.999, and an average value of 0.990. These findings indicate that the theoretical damage curves, which are based on wave velocity characterization and the rock damage equation using the segmented function, align well with the actual damage observed. Consequently, this approach effectively describes the evolution process between rock damage variables and strain.