## 3 Results  
### Defining Damage Variables and Initial Damage  
The internal state variables of a rock that change during the damage process and affect its mechanical behavior are known as the rock's damage variables. However, the definition of these damage variables is not uniform. They are defined in various ways, including elastic constants, yield stress, density, ultrasonic velocities, and acoustic emission. Among these variables, ultrasonic wave velocity has several advantageous qualities. Firstly, it causes minimal damage to the rock, thus reducing potential errors. Secondly, it is a technique that can be easily repeated, minimizing the likelihood of inconsistency. Thirdly, the required equipment is relatively simple and straightforward to operate. Fourthly, the data obtained from this technique are stable and easy to process. Therefore, ultrasonic wave velocity is regarded as an effective means of defining damage variables and initial damage. Assuming the rock is an isotropic body comprising a rock matrix and microcracks, the propagation of ultrasonic waves in the rock medium follows the law of elastic wave propagation. Consequently, the stress-strain constitutive equation, expressed by the displacement of the rock unit, can be derived. \[ \begin{split}&(\lambda+G)\frac{\partial c}{\partial x}+G\
abla^{2}u= \rho\frac{\partial^{2}u}{\partial t^{2}}\\ &(\lambda+G)\frac{\partial x}{\partial y}+G\
abla^{2}v=\rho \frac{\partial^{2}v}{\partial t^{2}}\\ &(\lambda+G)\frac{\partial c}{\partial z}+G\
abla^{2}w=\rho \frac{\partial^{2}w}{\partial t^{2}}\end{split}  \tag{1}\]  
\(\
abla^{2}\) is the Laplace operator; \(\rho\) is the density of the medium; \(e\) is the volumetric strain; \(G\) is the Ramey coefficient, \(G=E/(1+2\mu)\); and \(\lambda\) is the Ramey coefficient, \(\lambda=\mu E/[(1+\mu)\)\((1+2\mu)]\). In an elastic medium, the displacement \(U=(u,v,w)\) at any point satisfies Equation (1). In setting \(u=u(x,t)\), \(v=0\), and \(w=0\) and substituting into Equation (1), the longitudinal wave velocity expression can be obtained. \[V_{P}=\sqrt{\frac{\lambda+2G}{\rho}}=\sqrt{\frac{E(1-\mu)}{(1+\mu)(1-2\mu)\rho}} \tag{2}\]  
Setting \(u=0\), \(v=0\), and \(w=w(x,y)\) yields the transverse wave velocity expression  
\[V_{\text{S}}=\sqrt{\frac{G}{\rho}}=\sqrt{\frac{E}{(1+\mu)2\rho}} \tag{3}\]  
According to [PERSON]'s strain equivalence principle, rock damage can be represented by the damage variable \(D\)[26]. When the rock is considered an isotropic elastic damage material, the constitutive equation follows the theory of damage mechanics of continuous media. \[\sigma=E_{0}(1-D)\varepsilon=E\varepsilon \tag{4}\]  
\(E_{0}\) is the modulus of elasticity in the undamaged state; \(D\) is the damage variable; and \(E\) is the post-damage modulus of elasticity. Therefore, the damage variable for the rock can be derived as  
\[D=1-\frac{E}{E_{0}} \tag{5}\]  
From Equations (2) and (3), the elastic modulus can be characterized by longitudinal or transverse wave velocity, and since the transverse wave is more sensitive to the performance of damage variations within the rock, Equation (3) is transformed and substituted into Equation (5) to derive an expression for the damage variable based on the characterization of the transverse wave velocity  
\[D_{V}=1-\left(\frac{V_{\mathrm{S}}}{V_{\mathrm{Sf}}}\right)^{2} \tag{6}\]  
\(D_{\mathrm{V}}\) is the damage variable based on \(V_{\mathrm{S}}\); \(V_{\mathrm{S}}\) is the transverse wave velocity of the damaged rock; and \(V_{\mathrm{Sf}}\) is the transverse wave velocity of the intact undamaged rock. From Equation (6), the damage variable characterized by the wave velocity takes values in the range \([0,1]\); when \(V_{\mathrm{S}}=V_{\mathrm{Sf}},D_{\mathrm{V}}=0\), the rock is internally intact and undamaged, and when \(V_{\mathrm{S}}=0\), \(D_{\mathrm{V}}=1\), the rock is damaged.