## 3 Test Results and Discussion
### Results of the Single Impact Test
#### 3.1.3 Sandstone Damage Model under a Single Impact  
The continuous damage theory aims to quantitate the damage degree of materials or structures (the damage variable, \(D\)) [31, 32]. Damage is often defined as a decrease in the ultrasonic wave velocity of materials in engineering:  
\[D=1-(\frac{v_{n}}{v_{0}})^{2} \tag{11}\]  
where \(v_{0}\) and \(v_{n}\) are the ultrasonic wave velocities of the undamaged material and the damaged material, respectively. In this study, the longitudinal wave velocities of sandstone specimens before and after the impact test were measured using an RSM-SY5 ultrasonic detector and corresponding transducers. The statistical value of the sandstone damage under a single impact is shown in Figure 6.  
Figure 5: The stress–strain curves of sandstone specimens. (**a**) Dynamic stress–strain curves of sandstone specimens under different impact velocities. (**b**) Comparison between the typical dynamic and static stress–strain curves of sandstone specimens.  
Figure 4 shows that the strain rate has a linear positive correlation with the incident wave amplitude on the whole. At the same time, the greater the impact velocity, the greater the strain rate. Therefore, the strain rate is the reflection of the sandstone under an external load. The damage to the sandstone is a response of its own attributes to external loads. The damage evolution of the sandstone should be represented by the external load variable. Figure 5 shows that the sandstone damage is exponentially correlated with the peak stress, and the exponential function model should be used to represent the damage evolution model under a single impact. To maintain the dimensional unification of the damage model, the static compressive strength \(\sigma_{\mathrm{s}}\) of the sandstone is introduced. An exponential function that satisfies the above conditions is considered as follows: \(D=\alpha\cdot\exp(\beta\cdot\sigma/\sigma_{c})+\delta\), where \(\alpha\), \(\beta\) and \(\delta\) are undetermined parameters and \(\sigma\) is the amplitude of the incident stress wave. The damage evolution model of sandstone under a single impact using the above model is:  
\[D=0.0029\cdot\exp(5.4127\cdot\sigma/76.13)-0.0504,\ R^{2}=0.94 \tag{12}\]  
It can be seen from Figure 5 that the model can reflect the influence of the amplitude change of the incident stress wave on the damage.  
In addition, this paper introduces the energy method to determine the sandstone damage threshold. According to one-dimensional wave dynamics, the incident energy of the SHPB is:  
\[E_{I}=\frac{A_{0}}{\rho_{0}C_{0}}\int_{0}^{\tau}\sigma_{I}^{2}(t)d(t) \tag{13}\]  
where:  
\[\left\{\begin{array}{l}\sigma_{I}(t)=\frac{\rho_{0}C_{0}\
u(t)}{2}\\ \tau=\frac{\lambda}{\rho_{0}}\end{array}\right. \tag{14}\]  
Substituting Equation (14) into Equation (13) gives:  
\[E_{I}=\frac{A_{0}\rho_{0}}{4}\lambda\
u^{2} \tag{15}\]  
where \(A_{0}\) and \(\rho_{0}C_{0}\) are the cross-sectional area and wave impedance of the bar, respectively; \(\tau\) is the duration of the stress wave; and \(\lambda\) is the wavelength of the stress wave. The density \(\rho_{0}\) is 2.65 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), and the cross-sectional area \(A_{0}\) is 0.0019625 m\({}^{2}\). The  
Figure 6: Relationship between damage and incident stress wave amplitude under a single impact.  
wavelength, impact velocity and incident wave amplitude are 800 mm, 6.6 m/s and 45.21 MPa, respectively; thus, the damage threshold of sandstone is 45 J.