# Research on Seismic Wave Quality Factor of Marble Jointed Rock Mass under SHPB Impact
## 2 SHPB Test System
### SHPB Test Principle  
The power system is controlled by the built-in software of the computer, and a pressure threshold is set for the pressure chamber. After the pressure reaches a predetermined threshold, the projectile will be pushed forward. When the projectile is accelerated by high-pressure nitrogen inside the launching tube, it impacts the incident rod along the axial direction at a certain outlet speed. After the projectile is discharged from the chamber, the velocity of the projectile will be recorded by the infrared tachometer. A compressive stress wave will be generated in the incident rod when it is impacted. At this time, the incident rod will undergo one-dimensional elastic deformation, and a stress pulse will be formed in the rod and propagate forward along the incident rod. When the stress pulse propagates to the interface I-I between the specimen and the incident rod, the reflected tension unloading wave is formed at the interface because the incident rod wave impedance \(\rho_{1}v_{1}\) is larger than the specimen wave impedance \(\rho_{1}v_{1}\). The reflected tension wave propagates back into the incident rod and forms a transmitted compression wave into the specimen at the same time. Because the parameters of the rock specimen are different from those of the incident rod, the stress wave transmitted into the specimen does not propagate in the form of elastic wave but carries on energy transfer in the form of stress wave disturbance. After the compression wave propagates a specimen length in the specimen, it will be reflected again at the interface II-II between the specimen and the transmission rod to form the reflected compression wave into the specimen. The compression wave will be reflected and transmitted back and forth between the interfaces I-I and II-II, and finally reach the stress balance. In addition, a compressed transmitted wave will also form at the interface of II-II and enter the transmission rod, then enter the absorption rod and buffer through the projection rod and finally return to dissipation. The wave conversion at the interfaces of I-I and II-II is shown in Figure 2.  
According to the stress wave theory and one-dimensional wave hypothesis, if we ignore the dispersion effect of stress wave propagation in elastic rod, then the propagation of stress wave in elastic rod has no distortion and no attenuation. According to the displacement continuity, the compression pulse in the incident rod is called \(\varepsilon_{t}\), the reflected wave generated at the interface between the incident rod and the specimen I-I is regarded as \(\varepsilon_{r}\), and the transmitted wave generated at the interface between the transmission rod and the specimen II-II is regarded as \(\varepsilon_{t}\). Based on the above analysis, the strain rate, strain  
Figure 2: SHPB test principle.  
and average stress generated in the specimen at time T can be obtained, and the function expression is shown as follows.  
\[\dot{\epsilon}(t)=\frac{C_{0}}{L}[\epsilon_{i}(t)-\epsilon_{r}(t)-\epsilon_{t}(t)] \tag{1}\]  
\[\epsilon(t)=\frac{C_{0}}{L}\int_{0}^{t}[\epsilon_{i}(t)-\epsilon_{r}(t)- \epsilon_{t}(t)]dt \tag{2}\]  
\[\sigma(t)=\frac{A_{0}E_{0}}{2A}[\epsilon_{i}(t)-\epsilon_{r}(t)-\epsilon_{t}(t)] \tag{3}\]  
where \(L\) is the specimen length; \(C_{0}\) is the elastic wave velocity of the incident rod and transmission rod; \(E_{0}\) is the elastic modulus of the incident rod and transmission rod; \(A\) is the cross-sectional area of the specimen.  
The three waveforms \(\epsilon_{i}(t)\), \(\epsilon_{r}(t)\) and \(\epsilon_{t}(t)\) can be measured by SHPB system, and \(\dot{\epsilon}(t)\), \(\epsilon(t)\), \(\sigma(t)\) can be obtained by combining the above three Equation at the same time, which is commonly known as \"three-wave method\".