# Research on Seismic Wave Quality Factor of Marble Jointed Rock Mass under SHPB Impact
## 4 Rock Mass Seismic Wave Quality Factor  
### Quality Factor Calculation Method  
Rock mass seismic wave quality factor (Shorthand for the \(Q\)s, can be called quality factor) is the ratio of maximum variable performance to dissipated energy, and is a dimensionless parameter. Based on the experimental principle of the SHPB test system, it can be seen that in the test process the total energy of the separated Hopkinson pressure rod test system consists of four parts: incident wave energy \(W_{i}\), reflected wave energy \(W_{r}\), transmitted wave energy \(W_{t}\), and joint sample deformation energy \(W_{s}\). In the process of the SHPB test, the stress waves and deformation of joint samples in elastic rods such as projectile, incident rod, transmission rod and absorption rod are time dependent. Therefore, the above four energies can be expressed as time-dependent functions, namely \(W_{i}(t)\), \(W_{r}(t)\), \(W_{t}(t)\) and \(W_{s}(t)\). Under the action of incident stress wave, the incident elastic rod exerts dynamic load on the jointed rock mass sample, and the work done by the elastic rod on the jointed rock mass sample can be denoted as \(U(t)\). The energy of the work done by the incident rod on the jointed rock mass sample comes from the energy of the elastic wave in the rod, so \(U(t)=W_{i}(t)-W_{r}(t)-W_{t}(t)\). According to the first law of thermodynamics, the work done by incident elastic rod \(U(t)\) is all converted into the deformation energy \(W_{s}(t)\) of jointed rock mass sample. Therefore, the deformation energy \(W_{s}(t)\) of the joint  
Figure 8: Time\(-\)history curves of three\(-\)wave strain of specimens under different loading conditions. sample can be obtained from the stress wave energy in the incident elastic rod and can be expressed as the Equation (6). \[W_{s}(t)=U(t)=W_{i}(t)-W_{r}(t)-W_{l}(t)0\ll t\ll t_{0} \tag{6}\]  
where t is the independent variable time, \(t_{0}\) is the dynamic loading duration, which can be determined by the stress wave duration. According to the elastic theory, the energy of incident wave, reflected wave and transmitted wave can be calculated from the following Equation to obtain Equations (8)-(10). \[W_{i}\ (t) = Ac_{0}E\int_{0}^{t_{0}}\epsilon_{i}^{2}(t)dt \tag{7}\] \[W_{r}\ (t) = Ac_{0}E\int_{0}^{t_{0}}\epsilon_{r}^{2}(t)dt\] (8) \[W_{l}\ (t) = Ac_{0}E\int_{0}^{t_{0}}\epsilon_{l}^{2}(t)dt \tag{9}\]  
Substitute Equations (7)-(9) into Equation (6) to get Equation (10). \[W_{s}\ (t) = Ac_{0}E\int_{0}^{t_{0}}\epsilon_{i}^{2}(t)-\epsilon_{r}^{2}(t) dt-\epsilon_{t}^{2}(t) \tag{10}\]  
According to Equation (10), the stress wave data measured by SHPB experiment can be used to calculate the deformation energy of jointed rock samples. According to theoretical analysis and Equation (10), the deformation energy \(W_{s}(t)\) of jointed rock mass sample in the SHPB experiment is not a fixed value, but a quantity that changes with loading time. Under the action of stress wave in the incident rod, joint rock specimens under the action of stress wave deformation will occur. As stress wave energy is passed to the joint rock sample, sample deformation energy will also increase gradually, and reach the maximum value, at this time, the maximum stored energy is \(W_{s}max\). Then, with the continuous dynamic loading, the stress acting on the jointed rock samples will gradually decrease. The deformation of the sample gradually recovers, and the deformation energy released in the process of deformation recovery is transferred to the elastic rod in the form of elastic wave. The deformation energy of the sample gradually decreases until the end of dynamic loading. Due to the inelastic characteristics of jointed rock samples, in the process of dynamic loading and unloading part of the deformation energy will be consumed by jointed rock samples, resulting in a loss of stress wave energy; this part of the loss energy is equal to the energy consumed by the rock samples, which is called dissipation energy \(W_{l}\).