## 3 Results and Analysis
### Energy Analysis of jointed Specim under Impact Load  
The energy input of the SHPB apparatus is realized by a bullet impact. According to the assumption of the one-dimensional stress wave theory and ignoring the energy attenuation during the wave propagation in the bar, the incident wave energy \(E_{I}\), reflected wave energy \(E_{R}\), and transmitted wave energy \(E_{T}\) can be expressed as:  
\[E_{I}=A_{0}C_{0}E_{0}\int_{0}^{t}\epsilon_{t}^{2}(t)dt \tag{3}\]  
\[E_{R}=A_{0}C_{0}E_{0}\int_{0}^{t}\epsilon_{r}^{2}(t)dt \tag{4}\]  
\[E_{T}=A_{0}C_{0}E_{0}\int_{0}^{t}\epsilon_{t}^{2}(t)dt \tag{5}\]  
Figure 8: Reflection and transmission stress waves of jointed specimens with various joint inclination angles when JRC is 0. (**a**) Reflected wave, (**b**) Transmitted wave. Figure 9: Reflection and transmission stress waves of jointed specimens with various JRCs when the joint inclination angle is 60\({}^{\circ}\). (**a**) Reflected wave, (**b**) Transmitted wave. where \(E_{I},\ E_{R},\) and \(E_{T}\) representing the energy of incident, reflected, and transmitted waves, respectively; \(A_{0},\)\(C_{0},\) and \(E_{0}\) represent the cross-sectional area, longitudinal wave velocity, and elastic modulus of incident and transmission bar. According to the first law of thermodynamics, the energy during wave propagation should satisfy the following relationship:  
\[E_{I}=E_{R}+E_{D}+E_{T} \tag{6}\]  
where \(E_{D}\) represents the energy dissipated by the jointed specimen, which is the irreversible dissipation energy. In order to reduce the influence of the incident wave energy, reflected energy coefficient and transmitted energy coefficient are introduced to analyze the effect of serrated joints on stress wave propagation quantitatively, and they are calculated as follows:  
\[R_{r}=\frac{E_{R}}{E_{I}} \tag{7}\]  
\[R_{t}=\frac{E_{T}}{E_{I}} \tag{8}\]  
where \(R_{r}\) and \(R_{t}\) represent the reflected energy coefficient and transmitted energy coefficient, respectively. Figure 10 illustrates the relationship between the average values of the reflected energy coefficient and transmitted energy coefficient with the JRC and joint inclination angle when the stress wave propagates through jointed specimens, as obtained from Table 2. The following patterns can be observed by analyzing Figure 10:  
1. As the joint inclination angle increases from \(0^{\circ}\) to \(45^{\circ}\), the reflected energy coefficient progressively increases, while the transmitted energy coefficient gradually decreases. On the other hand, with the joint inclination angle increasing from \(45^{\circ}\) to \(90^{\circ}\), the reflected energy coefficient decreases, and the transmitted energy coefficient increases. At the joint inclination angle of \(45^{\circ}\), the reflected energy coefficient reaches its maximum value, and the transmitted energy coefficient reaches its minimum value. 2. With the increase of JRC from 0 to 20, the reflected energy coefficient gradually increases at joint inclination angles of \(0^{\circ}\) or \(90^{\circ}\), while the transmitted energy coefficient decreases progressively. Additionally, with the increase of JRC, the reflected energy coefficient gradually decreases when joint inclination angle is in the range of \(30^{\circ}\) to \(60^{\circ}\), while the transmitted energy coefficient gradually increases. \[E_{R}=A_{0}C_{0}E_{0}\int_{0}^{t}\epsilon_{r}^{2}(t)dt \tag{4}\]0 Energy coefficients variation of jointed specimens with various JRCs and angles. **(a)** Reflected energy coefficient, **(b)** Transmitted energy coefficient. behavior arises due to the combined effects of axial compressive stress and shear stress on the joint surface. At JRC of 20, the integrity of the jointed specimen is improved due to the large height of serrated joints, and cracks generate at the end of the joint surface and propagate along the direction of the joint inclination, exhibiting conspicuous tensile failure.