# Experimental Study of the Thermal and Dynamic Behaviors of Polypropylene Fiber-Reinforced Concrete
## 2 Materials and Methods
### Principles of Split Hopkinson Pressure Bar (SHPB)  
Figure 5 shows a diagram of the Hopkinson pressure bar during the dynamic test in order to illustrate the principle of the SHPB during HPC dynamic testing. Many researchers have explained the principles of the SHPB during the test in terms of mathematics [55; 56]. In the following paragraphs, the principles of the split Hopkinson pressure bar (SHPB) are explained in terms of wave propagation based on previous research [5; 55; 56; 57]. When the striker bar hits the incident bar, a compressive stress pulse wave of approximately one-dimensional propagation is generated in the incident bar. During the test, the compressive stress wave in the incident bar will propagate to the interface (the 1-1 interface in Figure 5). Some of the compressive stress waves can continue to propagate into the concrete samples, while the rest of the compressive stress waves will be reflected into the incident bar. As the compressive stress waves in the concrete sample reach the contact surface of the sample and the transmission bar (the 2-2 interface in Figure 5), the stress waves will partly be reflected back to the concrete sample and partly transmitted into the transmission bar, as happened in the 1-1 interface. As the compressive stress waves are reflected back and forth three to six times through the 1-1 and 2-2 interfaces in the rock sample, the stress equilibration is established in the rock sample. Figure 4: Schematic of conventional split Hopkinson pressure bar (SHPB) adapted from [57]. Figure 3: Split Hopkinson pressure bar (SHPB) system. As illustrated in Figure 3, the cross-sectional area of the incident bar and the transmission bar is \(A_{0}\), while the cross-sectional area and length of the concrete sample are \(A\) and \(L\), respectively. The stress at the interface of 1-1 and interface of 2-2 is \(\sigma_{1}(t)\) and \(\sigma_{2}(t)\), respectively. \(C_{0}\) and \(E_{0}\) indicate the wave velocity and the elastic modulus, respectively, of the incident bar and the transmission bar. \(\varepsilon_{i}\) and \(\varepsilon_{r}\) indicate the incident wave and the reflected wave in the incident bar. The transmission wave in the transmission rod is \(\varepsilon_{t}\). \(u_{1}\) and \(u_{2}\) indicate the mass velocities at the interface of specimen 1-1 and the interface of specimen 2-2, respectively. If the average strain in the specimen is \(\varepsilon\), the strain rate is \(\dot{\varepsilon}\). Based on the continuity condition of the displacement and the one-dimensional stress hypothesis of stress waves, the following equations can be constructed. Velocity at the interface 1-1:  
\[u_{1}(t)=C_{0}[\varepsilon_{i}(t)-\varepsilon_{r}(t)] \tag{1}\]  
Velocity at the interface 2-2:  
\[u_{2}(t)=C_{0}\varepsilon_{t}(t) \tag{2}\]  
Strain rate in the rock sample:  
\[\dot{\varepsilon}(t)=\frac{u_{1}(t)-u_{2}(t)}{L}=\frac{C_{0}}{L}[\varepsilon_ {i}(t)-\varepsilon_{r}(t)-\varepsilon_{t}(t)] \tag{3}\]  
Strain during time \(t\):  
\[\varepsilon(t)\;=\frac{c_{0}}{L}\int_{0}^{t}[\varepsilon_{i}(t)\;-\;\varepsilon _{r}(t)\;-\;\varepsilon_{t}(t)]dt \tag{4}\]  
Stress at 1-1 interface:  
\[A\sigma_{1}(t)=A_{0}E_{0}[\varepsilon_{i}(t)+\varepsilon_{r}(t)] \tag{5}\]  
Stress at 1-1 interface:  
\[A\sigma_{2}(t)=A_{0}E_{0}\varepsilon_{t} \tag{6}\]  
Average stress in specimen:  
\[\sigma(t)=\frac{[\sigma_{1}(t)+\sigma_{2}(t)]}{2}=\frac{A_{0}E_{0}}{2A}[ \varepsilon_{i}(t)+\varepsilon_{r}(t)+\varepsilon_{t}(t)] \tag{7}\]  
When the stress pulse wave propagates to and fro several times in the specimen, a stress equilibrium state is established. In this case, the three strains are equal, as shown in Equation (8). \[\varepsilon_{i}+\varepsilon_{r}=\varepsilon_{t} \tag{8}\]  
Figure 3:0 Schematic diagram of rock sample under impact by SHPB adapted from [57]. Thus, by substituting Equation (8) into Equations (1)-(7), the following equations can be achieved:  
Figure 3:1  
Figure 3:2  
Figure 3:3  
Equations (8) and (9) can be used to calculate the stress, strain, and strain rate in this research.