## 3 Experimental Results and Discussion
### Toughness Index  
In the evaluation of the post-peak mechanical properties of concrete, the use of the ductility index-peak strain to reflect the toughness of the material has a certain one-sidedness. Due to the large dispersion of concrete materials, there may be some errors in the peak strain of the experiment results;  
Figure 16: Typical failure modes of FRC under a dynamic load. Figure 15: Typical failure modes of plain concrete under a dynamic load. hence, the use of a non-dimensional relative index to describe the toughness of the material is more reasonable. According to the method of defining the toughness of steel FRC in the literature [32; 33], the formula for the toughness index can be defined for PVA-reinforced concrete as follows:  
\[\eta=\frac{W_{f}}{W_{e}} \tag{5}\]  
where \(W_{f}\) is the area of OCD, which is defined the area surrounded by the limit strain of \(20\times\mathrm{E}^{-3}\) in this paper; \(W_{e}\) is the elasticity energy consumed by concrete materials, which is defined by the area of OAB at the strain relative to \(0.85\lx@math@degree_{C}\); and \(f_{c}\) is the peak stress. A detailed diagram is shown in Figure 17.  
The above formula does not only account for both the elastic energy absorbed during the elastic phase and the plastic energy absorbed during the plastic phase of the fiber concrete, but also eliminates the energy calculation error caused by the dispersion of the concrete, which is reasonable for describing the toughness of the concrete [32]. The calculated results of the toughness indices according to the formula are shown in Table 4 and Figures 18 and 19 below. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Strain Rate (s\({}^{-1}\))** & \(W_{f}\) (**/m\({}^{3}\))** & \(W_{e}\) (**/m\({}^{3}\))** & **Toughness Index** \\ \hline C30 PVA0 & \(10^{-5}\) & 201.21 & 29.02 & 6.93 \\ C30 PVA0 & \(10^{-4}\) & 157.87 & 25.06 & 6.30 \\ C30 PVA0 & \(10^{-3}\) & 160.90 & 29.24 & 5.50 \\ C30 PVA0 & \(10^{-2}\) & 112.61 & 28.41 & 3.96 \\ C30 PVA0.2 & \(10^{-5}\) & 237.59 & 28.59 & 8.31 \\ C30 PVA0.2 & \(10^{-4}\) & 203.66 & 28.95 & 7.03 \\ C30 PVA0.2 & \(10^{-3}\) & 252.82 & 40.15 & 6.30 \\ C30 PVA0.2 & \(10^{-2}\) & 244.93 & 50.11 & 4.89 \\ C30 PVA0.4 & \(10^{-5}\) & 286.49 & 29.64 & 9.66 \\ C30 PVA0.4 & \(10^{-4}\) & 288.44 & 34.27 & 8.42 \\ C30 PVA0.4 & \(10^{-3}\) & 259.54 & 34.39 & 7.60 \\ C30 PVA0.4 & \(10^{-2}\) & 233.84 & 36.57 & 6.39 \\ C30 PVA0.6 & \(10^{-5}\) & 266.11 & 25.28 & 10.53 \\ C30 PVA0.6 & \(10^{-4}\) & 317.60 & 35.56 & 8.93 \\ C30 PVA0.6 & \(10^{-3}\) & 263.17 & 31.09 & 8.47 \\ C40 PVA0.6 & \(10^{-2}\) & 259.54 & 34.79 & 7.46 \\ C40 PVA0 & \(10^{-5}\) & 198.86 & 30.50 & 6.52 \\ C40 PVA0 & \(10^{-4}\) & 200.75 & 35.28 & 5.69 \\ C40 PVA0 & \(10^{-3}\) & 190.13 & 38.66 & 4.92 \\ C40 PVA0 & \(10^{-2}\) & 163.00 & 44.82 & 3.64 \\ C40 PVA0.2 & \(10^{-5}\) & 269.46 & 35.58 & 7.57 \\ C40 PVA0.2 & \(10^{-4}\) & 297.19 & 41.69 & 7.13 \\ C40 PVA0.2 & \(10^{-3}\) & 327.58 & 47.43 & 6.91 \\ C40 PVA0.2 & \(10^{-2}\) & 298.17 & 52.14 & 5.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculated results of toughness index for FRC at different strain rates. Figure 17: Diagram of the toughness index of concrete. from Table 4 that the elastic strain energy absorbed by the FRC increases with the increase of the strain rate. The increase of the strain rate leads to the increase of the strength and elastic modulus of the concrete. There is a corresponding increase in the area surrounded by the elastic phase of the concrete stress-strain curve, indicating that the elastic strain energy of the concrete increased. The addition of PVA can significantly enhance the toughness of concrete. In the quasi-static state (i.e., a strain rate of \(10^{-5}\) s\({}^{-1}\)), the toughness indices of FRC (C30) with 0.2, 0.4, and 0.6% content are increased by 19.9, 39.4, and 51.8%, respectively, compared to those of the plain concrete. The values are 16.2, 49.4, and 57.1%, respectively, for C40.