## 4 Numerical Simulation
### Determination of Concrete Material Parameters
Parameters such as density \(\rho\) and static uniaxial compressive strength \(f_{c}\) can be accurately obtained from the tests, and some of the parameters can be calculated from Equations (10)-(12):  
\[G=\frac{E}{2(1+v)} \tag{10}\]  
\[P_{c}=\frac{f_{c}}{3} \tag{11}\]  
Figure 17:0  
where \(G\) and \(E\) are shear modulus and elastic modulus, respectively; \(v\) is Poisson's ratio; and \(T\) is static tensile strength. The remaining parameters are less sensitive to the strength of the concrete material, and their values are determined according to the relevant literature [36; 37; 38]. Based on the test data of C40 plain concrete and S1 PP0.2 HyFRC at 800 \({}^{\circ}\)C and an impact air pressure of 0.3 MPa, the values of the \(A\), \(B\), \(C\), and \(N\) parameters were corrected by repeated calculations after substituting them into the constitutive model. It is worth mentioning that because there is no change in heat during the impact compression test of heated concrete, the damage caused by temperature to concrete specimens is macroscopically manifested in the deterioration of the physical and mechanical properties of concrete materials, which can be characterized by the change in material model parameters [39]. The basic mechanical parameters of the specimens at room temperature and after heating were obtained by static experiments. In this way, the numerical simulation results derived from the modified parameters of the HJC constitutive model can more accurately reflect the test results (see Tables 6 and 7). \begin{table}
\begin{tabular}{c c c c} Density \(\rho\) (kg/m\({}^{3}\)) & 2181 & Pressure at the crushing point \(P_{c}\) (MPa) & 4.9 \\ Shear modulus \(G\) (GPa) & 2.75 & Volumetric strain at the collapsing point \(\mu_{c}\) & 0.000675 \\ Normalized viscous strength \(A\) & 0.45 & Pressure at the compaction point \(P_{l}\) (MPa) & 810 \\ Normalized pressure hardening coefficient \(B\) & 1.28 & Volumetric strain at the compaction point \(\mu_{l}\) & 0.1 \\ Strain rate influence coefficient \(C\) & 0.006 & Damage factor \(D_{1}\) & 0.04 \\ Normalized pressure hardening coefficient \(N\) & 0.61 & Damage factor \(D_{2}\) & 1.0 \\ Static uniaxial compressive strength \(f_{c}\) (MPa) & 14.8 & Pressure parameter \(K_{1}\) (GPa) & 85 \\ Static tensile strength \(T\) (MPa) & 2.38 & Pressure parameter \(K_{2}\) (GPa) & \(-\)171 \\ Reference strain rate \(\hat{\epsilon}_{0}\) (ms\({}^{-1}\)) & 0.001 & Pressure parameter \(K_{3}\) (GPa) & 208 \\ Total plastic strain before damage EF\({}_{\text{min}}\) & 0.01 & Failure mode \(FS\) & 0.004 \\ Normalized maximum strength \(SF_{\text{max}}\) & 7 & & \\ \end{tabular}
\end{table} 
Table 6: Main parameters of the modified HJC constitutive model for C40 plain concrete. The velocities measured by the velocimetry system under 0.3 MPa, 0.4 MPa, and 0.5 MPa air pressure are 9.6 m/s, 11.8 m/s, and 14.1 m/s, respectively. The measured bullet velocities in the test were consistent with the bullet velocity settings in the numerical simulation. The modified waveforms and the comparisons of stress-strain curves can be seen to be in very good agreement (see Figures 21 and 22). The modified HJC model accurately reflects the mechanical properties of C40 plain concrete and S1 PP0.2 HyFRC. Numerical simulations considering the homogeneity of the specimens and the reference values taken for the parameters of the modified HJC model introduce errors. However, the errors of numerical simulation and test results are small, and the error range of peak stress and peak strain at different impact velocities is within 6% (see Table 8). The failure modes of C40 plain concrete and S1 PP0.2 HyFRC in the 800 \({}^{\circ}\)C temperature group at impact velocities of 9.6 m/s, 11.8 m/s, and 14.1 m/s were analyzed by numerical simulation (see Figures 23 and 24). The comparison reveals that the numerical simulation results are consistent with the failure modes obtained from practical tests.