## 5 Verification of Collision Finite Element Method (CFEM)
### Verification and Parameter Determination of J-C Constitutive Model  
In order to verify the accuracy of the J-C constitutive model and the presented LSM parameter determination method, this paper firstly obtains the parameters of the J-C constitutive equation by normalizing the tensile test data of 50 SiMnVB steel at different strain rates, and, through the theory and data processing analysis in Section 3, as shown in Table 3, the model parameters corresponding to the strain rate of \(10^{-3}\) s\({}^{-1}\) are given. \begin{table}
\begin{tabular}{c c} \hline
**Position** & **Boundary State** \\ \hline \(x=-a/2\) & \(u_{x}=u_{y}=u_{z}=\gamma_{x}=\gamma_{y}=\gamma_{z}=0\) \\ \(y=-b/2\) & \(u_{x}=u_{y}=u_{z}=\gamma_{x}=\gamma_{y}=\gamma_{z}=0\) \\ \(x=a/2\) & \(u_{z}=\gamma_{x}=0\) \\ \(y=b/2\) & \(u_{z}=\gamma_{y}=0\) \\ \hline \end{tabular}
\end{table} 
Table 2: Boundary conditions used in numerical model. Figure 2: PMMA impact model and mesh generation. (**a**) Local mesh (a quarter part). (**b**) Finite element overall model. By testing SHPB at different strain rates, [PERSON] [8] obtained quasi-static and dynamic curves of normalized 50 SiMnVB steel at room temperature, and carried out the J-C model parameters by fixing the strain rate and fitting coefficient from test curves, shown as Figure 3. Then, in this paper, the presented LSM is carried out to confirm the J-C model parameters, as seen in Table 3 and Figure 3.  
Similarly, for the PMMA compression test data [13], the corresponding J-C constitutive model parameters can be obtained, as shown in Table 3 and Figure 4. From the figure, it can be seen that the J-C model has a good fit with the experimental values, especially in the initial stage of plastic strain. Due to the varying speed of strain changes during the process of structural breakdown during impact, the strain rate varies. Therefore, for PMMA materials with different strain rates, the J-C model was compared and analyzed with experimental values, as shown in Figure 5. As shown in the figure, for PMMA materials, the J-C model can characterize the plastic or viscoelastic properties of the material within a certain range under different strain rates, with good accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & & \(A\) & \(B\) & \(n\) & \(C\) \\ \hline
50 SiMnVB steel & reference [8] & 615 & 588 & 0.408 & 0.034 \\  & Presented J–C model & 640 & 547 & 0.439 & 0.033 \\ PMMA & Presented J–C model & 55 & 312.78 & 0.62 & 0.105 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter determination of the J–C constitutive model for steel and PMMA. Figure 3: Steel J–C constitutive model verification (\(\dot{\epsilon}=10^{-3}\) s\({}^{-1}\)) [8]. Figure 5: PMMA J–C constitutive model verification for different strain ratios [13]. ### Verification and Parameter Determination of J-C Constitutive Model  
For the J-C damage failure model, this paper uses the previous least squares method to determine the parameters of the J-C damage failure model from the quasi-static mechanical property experiment and compression SHPB experiment data of material specimens under different stress states and temperatures; that is, \(D_{1}\), \(D_{2}\), and \(D_{3}\) are determined from the quasi-static triaxial stress fracture test, and \(D_{4}\) and \(D_{5}\) are determined from the fracture test under different strain rates and temperatures. The J-C damage failure model parameters of 45 steel obtained in this article are shown in Table 3.  
From Table 3, it can be seen that the parameters determined by the least squares method in this article are basically consistent with the literature values. At the same time, based on the parameters and experimental data determined in this article, a comparative analysis was conducted on the J-C damage failure model and experimental data of 45 steel, as shown in Figures 6-8. From the figures, it can be seen that the J-C damage model has a good agreement with the experimental values.