## 3 The Inversion of J-C Constitutive Parameters for Powder Liner
### Numerical Simulation of Penetrating Steel Target
#### 3.1.3 Numerical Simulation Result  
Based on the above model and known parameters, the numerical simulation of the penetrating steel target can obtain the perforation results shown in Figure 6, The green part is the metal jet. As can be seen from the figure, according to the constitutive parameters of copper, the continuity of the metal jet is poor, the final perforation depth in the steel  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Density/g\(\cdot\)cm\({}^{-3}\)** & **Elasticity Modulus/MPa** & **Poisson Ratio** & **Yield Strength/MPa** \\ \hline Steel target [22] & 7.83 & 2.06 \(\times\) 10\({}^{5}\) & 0.269 & 500 \\ \hline Shell case [23] & 7.83 & 2.07 \(\times\) 10\({}^{5}\) & 0.3 & 400 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Material parameters of steel target and shell case.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\rho\)/g\(\cdot\)cm\({}^{-3}\) & **\(E_{0}\)/MPa** & \(A_{0}\)/MPa & \(B_{0}\)/MPa & \(R_{1}\) & \(R_{2}\) & \(\omega\) \\ \hline
1.82 & 9.2 \(\times\) 10\({}^{3}\) & 8.5 \(\times\) 10\({}^{5}\) & 1.8 \(\times\) 10\({}^{4}\) & 4.6 & 1.3 & 0.38 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters of the explosive [20].  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(C_{0}\) & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(C_{6}\) & \(V_{0}\) & \(E_{1}\)/MPa \\ \hline
0 & 0 & 0 & 0 & 0.4 & 0.4 & 0 & 1.0 & 0.253 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material parameters of air [21].  
target is only 19.85 cm, and the perforation aperture is 1.31 cm. However, the average perforation depth obtained in the penetrating steel target experiment of shaped charge is 30.46 cm, so the error between the simulated results and the experimental results using copper constitutive parameters is 34.83%. It can be seen that the constitutive parameters of copper cannot accurately describe the deformation and flow behavior of the new powder liner during perforation. Therefore, to improve the accuracy and reliability of the numerical simulation and make it better serve the perforation research of oil and gas wells, it is necessary to invert the constitutive parameters of the new powder liner under detonation.  
Although parameters \(A,B,C,n\), and \(m\) in the J-C constitutive model have the potential to be considered as design variables, during the inversion process, the computational workload increases exponentially with an increasing number of inversion parameters. To effectively conduct inversion design, this study initially focuses on analyzing the individual effects of five constitutive parameters on perforation simulation results (perforation depth and aperture). The parameters with more significant impacts are then selected as inversion variables. Subsequently, the expressions of the inversion variables of perforation depth and perforation aperture are fitted by the response surface method. Based on the expression, the distribution of the Pareto optimal solution is iterated by a multi-objective genetic algorithm. Finally, by considering the distribution of Pareto optimal solutions and selecting the solution closest to the experimental results, we determine corresponding parameter values that can be regarded as J-C constitutive parameters for powder liner in this paper.