## 3 The Inversion of J-C Constitutive Parameters for Powder Liner
### Numerical Simulation of Penetrating Steel Target
#### 3.1.2 Material Model and Parameters  
1. Powder liner  
Currently, the Johnson-Cook (J-C) constitutive model and Grunensin equation of state are commonly employed in numerical simulations of perforation to characterize the mechanical behavior of the liner during jet molding [16]. Specifically, the Grunensin equation of state is primarily utilized to describe the thermodynamic properties of a metal jet under high temperature and pressure conditions, encompassing relationships between physical quantities such as density, pressure, and temperature. The expression is presented below:  
\[P=\frac{\rho_{0}c^{2}\mu\big{[}1+\big{(}1-\frac{\gamma_{0}}{2}\big{)}\mu-\frac {\pi}{2}\mu^{2}\big{]}}{\big{[}1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{3} \frac{\mu_{3}}{\mu+1}\big{]}^{2}}+(\gamma_{0}+\alpha\mu)E \tag{1}\]  
where \(\rho_{0}\) is the initial density of the material, \(C_{0}\) is the speed of sound, \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(\alpha\) are material characteristic parameters, \(E\) is the initial internal energy, \(\mu=\rho/\rho_{0}-1\) is the relative compression degree, \(\gamma_{0}\) is the Gaussian coefficient, and \(\rho\) is the density of the material. For the powder liner used in this paper, its initial density \(\rho_{0}\) is 14.96 g \(\cdot\) cm\({}^{-3}\), and \(C_{0}\) = 394 m/s, \(S_{1}\) = 1.49, \(S_{2}\) = \(S_{3}\) = 0, \(\gamma_{0}\) = 1.99, which are selected according to the literature [17]. The J-C constitutive model, proposed by Johnson and Cook in 1983, is designed to accurately capture the deformation flow behavior of metal materials under high temperatures, high strain rates, and large deformation conditions [18]. The mathematical expression is presented as follows:  
\[\overline{\sigma}=\big{[}A+(B\bar{\tau})^{n}\big{]}\bigg{[}1+C\ln\bigg{(} \frac{\overline{\tau}\overline{\tau}}{\overline{\tau}_{0}}\bigg{)}\bigg{]} \bigg{[}1-\bigg{(}\frac{T-T_{\rm r}}{T_{\rm m}-T_{\rm r}}\bigg{)}^{m}\bigg{]} \tag{2}\]  
where \(\overline{\sigma}\) is the flow stress, \(A\) is the initial yield strength, \(B\) is the strain hardening coefficient, \(C\) is the strain rate sensitivity coefficient, \(m\) is the temperature coefficient, \(n\) is the strain hardening index, \(\bar{\tau}^{\rm p}\) is equivalent plastic strain, \(\overline{\bar{\tau}}^{\rm p}\) is equivalent plastic strain rate, \(\overline{\bar{\epsilon}}_{0}\) is reference strain rate, \(T\) is temperature, \(T_{\rm r}\) is the reference temperature (generally room temperature), and \(T_{m}\) is the melting temperature of the material under normal conditions. Among them, \(A\), \(B\), \(C\), \(m\), and \(n\) are the key parameters of J-C constitutive, and also the potential, inversion parameters. The parameter values are shown in Table 1.  
Since the constitutive parameters of the powder liner in this type of shaped charge are unknown, we utilize the constitutive parameters of the copper liner as reference values for conducting numerical simulation analysis. It should be pointed out that in current numerical simulation studies involving perforation, researchers mostly choose the constitutive parameters of copper as the constitutive parameters of the liner, regardless of the material of the liner, because copper liner was widely used in perforating operations in the early days, and its constitutive parameters have been well developed and clear (see Table 1). Therefore, in this paper, the mature constitutive parameters of copper liner are used as reference values to carry out numerical simulation analysis in order to evaluate the gap between the simulation results and the experimental results. \begin{table}
\begin{tabular}{c c c c c} \hline \(A\)/MPa & \(B\)/MPa & \(C\) & \(n\) & \(m\) \\ \hline
90 & 292 & 0.025 & 0.31 & 1.09 \\ \hline \end{tabular}
\end{table} 
Table 1: Key parameters of the constitutive equation of copper liner [19].