## 2 Numerical Simulation Methods
### Numerical Simulation Models and Parameters
#### 2.1.2 Numerical Simulation Parameters  
In order to numerically simulate the damage process and damage characteristics of the reactive fragment impacting steel plate, AUTODYN-3D is used in this paper. Following convergence testing, the boundary condition was chosen to be the transmit non-reflective boundary condition, and the smooth particle length was chosen to be 0.04 mm. Both reactive fragments and steel plates used the J-C failure model as a criterion for material failure. Both the Q345 steel and 304 SS plates were studied using the Zerilli-Armstrong (Z-A) strength model [37]. In contrast to 304 SS, which is a face-centered cubic (FCC) metal, Q345S is a body-centered cubic (BCC) metal. Different versions of the Z-A strength model exist for both face- and body-centered cubic metals [38], and they are more useful for describing the dynamic yield strength properties of metals with various crystal structures. Part of the model parameters came from the ANSYS18.2/AUTODYN software's built-in parameters for the same type of material, such as the constant class parameters of the material model, and part of them came from the basic physical properties of the material, such as the static yield strength, shear modulus, density, melting point, and other parameters. For FCC, the Z-A strength model is stated as  
\[Y=Y_{0}+C_{2}\sqrt{\epsilon}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right] \tag{2}\]  
For BCC, the Z-A strength model is stated as  
\[Y=Y_{0}+C_{1}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right]+C_{5}\epsilon^{n} \tag{3}\]  
where \(\epsilon\) is the equivalent plastic strain, \(\dot{\epsilon}\) is the equivalent plastic strain rate, and \(T\) is temperature. \(Y_{0},C_{1},C_{2},C_{3},C_{4},C_{5}\) and \(n\) are material constants. The Z-A model parameters of Q345S plate are shown in Table 1. The Z-A model parameters of 304 SS plate are shown in Table 2.  
Figure 1: Schematic diagrams of numerical simulation models: (**a**) spherical reactive fragment impacting 304 SS; (**b**) cylindrical reactive fragment impacting Q345S. The shock equations [39] of state are stated as  
\[P=P_{H}+\Gamma\rho(e-e_{H}) \tag{4}\]  
\[P_{H}=\frac{\rho_{0}c_{0}\mu(1+\mu)}{[1-(s-1)\mu]^{2}} \tag{5}\]  
\[e_{H}=\frac{1}{2}\frac{P_{H}}{P_{0}}\bigg{(}\frac{\mu}{1+\mu}\bigg{)} \tag{6}\]  
\[\mu=\frac{\rho}{\rho_{0}}-1 \tag{7}\]  
where \(P\) is the hydrostatic pressure, \(\rho_{0}\) is the initial density, \(C\) is the sound speed, \(e\) is the energy, and \(\Gamma\) and \(S\) are the material constants in the equation. Parameters of steel for shock equation of state are shown in Table 3.  
The ignition growth reaction rate equation, the shock equation of state when unreacted, and the JWL equation of state when reacted were integrated with the Steinberg-Guinan (S-G) strength model for the reactive fragments. S-G strength model parameters of reactive fragment are shown in \[Y=Y_{0}+C_{1}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right]+C_{5}\epsilon^{n} \tag{3}\]0 According to the S-G strength model [40]:  
\[Y=Y_{0}+C_{1}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right]+C_{5}\epsilon^{n} \tag{3}\]1  
\[Y=Y_{0}+C_{1}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right]+C_{5}\epsilon^{n} \tag{3}\]2  
The JWL equation of state is stated as  
\[Y=Y_{0}+C_{1}\exp\left[-C_{3}T+C_{4}T\log\dot{\epsilon}\right]+C_{5}\epsilon^{n} \tag{3}\]3 \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{Y_{0}}\) (Mbar) & \(\mathbf{C_{1}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{C_{5}}\) & \(\mathbf{n}\) & \(\mathbf{G}\) (Mbar) \\ \hline \(3.45\times 10^{-3}\) & 0.01033 & 0.00698 & \(4.15\times 10^{-4}\) & 0.00266 & 0.289 & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Q345S Z–A strength model parameters. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{Y_{0}}\) (Mbar) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{G}\) (Mbar) \\ \hline \(2.05\times 10^{-3}\) & 0.0089 & 0.0028 & \(1.15\times 10^{-4}\) & 0.85 \\ \hline \end{tabular}
\end{table} 
Table 2: 3045S Z–A strength model parameters.