# Modeling of Impact Energy Release of PTFE/Al Reactive Material
## 2 Modeling of PTFE/Al Impact Energy Release  
### Parameter Determination of JWL Equation for Unreacted PTFE/Al  
[PERSON] [22] modified the isentropic equation based on the work of [PERSON] and [PERSON] and found that the better form of the C-J isentropic equation is:  
\[P_{s}=Ae^{-R_{1}\overline{v}}+Be^{-R_{2}\overline{v}}+C\overline{v}^{-(\omega +1)} \tag{1}\]  
In the formula, \(A\), \(B\), \(C\), \(R_{1}\), \(R_{2}\), and \(\omega\) are constant coefficients, \(P_{s}\) is pressure, \(\overline{v}=\
u/v_{0}\) is the relative specific volume, and \(v_{0}\) is the initial specific volume. On the isentropic line, the change of internal energy \(E\) with \(\overline{v}\) is:  
\[E_{s}=-\int P_{s}d\overline{v}=\frac{A}{R_{1}}e^{-R_{1}\overline{v}}+\frac{B} {R_{2}}e^{-R_{2}\overline{v}}+\frac{C}{\omega}\overline{v} \tag{2}\]  
The Gruneisen equation of state can be written as [23]:  
\[P(E,\overline{v})=P_{s}+\frac{\Gamma}{\overline{v}}(E-E_{s}) \tag{3}\]  
In the formula, \(\Gamma\) is the Gruneisen coefficient. Substitute Equations (1) and (2) into Equation (3) to get:  
\[P(E,\overline{v})=A\bigg{(}1-\frac{\omega}{R_{1}\overline{v}}\bigg{)}e^{-R_{ 1}\overline{v}}+B\bigg{(}1-\frac{\omega}{R_{2}\overline{v}}\bigg{)}e^{-R_{2} \overline{v}}+\frac{\omega E}{\overline{v}} \tag{4}\]  
Equation (4) is the standard JWL equation of state. In LS-dyna, the JWL state equation adopts the expression with the temperature form:  
\[P=Ae^{-R_{1}\overline{v}}+Be^{-R_{2}\overline{v}}+\frac{\omega\overline{c_{v} }}{\overline{v}}T \tag{5}\]  
where \(T\) is the temperature in energetic material after the shock wave and \(\overline{c_{v}}\) is the specific heat capacity per unit volume. According to the mass, momentum, and energy conservation relations before and after the shock wave, the physical quantities of the reactive material after the shock wave can be calculated [24]:  
\[\rho_{0}(D-u_{0})=\rho(D-u) \tag{6}\]  
\[P-P_{0}=\rho_{0}(D-u_{0})(u-u_{0}) \tag{7}\]  
\[Pu-P_{0}u_{0}=\rho_{0}(D-u_{0})\Bigg{[}\bigg{(}E+\frac{u^{2}}{2}\bigg{)}- \bigg{(}E_{0}+\frac{u_{0}^{2}}{2}\bigg{)}\Bigg{]} \tag{8}\]In the formula, \(P\), \(\rho\), \(and\)\(u\) are pressure, density, and particle velocity after the shock wave, respectively; \(P_{0}\), \(\rho_{0}\), and \(u_{0}\) are the initial pressure, density, and particle velocity, respectively; \(D\) is the shock wave velocity; \(E\) is the total energy, including internal energy and the energy released by chemical reactions; \(E_{0}\) is the initial internal energy. In a wide pressure range, the shock wave velocity \(D\) in the condensing medium has the following relationship with the post-wave particle velocity \(u\)[25]:  
\[D=C_{0}+Su \tag{9}\]  
Assuming \(u_{0}\) = 0 and \(P_{0}\) = 0, the two conservation relations Equations (6) and (7) and the \(D\)-\(u\) relation can be used to calculate the impact compression line of the unreacted reactive material:  
\[P_{H}=\frac{C_{0}^{2}(v_{0}-v)}{\left[v_{0}-S(v_{0}-v)\right]^{2}} \tag{10}\]  
There are only two unknown constants, \(C_{0}\) and \(S\), in Equation (10), so as long as the material parameters of PTFE/Al are obtained, the impact compression line of PTFE/Al reactive material can be obtained. The material parameters of PTFE/Al can be calculated by the Hill mixture rule, and Equation (11) is the Hill mixture rule. According to Equation (11) and material parameters of PTFE and Al (see in Table 1), the density, sound velocity, and Hugoniot parameters of the PTFE/Al(73.5/26.5) reactive material were calculated and the parameters are recorded in \[E_{s}=-\int P_{s}d\overline{v}=\frac{A}{R_{1}}e^{-R_{1}\overline{v}}+\frac{B} {R_{2}}e^{-R_{2}\overline{v}}+\frac{C}{\omega}\overline{v} \tag{2}\]0 According to the material parameters of PTFE/Al, the impact compression line of the PTFE/Al reactive material was obtained when PTFE/Al did not react.