## 3 Modifications to the NNO BLE  
In this study we developed a set of functions that, when incorporated into the NNO BLE, significantly improve its predictive ability for a foam-filled dual-wall system. These functions were intended to more properly take into account the material properties of the impacting projectile and the walls in the dual-wall system. In the modified NNO BLE, the presence of the foam in the dual-wall system under consideration is taken into account in the same manner in which they are traditionally included when the original NNO BLE is applied to such dual-wall systems. Namely, in both cases, the thicknesses of the outer and inner walls are increased slightly using a mass equivalence calculation that allocated 50% of the foam filler mass to the outer wall and 50% to the inner wall. The following equation gives a top-level perspective of how the original NNO BLE is to be modified for these types of wall configurations:  
\[d_{crit}^{mod}=d_{crit}^{orig}*f_{1}(\rho_{rw},\sigma_{rw})*f_{2}(\rho_{p})*f_{ 3}(\theta_{p}) \tag{1}\]  
where \(d_{crit}^{mod}\) and \(d_{crit}^{orig}\) are the modified and original critical, or ballistic limit, projectile diameters as predicted by the modified and original NNO BLE, respectively. In this equation, the function \(f_{1}\) accounts for the inner wall density and strength, if different from aluminum, the function \(f_{2}\) accounts for the density of non-aluminum projectiles, and the function \(f_{3}\) accounts for the effects of impact obliquity. The forms of the modifier functions \(f_{1}\), \(f_{2}\),and \(f_{3}\) in Equation (1) are guided by expected asymptotic function values or the roles played by those functions in modifying the original NNO BLE. For example, as projectile density approaches that of aluminum (from below, that is, when \(\rho_{p}\) becomes greater than \(\sim\)2.0 gm/cm\({}^{3}\)), all modification functions should approach unity (i.e., the modifiers should all approach unity when aluminum projectiles are considered because that is the projectile material on which-for the most part-the NNO BLE is based). The same should be true when aluminum walls are used, that is, when \(\rho_{rw}\) and \(\sigma_{rw}\) take on values corresponding to those of aluminum-in these cases, the values of the modifier functions should then also all approach unity. The following equations define the modifying functions \(f_{1}\), \(f_{2}\), and \(f_{3}\):  
\[f_{1}(\rho_{rw},\sigma_{rw})=1-exp\Big{\{}-15.73\Big{[}(f_{RW}*f_{RWD})^{-9.8 26}\Big{]}\Big{\}} \tag{2}\]  
where  
\[f_{\text{RWS}}=(\sigma_{RW}/50)^{\{1-exp[-7,158(\sigma_{RW}/50)^{-3.949}]\}} \tag{3a}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Wall** & **Material** & **Density** & **Source** & **Strength** & **Source** \\
**Type** & & **(gm/cm\({}^{3}\))** & **Source** & **(ksi)** & **Source** \\ \hline \multirow{4}{*}{Outer Wall} & PETP Fabric & 0.769 & \multirow{4}{*}{\
\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{
\begin{tabular}{} \end{tabular} } \\  & Rayon Fabric & 0.713 , 0.832 [FOOTNOTE] & & 56.8 & Ref. [13] \\  & Rayon Fabric & 0.549 & & 60.0 & Ref. [14] \\  & T300/Epoxy & 1.53, 1.64 & references where used & 290, 264 & Toray data sheets \\  & IM7/Epoxy & 1.58 & Calculated & 397 & Hexcel data sheets \\ \hline \end{tabular}  
* \({}^{1}\) Laminated Fabric, \({}^{2}\) Unaminated fabric, \({}^{3}\) Elastomer coated fabric, \({}^{4}\) Uncoated fabric.  
\end{table} 
Table 3: Density and Strength Values for Inner Wall Materials. \[f_{\text{RWD}}=(\rho_{\text{RW}}/2.71)^{\{3.0[1-exp(-0.007967\rho_{\text{RW}} \text{\@@cite[cite]{[\@@bibref{}{RWD}{}{}]}}^{2.34)}]\}} \tag{3b}\] \[f_{2}\big{(}\rho_{p}\big{)}=1+(MF_{0}-1)*\ \Big{[}1-exp\Big{(}-f_{PD}(1-V_{P}/72)^{ f_{VP}}\Big{)}\Big{]}/[1-exp(-f_{PD})]\] (4) with \[\left\{ \begin{aligned} & MF_{0}\\ & f_{PD}\\ & f_{VP}\end{aligned}