## 2 Numerical Modelling
### Blast Load Modelling  
Modelling the blast loading can be approached in a number of different ways and methods. Two different approximations of the blast are explored in this work: (i) the Load Blast Enhanced (LBE) method and (ii) the Arbitrary Lagrangian Eulerian (ALE) approach. Both of the methods are implemented in LS-DYNA and they are commonly used for reproducing blast events. Numerous examples can be found in the literature using either ALE [30; 31; 32; 33; 34; 35] or LBE [36; 37; 38; 39; 40; 41; 42]. The Load Blast Enhanced (LBE) approach is an empirical method in which the explosive is simplified to a spherical air burst, the pressure imposed by the wave on exposed surfaces is calculated while taking the mass of the high-explosive, the distance to the target, and the angle of incidence relative to the location of the explosive into account [43]. This approach is only valid on a limited range of scaled distances (\(0.147<Z<40\) [m/kg\({}^{1/3}\)]) [44]. The work reported here is covered by this range as \(Z=0.965\) m/kg\({}^{1/3}\). One of the main advantages of this approach is the low computational effort required to define the blast, because the air domain is not explicitly modelled. Conversely,  
Figure 3: Mechanical behaviour of the NL20 agglomerated cork compound [10; 29]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Units** & **Value** \\ \hline Density, \(\rho\) & kg/m\({}^{3}\) & 200 \\ [PERSON]’s modulus, \(E\) & GPa & 22.8 \\ [PERSON]’s modulus of compacted material, \(E_{\text{c}}\) & GPa & 90 \\ [PERSON]’s ratio of compacted material, \(\
u_{\text{c}}\) & – & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material parameters for the NL20 agglomerated cork compound [10; 29]. as a simplification, the approach does not take into account effects, such as shadowed surfaces, reflected waves, etc. The blast is uniquely defined by the mass and location of the explosive, and the surfaces onto which the wave will impact. The Arbitrary Lagrangian Eulerian (ALE) approach explicitly defines the high-explosive, including its shape, the surrounding air, and the coupling between the fluids and the structure. This significantly increases the computational cost, but takes into account all possible reflections and interactions between the shock waves from the detonation and structures. To this end, it is necessary to define not only the high-explosive material and its equation of state, but also the behaviour of the surrounding air. The air domain must involve the structures that will be subjected to the blast. This region is shown in Figure 2 and labelled as ALE domain. This domain has a volume \(200\times 400\times 450\) [mm\({}^{3}\)] and an element size of 4 mm was used, which resulted in a total of \(5.65\times 10^{5}\) elements. The external surfaces of this volume are defined as non-reflecting, following the observations and recommendations of [PERSON] et al. [30]. The explosive is a 16.5 mm radius sphere and its behaviour is described by a high-explosive material using a Jones-Wilkins-Lee (JWL) equation of state (EoS). The release of chemical energy from the detonation is accounted for in this EoS, both from the resulting compression and the programmed detonation [18; 45]. The former defines the detonation when the compression in the material reaches the Chapmant-Jouget pressure (\(P_{\rm C}\)), whilst the latter defines the initial detonation time as a function of the detonation velocity \(D\) and the location of the detonation point [46]. The JWL model defines the pressure as  
\[p=A\left(1-\frac{\omega}{R_{1}\
u}\right)e^{-R_{1}\
u}+B\left(1-\frac{\omega} {R_{2}\
u}\right)e^{-R_{2}\
u}+\frac{\omega\bar{E}}{\
u}, \tag{5}\]  
where \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are material constants, \(\
u=\rho_{0}/\rho\) is the ratio of the initial to the current densities (i.e., the relative volume) and \(\bar{E}\) is the internal specific energy (i.e., per unit volume).