## 4 Modelling Approach and Results  
### Computational Simulation Approach  
To provide additional insight into the spatial impulse and transient behaviour of the structures, the tests were simulated using LS-Dyna R8 [31]. The model geometry is shown schematically in Figure 17. The V-structure, clamp frame, air block, and explosive were modelled using the multi-material arbitrary Lagrange-Eulerian (MMALE) fluid structure interaction (FSI) approach. The tests were modelled in quarter-symmetry using the same geometry of V-structure specimens and clamp frames as the physical experiment. A termination time of 300 us was used, following previous work by the authors [4]. The V-structure was modelled using Lagrangian shell elements with a Belytschko-Tsay element formulation (ELFORM 2) and a Johnson Cook (MAT15) material model. The element size was set to 1.0 mm to best capture the curvature of the V-tip geometry and to ensure the best coupling with the Eulerian grid. Quasi-static uniaxial tensile tests were performed on samples from the Domex sheeting, following ASTM A370 [32]. Tests were performed at three engineering strain rates (10.4 \(\times\) 10\({}^{-4}\), 41.6 \(\times\) 10\({}^{-4}\), and 208.3 \(\times\) 10\({}^{-4}\) s\({}^{-1}\)). The results were used to obtain parameters for the Johnson-Cook material model [33] of A = 770 MPa, B = 420 MPa, \(n\) = 0.5 and c = 0.0009 at the reference strain rate of 10.4 \(\times\) 10\({}^{-4}\) s\({}^{-1}\), defined using the standard notation. A Taylor-Quinney coefficient of 0.9, a Poisson's ratio of 0.33, a density of 7870 kg/m\({}^{3}\) and a Young's modulus of 210 GPa were assumed for the steel. Figure 16: Transient deformation profiles at discrete times, along with a ridge, in 105\({}^{\circ}\) V-structures with (**a**) 10 g PE4, 32 mm tip radius, (**b**) 15 g PE4, 62 mm tip radius. Based on previous work by the authors [4], the air was represented as a 200 mm by a 200 mm block (of varying height depending on the V-structure geometry) divided into 2 mm cubed elements, thereby minimising leakage forces and giving reasonable computational times. The volume of the explosive disc was discretised into elements that filled the air mesh using an initial volume fraction description with the detonation point placed 1 mm from the top of the explosive charge. The coupling between the explosive products and the test plate was modelled using a penalty-based approach. The air was modelled using an ideal gas equation of state and a null material model. Boundary conditions were imposed on the symmetry boundaries, and all other surfaces of the air block were defined as free boundaries, allowing outflow of the Eulerian air formulation. The PE4 was modelled using the JWL equation of state and the high explosive burn material model, with the properties given in Table 1. Recent work by [PERSON] et al. [34] on cylinder expansion tests showed that an inverse modelling approach for deriving JWL parameters for PE4 could lead to some improved fits with their experimental data, but to keep the simulations independent of the experiments herein, this approach was not adopted. Instead, the JWL parameters for PE4 were based on those commonly used for C4, following the same approach as [PERSON] [35] and [PERSON] and Langdon [36]. The clamping frame and bolts were modelled with solid elements using a constant stress element formulation (ELFORM 1) with an elastic material model for mild steel. A \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\rho_{0}\) & **D** & **P\({}_{\text{CJ}}\)** & **A** & **B** & **\(R_{1}\)** & \(R_{2}\) & \(\omega\) &
\begin{tabular}{c} \(E_{0}\) \\ **(GPa)** \\ \end{tabular} \\ \hline
1616 & 8193 & 28 & 609.77 & 12.95 & 4.5 & 1.4 & 0.25 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of PE4 used in the computational simulations [34; 35]. Figure 17: Schematic showing the computational model.