## 4 Modeling of the Blast Loading
\(\rho_{f}\) is the density of the fluid on the impact side, and \(c_{f}\) the speed of sound of the fluid. The decay time of the positive pressure phase is denoted by \(t_{0}\). Figure 12 shows the momentum ratio as a function of \(q\), as well as the representative temporal magnitudes for a homogeneous steel plate of thickness h = 0.0254 m. See also [67]. Figure 12 provides simple interpretations for the two limits \(t_{0}/t_{*}\ll 1\) and \(t_{0}/t_{*}\gg 1\). Massive plates that are hardly set in motion by the (free-field) momentum \(I_{0}\) (\(t_{0}/t_{*}\ll 1\)) generally ensure an almost perfect rebound of the impacting pressure wave (\(-I_{0}\)) so that the final force impact exerted on the plate is approximately \(2I_{0}\). In general, the influence of the structure on the fluid movement decreases with decreasing plate thickness. In the limiting case of vanishing thickness, the plate acquires the velocity of the fluid and, like a free surface, experiences no momentum load. The significant effects of the ambient medium and the plate thickness on the transmitted momentum result from the specific time variable relationships. For a homogeneous plate in water, a value of \(q~{}=~{}0.98\) results in a transmitted momentum of about \(0.75I_{0}\). In contrast, with the distribution of the total plate mass that is possible for sandwich construction plates with correspondingly thin cover layers, significantly reduced momentum loads can be achieved. For example, with a mass distribution of one-third each for the upper cover surface, the core, and the bottom cover surface, the corresponding cover surface is only hit by a force impact of approximately \(0.38I_{0}\) (\(t_{*}~{}=~{}0.34\to q\approx 3.0\)). In the case of structures subjected to air-blast loading, this advantage is essentially non-existent due to the low density of the fluid. In that case, both a solid plate and a sandwich structure experience impulse changes of almost \(2I_{0}\) since the times of decay and fluid-structure interaction differ by several orders of magnitude, and a change in the thickness ratios has no significant effect. [PERSON] and [PERSON] [69] employed a simplified sequential model to describe the typical structural response of a sandwich construction under explosive loading with three successive phases. Phase I involves the fluid-structure interaction already described above and the transfer of momentum \(I_{t}\) to the upper surface layer. The resulting velocity  
\[v_{1}~{}=~{}\frac{I_{t}}{\rho h} \tag{5}\]  
Figure 12: Plot of the momentum ratio \(I_{t}/I_{0}\) as a function of the time scale ratio \(q\) for assessing fluid–structure interaction according to the Taylor relationship. of the upper surface leads to the continuous compression and densification of the core (phase II). In the case of weak explosive loading, the densification front can be stopped within the core, and the velocity of the upper surface layer can be completely slowed down. In such a case, the kinetic energy is dissipated mainly by plastic deformations of the core structure. Larger impinging pulses ultimately cause the acceleration of the entire structure, with similar velocities at the upper and bottom sides (except for the edge areas). In phase III, the sandwich component bends, and additional dissipation processes occur due to plastic membrane distortions in the cover layers. The spatial dependence of the pressure distribution due to different angles of incidence and propagation times of the shock wave remained mostly unconsidered in the previous explanations. However, such effects are included in the ConWep simulation method for explosion loads in air developed by the United States Army Corps of Engineers and implemented in ABAQUS/Explicit [70,71].