## 4 Numerical Work
### Shock Tube Tests  
#### 4.3.1 Numerical Model  
Figure 15 shows the numerical model used in the simulations of the shock tube tests. One-quarter of the test setup was modelled to decrease computational costs. The aluminium skins were modelled with quadrilateral four-node reduced integration Belytscko-Tsay (type 2) elements, with five integration points across the thickness and stiffness-based hourglass control. In the central blast-exposed area of the plate, an element size of 1 mm \(\times\) 1 mm was used, while for the rest of the skin an element size of 4 mm \(\times\) 4 mm was applied. MAT_107, i.e., the Modified Johnson-Cook model in LS-DYNA, with parameters as described in Section 4.2.1, was used to model the behaviour of the aluminium skins. The core of the sandwich panel was modelled using eight-node hexahedral elements with reduced integration (type 1). The same mesh as used for the skins was extruded in 1 mm intervals through the thickness of the core. This resulted in an element size of 1 mm \(\times\) 1 mm \(\times\) 1 mm in the blast-exposed area, and an element size of 4 mm \(\times\) 4 mm \(\times\) 1 mm in the periphery. The core materials were modelled with the modified version of the Deshpande-Fleck model described in Section 4.1 and the material parameters given in Section 4.2.2. Automatic surface-to-surface contact with a friction coefficient \(\mu=0.1\) was used between all parts of the model. To account for the significant difference in material stiffness, the option SOFT = 1 was used in the contact formulation between the core and the skins/clamping frame. The clamping frame and bolts were modelled as elastic using MAT_001 with density \(\rho=7800\) kg/m\({}^{3}\), [PERSON]'s modulus \(E=210\) GPa and Poisson's ratio \(\
u_{\mathrm{s}}=0.3\). Eight-node hexahedral elements with reduced integration and stiffness-based hourglass control (type 1) with an element size of about 5 mm  
Figure 15: Illustrations of the numerical model: (**a**) Complete assembly and (**b**) sandwich specimen. Note that the mesh for flanges and bolts is omitted for clarity. were used for the clamping frame and bolts. Preliminary simulations of the setup included preloading of the bolts by applying a stress to the bolts' cross-section and dynamic relaxation. The applied preload showed no significant increase in accuracy and was omitted in the numerical study due to the high computational cost. Based on the measured densities, shown in Figure 2b, the core materials in the blast simulations of the sandwich components were given a density distribution, which is shown in Figure 16. As the density must be constant within each element, the applied density distribution is discretised to fit the element size of the model. This discretisation of the density distribution does introduce a mesh dependence in the models, but with the chosen element size of 1 mm \(\times\) 1 mm \(\times\) 1 mm the discretisation is sufficiently fine to capture the measured distributions. The blast load was applied to the exposed area of 300 mm \(\times\) 300 mm of the front skin as a tabulated pressure-time curve based on experiments using massive plates [26]. Pressure-time histories, as measured by pressure sensors mounted on the surface of massive steel plates, were fitted to the Friedlander equation given as  
\[P_{r}(t)=P_{r,max}\left(1-\frac{t-t_{a}}{t_{d+}}\right)\exp\left(\frac{-b(t-t_ {a})}{t_{d+}}\right) \tag{20}\]  
where \(P_{r,max}\) is the peak reflected overpressure, \(t_{a}\) is the arrival time of the blast wave at the specimen, \(t_{d+}\) is the duration of the positive phase, and b is the exponential decay coefficient. The results from tests on massive plates and the subsequent fitted parameters of the Friedlander equation were taken from [PERSON] et al. [26] for tests with 10 and 15 bar firing pressure. For the tests with 12 bar driver pressure, no tests on massive plates were available.