## 4 Numerical Work
### Shock Tube Tests
#### 4.3.2 Blast Simulation Results
As already seen in the results from the simulations of the low-velocity impact tests (Figure 14b) the model with XPS-250 has a noticeably higher energy absorption than what is observed experimentally. The results from the blast simulations could be affected both by the material calibration and the formulation of the Deshpande-Fleck model. As shown by [PERSON] and [PERSON] in [45], polymeric foams do not necessarily follow the Deshpande-Fleck yield surface, since they generally have a lower yield stress in compression than in tension. This means that as the current model is calibrated from tests in uniaxial compression, the yield stress in tension may not be accurately represented. The same trends as described above are observed in the results from the blast simulations with 12 bar firing pressure, shown in Figure A5, and 15 bar firing pressure, shown in Figure A10, where the difference in displacement is as expected between XPS-400 and XPS-700. However, XPS-250 has a lower maximum displacement than expected at these load levels. The numerical results for the different core material formulations from simulations of tests at 10 bar firing pressure are presented in Figures 18-24. These consist of simulations with constant density (CD), distributed density (DD), constant density with viscoplasticity (CD-VP) and distributed density with viscoplasticity (DD-VP). As shown in Figure 19, the back-skin displacement is not much affected by changing from the CD to the DD model. The core compression, shown in Figure 20, is on the other hand much more affected by the density distribution of the core material. Especially for the simulations of XPS-250, the core compression is nearly doubled for the DD, compared to the CD model. For XPS-400 and XPS-700 cores, which have much less plastic deformation at this load level, the difference between the CD and the DD model is much smaller. Note that the core compression reported in these plots is the difference in mid-point displacement of the back and front skin and does not represent the  
Figure 18: Comparison between experimental and numerical midpoint displacements for tests with 10 bar firing pressure. core compression of the whole specimen. However, the distribution of the internal energy, shown in Figure 21, implies that for XPS-250 and XPS-400 there is a significant increase in core internal energy when changing from the CD to the DD model. This increase in internal energy indicates that for the XPS-250 and XPS-400 materials, the core is dissipating more energy when the DD model is applied. For XPS-700, the same change in core internal energy from CD to DD is not observed at this load level. There are two possible reasons for this. First, the plateau stress of the XPS-700 is higher than the applied pressure, and thus there is not much plastic deformation of the core. Second, the density distribution found for XPS-700 is narrower (see Figure 2b), and therefore the effect of the distribution is less pronounced. These observations are further illustrated by the deformation profiles and plastic strain fields shown in Figure 22 for XPS-250, Figure 23 for XPS-400 and Figure 24 for XPS-700. For XPS-250, it is seen that the plastic strains in the core material is much higher at the centre of the cross-section with the DD model because the density is lowest there. An increase in plastic strain towards the centre of the plate is also seen. Both these effects give a better utilisation of the core material in terms of energy absorption. Note also that the introduction of a distributed density reduces the plastic strains in the core at the edge of the blast-exposed area, where the back skin meets the frame. This increase in plastic strain in the middle of the cross-section seems to be beneficial as it both lowers the forces transferred to the back skin and reduces the possibility of core fracture in this area. For XPS-400, the difference in behaviour between the CD and the DD model is similar to that of XPS-250, but less pronounced, as the material plateau stress is higher for XPS-400.