## 2 Materials
### Foam Cores
#### 2.2.2 Low-Velocity Impact Tests  
The compressive response of the three XPS foams at elevated strain rates was investigated by conducting low-velocity impact tests in an Instron CEAST 9350 drop tower system (Instron, Norwood, MA, USA) using cubes with dimensions 50 mm \(\times\) 50 mm \(\times\) 50 mm as samples. The test setup was similar to that used by [PERSON] and [PERSON] [7], with some minor modifications to account for the test sample geometry. Here, a conical steel nose with a maximum diameter of 80 mm was attached to the instrumented striker to create a rigid, flat impact surface. Series of five tests were carried out for each of the three foam densities at impact velocities ranging from 2.6 to 6 m/s. The total impacting mass was 5.339 kg in all tests. The impact force was measured by a calibrated load cell placed 220 mm from the impactor nose. As the load cell is not located at the tip of the striker, the recorded force is not equal to the contact force between the impactor nose and the test sample. Based on dynamic equilibrium, the actual impact force (\(F\)) was found from the recorded force (\(P\)) according to Equation (1), accounting for the mass \(m_{1}\) above and the mass \(m_{2}\) below the load cell [7].  
\[F=\left(1+\frac{m_{2}}{m_{1}}\right)P \tag{1}\]  
A Phantom v2511 high-speed camera (Vision Research Inc., Wayne, NJ, USA) placed perpendicular to one side of the specimen recorded deformations during impact at a frequency of 33 kHz. Before the tests, the imaged surface of the samples was painted with a speckle pattern to enable use of 2D-DIC for  
Figure 2: (a) Typical true stress-strain curves from material compression tests of extruded polystyrene (XPS) foams in the thickness direction (ND) as a function of density and strain rate, and (b) density variation through the ND direction of the XPS foams. All plots are reproduced from [6].  
surface strain measurements during impact. Each sample was carefully measured and weighed before testing, and the density for all samples were found to be within the standard deviation given in Table 2.  
Figure 3a shows unfiltered force-displacement curves for XPS-250 samples impacted at velocities ranging from 3 to 6 m/s. Within this velocity range, the difference in yield and plateau stress is negligible when the impact velocity is increased. The peak force increases, however, with initial velocity because of the increase in kinetic energy. Similar test series were conducted for XPS-400 and XPS-700. These results are omitted for brevity as they exhibited the same trends as for XPS-250, i.e., no noticeable increase in yield and plateau stress within the chosen velocity range. As expected, the amount of compression of the samples decreased with increasing density. Figure 3b shows a comparison of force-displacement curves for XPS-250, XPS-400 and XPS-700 impacted at an initial velocity of 5 m/s.  
A comparison of Cauchy stress-logarithmic strain curves obtained from quasi-static tests and impact tests are shown in Figure 4. For all samples, global strains were computed based on the relative displacement of the top and bottom contact point between the sample and the test rig. Stresses were calculated assuming a negligible plastic Poisson's ratio for the foam. Note that the strain rate in the impact tests is not constant but decreases as the samples are compressed because the impactor is slowed down towards the maximum displacement. Thus, the strain rates given in Figure 4 are based on the impactor velocity at contact with the foam samples. The strain-rate sensitivity for XPS-250 seems to be less than the strain-rate sensitivity for the two other foams in this plot, and the reason for this result is unclear and require further investigations.  
Figure 4: Stress-strain curves for the three XPS-foams at strain rates from 0.001 s\({}^{-1}\) to 100 s\({}^{-1}\).  
Figure 3: Force-displacement curves for (**a**) XPS 250 at four impact velocities and (**b**) comparison of the three XPS densities at 5 m/s impact velocity.